CURRENT ELECTRICITY

CURRENT ELECTRICITY

1. ELECTRIC CURRENT 3. DRIFT VELOCITY

“The flow of charge in a definite direction constitutes the G G G G

“If u1 , u2 , u3 , ...u n are random thermal velocities of n free
electric current and the time rate of flow of charge through
electrons in the metal conductor, then the average thermal
any cross-section of a conductor is the measure of current”.
velocity of electrons is
i.e.,
G G G G
u1  u  u3  ...  u n G
net charge flown q dq 0
Electric current , I= n
time taken t dt
As a result, there will be no net flow of electrons of charge
1. Though the “electric current represents the direction of flow
in one particular direction in a metal conductor, hence no
of positive charge”.
current”.
2. Yet it is treated as a scalar quantity.
“Drift velocity is defined as the average velocity with which
3. Current follows, the laws of scalar addition and not the laws
the free electrons get drifted towards the positive end of
of vector addition.
the conductor under the influence of an external electric
4. Because the angle between the wires carrying currents does
field applied”.
not affect the total current in the circuit.
–4 –1
1. The drift velocity of electons is of the order of 10 ms .
2. CURRENT CARRIERS 2. If V is the potential difference applied across the ends of the
conductor of length l, the magnitude of electric field set up is
(a) Current carriers in solid conductors :
1. In solid conductors like metals, the valence electrons of the Potential difference V
E
atoms do not remain attached to individual atoms but are length A
free to move throughout the volume of the conductor.
2. Under the effect of an external electric field, the valence
electrons move in a definite direction causing electric current
in the conductors.
3. Thus, valence electrons are the current carriers in solid
conductors.
(b) Current carriers in liquids :
1. In an electrolyte like CuSO4, NaCl etc., there are positively 3. Each free electrons in the conductor experience a force,
 
and negatively charged ions (like Cu , SO , Na , Cl ).   G G
4 F e E.
2. These are forced to move in definite directions under the 4. The acceleration of each electron is
effect of an external electric field, causing electric current. G
3. Thus, in liquids, the current carriers are positively and G eE
a .
negatively charged ions. m
(c) Current carriers in gases : 5. At any instant of time, the velocity acquired by electron
G
1. Ordinarily, the gases are insulators of electricity. having thermal velocity u1 will be
2. They can be ionized by applying a high potential difference G G G
v1 u1  aW1
at low pressure
3. Thus, positive ions and electrons are the current carriers in where W1 is the time elapsed since it has suffered its last
gases. collision with ion/atom of the conductor.
 CURRENT ELECTRICITY

6. Similarly, the velocities acquired by other electrons in the

“ The drift velocity of electrons is small because of the frequent
conductor will be collisions suffered by electrons.
G G G G G G G G G
v 2 u 2  aW 2 , v3 u 3  aW3 , ....., v n u n  aW n . “ The small value of drift velocity produces a large amount of
electric current, due to the presence of extremely large
7. The average velocity of all the free electrons in the conductor
number of free electrons in a conductor. The propagation of
under the effect of external electric field is the drift velocity
G current is almost at the speed of light and involves
vd of the free electrons. electromagnetic process. It is due to this reason that the
G G G electric bulb glows immediately when switch is on.
G v  v 2  ...  v n
Thus, v d “ In the absence of electric field, the paths of electrons between
n
successive collisions are straight line while in presence of
G G G G G G electric field the paths are generally curved.
u  aW1  u 2  aW 2  ... u n  aWn
n NA x d
“ Free electron density in a metal is given by n
G G G A
§ u1  u 2  ...  u n · G W  W2  ...  W n G G
¨ ¸a 0  aW aW where N A = Avogrado number, x = number of free
© n ¹ n
electrons per atom, d = density of metal and A = Atomic
weight of metal.
W  W2  ...  Wn
where, W = average time that has elapsed 1. Mobility of charge carrier (P), responsible for current is
n
defined as the magnitude of drift velocity of charge per unit
since each electron suffered its last collision with the ion/
electic filed applied, i.e.,
atom of conductor and is called average relaxation time.
8.
–14
Its value is the order of 10 second. drift velocity vd qEW/m qW
P
G electric field E E m
9. Putting the value of a in the above relation, we have
G e We
G e EW 2. Mobility of electron, Pe
vd me
m
3. The total current in the conducting material is the sum of
eE the currents due to positive current carriers and negative
Average drift speed, vd W
m current carriers.
G
The negative sign show that vd is opposite to the direction vd Pe E
2 –1 –1 –1 –1
G 4. SI unit of mobility is m S V or ms N C
of E .
3.3 Relation between current and Drift Velocity
3.1 Relaxation time (W)

The time interval between two successive collisions of 1. Consider a conductor (say a copper wire) of length l and of
electrons with the positive ions in the metallic lattice is defined uniform area of cross-section
? Volume of the conductor = Al.
mean free path O
as relaxation time W . With 2. If n is the number density of electrons, i.e., the number of
r.m.s. velocity of electrons v rms
free electrons perunit volume of the conductor, then total
rise in temperature vrms increases consequently W decreases. number of free electrons in the conducture = Aln.

3.2 Mobility 3. Then total charge on all the free electrons in the conductor,

Drift velocity per unit electric field is called mobility of electron i.e. q AAne
4. The electric field set up across the conductor is given by
vd m2
P It’s unit is E = V/l (in magnitude)
E volt  sec
5. Due to this field, the free electrons present in the conductor
“ If cross-section is constant, I v J i.e. for a given cross-sectional will begin to move with a drift velocity vd towards the left
area, greater the current density, larger will be current. hand side as shown in figure
 CURRENT ELECTRICITY

V mA
or = R = a constant for a given conductor for a
I A n e2 W
given value of n, l and at a given temperature. It is known as
the electrical resistance of the conductor.
Thus, V = RI

6. Time taken by the free electrons to cross the conductors, this is Ohm’s law.
t = l/vd (1) Ohm’s law is not a universal law, the substances, which
q AAne obey ohm’s law are known as ohmic substance.
Hence, current, dI
t A (2) Graph between V and i for a metallic conductor is a straight
t
vd line as shown. At different temperatures V-i curves are
different.
or I A n e vd

§ e EW ·
7. Putting the value of vd ¨ ¸ , we have
© m ¹

Ane 2 WE
I
m

4. OHM’S LAW
(A) Slope of the line (B) Here tanT1 > tanT2
Ohm’s law states that the current (I) flowing through a
V
conductor is directly proportional to the potential difference = tan T R So, R1 > R2 i.e., T1 > T2
i
(V) across the ends of the conductor”.
(3) The device or substances which don’t obey ohm’s law
e.g. gases, crystal rectifiers, thermoionic valve, transistors
etc. are known as non-ohmic or non-linear conductors.
For these V-i curve is not linear.

V 1
Static resistance R st
i tan T

'V 1
i.e., I v V or V v I or V = RI Dynamic resistance R dyn
'I tan I

V
or R constant
I

4.1 Deduction of Ohm’s law

eE
We know that v d W
m

eV 5. ELECTRICAL RESISTANCE
But E = V/l ? v d W
mA
“The electrical resistance of a conductor is the obstruction
Also, I = A n e vd
posed by the conductor to the flow of electric current
§ eV · § A n e2 W · through it”.
? I = An e ¨ W¸ ¨ ¸V
© mA ¹ © mA ¹ 1. i.e., R = V/I
 CURRENT ELECTRICITY

(vi) Resistivity increases with impurity and mechanical stress.

volt
2. The SI unit of electrical resistance is ohm or . (vii) Magnetic field increases the resistivity of all metals except
amp
iron, cobalt and nickel.
3. Dimensions of electric resistance (viii) Resistivity of certain substances like selenium, cadmium,
sulphides is inversely proportional to intensity of light
Pot. diff. work done/charge falling upon them.
current current
V mA m A
3. We have, R u
2
ML T / AT2 I Ane2 W ne W A
2
ª¬ M1 2 T 3 A 2 º¼
A
A
comparing the above relation with the relation, R U .
5.1 Electrical, Resistivity or Specific Resistance A
4. We have, the resistivity of the material of a conductor,
“The resistance of a conductor depends upon the following
factors : m
U
(i) Length (l) : The resistance (R) of a conductor is directly ne2 W
proportional to its length (l), i.e., R v l
(ii) Area of cross-section (A) : The resistance (R) of a 5.3 Conductivity
conductor is inversely proportional to the area of cross- 1
section (A). of the conductor, i.e., R v 1/A Reciprocal of resistivity is called conductivity (V) i.e. V with
U
(iii) The resistance of conductor also depends upon the nature unit mho/m and dimensions [M 1 L3 T 3 A 2 ] .
of material and temperature of the conductor.
5.4 Conductance
A UA
From above ; R v or R .” 1
A A Reciprocal of resistance is known as conductance. C It’ss
R
U)
5.2 Resistivity (U 1
unit is or :–1 or “Siemen”.
:
1 Where U is constant of proportionality and is known as
specific resistance or electrical resistivity of the material of
the conductor
2. Specific resistance (or electrical resistivity) of the material
of a conductor is defined as the resistance of a unit length
with unit areas of cross section of the material of the
conductor.
(i) Unit and dimension : It’s S.I. unit is ohm × m and
5.5 Stretching of Wire
dimension is [ML3T–3A–2]
If a conducting wire stretches, it’s length increases, area of cross-
m
(ii) It’s formula : U section decreases so resistance increases but volume remain
ne 2 W constant.
(iii) Resistivity is the intrinsic property of the substance. It is Suppose for a conducting wire before stretching it’s length = l1,
independent of shape and size of the body (i.e. l and A). area of cross–section = A1, radius = r1, diameter = d1, and
(iv) For different substances their resistivity is also different l1
e.g. Usilver = minimum = 1.6 u 10 –8 : -m and resistance = R1 U
A1
Ufused quartz = maximum | 1016 : -m
Uinsulator ! Ualloy ! Usemi -conductor ! Uconductor
(Maximum for fused quartz) (Minimum for silver )

(v) Resistivity depends on the temperature. For metals

Ut U 0 (1  D't) i.e. resitivity increases with temperature.
 CURRENT ELECTRICITY

Volume remains constant i.e., A1l1 = A2l2 3. Semiconductors : These are those material whose electrical
After stretching length = l2, area of cross-section = A2, conductivity lies inbetween that of insulators and conductors.
Semiconductors can conduct charges but not so easily as is
l
radius = r2, diameter = d2 and resistance R2 U 2 in case of conductors. When a small potential difference is
A2 applied across the ends of a semiconductor, a weak current
Ratio of resistances before and after stretching flows through semiconductor due to motion of electrons and
holes.
2 2 4 4
R1 l1 A 2 § l1 · § A2 · § r2 · § d 2 · Examples of semiconductors are germanium, silicon etc.
= × =¨ ¸ =¨ ¸ =¨ ¸ =¨ ¸
R 2 l2 A1 © l2 ¹ © A1 ¹ © r1 ¹ © d1 ¹ The value of elecrical resistance R increases with rise of
temperature.
2
R1 § l1 ·
(1) If length is given then R v l 2 Ÿ ¨ ¸ Rt  R0 increase in resistance
R2 © l2 ¹ D
R0 u t original resistance × rise of temp.

4
1 R § r2 · Thus, temperature coefficient of resistance is defined as the
(2) If radius is given then R v 4 Ÿ 1 ¨ ¸ increase in resistance per unit original resistance per degree
r R2 © r1 ¹
celsium or kelvin rise of temperature.

6. CURRENT DENSITY, CONDUCTANCE 1. For metals like silver, copper, etc., the value of a is positive,
therefore, resistance of a metal increases with rise in
AND ELECTRIAL CONDUCTIVITY –1 –1
temperature. The unit of D is K or °C .

6.1 Relation between J, V and E 2. For insulators and semiconductors D is negative,

therefore, the resistance decreases with rise in temperature.
n Ae2 WE
We know, I = n Aevd = nAe §¨ W ¸·
eE 6.2 Non-Ohmic Devices
©m ¹ m
Those devices which do not obey Ohm’s law are called non-
ne 2 WE 1 ohmic devices. For example, vaccum tubes, semiconductor
or or J E
A m U diode, liquid electrolyte, transistor etc.
For all non-ohmic devices (where there will be failure of
? J VE
Ohm’s law), V–I graph has one or more of the following
1. Insulators : These are those materials whose electrical characteristics :
conducticity is either very very small or nil. (a) The relation between V and I is non-linear, figure
Insulators do not conduct charges. When a small potential
difference is applied across the two ends of an insulator, the
current through the insulator is zero.
Examples of insulators are glass, rubber, wood etc.
Variation of R, U with T
2. Conductors : These are those materials whose electrical
conductivity is very high
Conductor conduct charges very easily. When a small
potential difference is applied across the two ends of
(b) The relation between V and I depends on the sign of V. It
conductor, a strong current flows through the conductor.
means, if I is the current for a certain value of V, then
For super-conductor, the value of electrical conductivity is
reversing the direction of V, keeping its magnitude fixed,
infinite and electrical resistivity is zero.
does not produce a current of same magnitude I, in the
Examples of conductors are all metals like copper, silver, opposite direction, figure.
aluminium, tungsten etc.
 CURRENT ELECTRICITY

To remember the value of colour coding used for carbon

resistor, the following sentences are found to be of great
help (where bold letters stand for colours).
B B ROY Green, Britain Very Good Wife Gold Silver.
Way of finding the resistance of carbon resistor from its
colour coding.
In the system of colour coding, Strips of different colours
are given on the body of the resistor, figure. The colours on
strips are noted from left to right.
(c) Therelation between V and I isnot unique, i.e., there is more
than one value of V for the same current I, figure.

(i) Colour of the first stip A from the end indicates the first
significant figure of resistance in ohm.
(ii) Colour of the second strip B indicate the second significant
figure of resistance in ohm.
(iii) The colour of the third strip C indicates the multiplier,
i.e., the number of zeros that will follow after the two
7. COLOUR CODE FOR CARBON RESISTORS significant figure.
(iv) The colour of fourth strip R indicates the tolerance limit
The colour code for carbon resistance is given in the
of the resistance value of percentage accuracy of resistance.
following table.

Colour code of carbon resistors

8. COMBINATION OF RESISTORS

8.1 Resistances in Series

Colour Letter as No. Mulitplier Colour Tolerance
anAid to Resistors are said to be connected in series, if the same
memory current is flowing through each resistor when some poential
difference is applied across the combination.
0
Black B 0 10 Gold 5%
1
Brown B 1 10 Silver 10%
2
Red R 2 10 No colour 20%
3
Orange O 3 10
4
Yellow Y 4 10
5
Green 5 10
6
Blue B 6 10 1. Let V be the potential difference applied across A and B
7 using the battery H. In series combination, the same current
Violet V 7 10
(say I) will be passing through each resistance.
8
Grey 8 10
2. Let V1, V2, V3 be the potential difference across R1, R2 and
9
White W 9 10 R3 respectively. According to Ohm’s law
–1
Gold 10 V1 = IR1, V2 = IR2, V3 = IR3
–2
Silver 10 3. Here, V = V1 + V2 + V3 = IR1 + IR2 + IR3 = I (R1 + R2 + R3)
 CURRENT ELECTRICITY

branches are different and I1, I2, I3 be the current through

the resistances R1, R2 and R3 respectively. Then,
I = I2 + I2 + I3
3. Here, potential difference across each resistor is V, therefore
4. If R s is the equivalent resistance of the given series V = I1R1 = I2 R2 = I3R3
combination of resistances, figure, then the potential
difference across A and B is, V V V
or I1 , I2 , I3
R1 R2 R3
V = IRs.
We have Putting values, we get
IRs = I (R1 + R2 + R3)
V V V
I  
or Rs R1  R 2  R 3 R1 R 2 R 3

Memory note 4. If Rp is the equivalent resistance of the given parallel

combination of resistance, figure, then
In a series resistance circuit, it should be noted that :
(i) the current is same in every resistor.
(ii) the current in the circuit is independent of the relative
positions of the various resistors in the series.
(iii) the voltage across any resistor is directly proportional to
the resistance of the resistor.
(iv) the total resistance of the circuit is equal to the sum of the V = IRp or I = V/Rp
individual resistances, plus the internal resistance of a cell we have
if any.
(v) The total resistance in the series circuit is obviously more V V V V 1 1 1 1
   
than the greatest resistance in the circuit. Rp R 1 R 2 R 3 or R p R1 R 2 R 3

Thus, the reciprocal of equivalent resistance of a number of

8.2 Resistances in Parallel
resistor connected in parallel is equal to the sum of the
Any number of resistors are said to be connected in parallel reciprocals of the individual resistances.
if potential difference across each of them is the same and
is equal to the applied potential difference. Memory note

In a parallel resistance circuit, it should be noted that :

(i) the potential difference across each resistor is the same
and is equal to the applied potential difference.
(ii) the current through each resistor is inversely proportional
to the resistance of that resistor.
(iii) total current through the parallel combination is the sum
of the individual currents through the various resistors.
(iv) The reciprocal of the total resistance of the parallel
combination is equal to the sum of the reciprocals of the
individual resistances.
1. Let V be the potential difference applied across A and B (v) The total resistances are connected in series, the current
with the help of a battery H. through each resistance is same. When the resistance are
in parallel, the pot-diff. accross each resistance is the same
2. Let I be the main current in the circuit from battery. I divides
and not the current.
itself into three unequal parts because the resistances of these
 CURRENT ELECTRICITY

(iii) Potential drop inside the cell = ir

9. CELL
(iv) Equation of cell E V  ir (E > V)
The device which converts chemical energy into electrical energy
is known as electric cell. Cell is a source of constant emf but not §E ·
(v) Internal resistance of the cell r ¨  1¸ ˜ R
constant current. ©V ¹
(vi) Power dissipated in external resistance (load)
2
V2 § E ·
P Vi i2R ¨Rr¸ . R
R © ¹

E2
Power delivered will be maximum when R r so Pmax .
4r
This statement in generalised from is called “maximum
power transfer theorem”.

(1) Emf of cell (E) : The potential difference across the

terminals of a cell when it is not supplying any current is
called it’s emf.
(2) Potential difference (V) : The voltage across the
terminals of a cell when it is supplying current to external
resistance is called potential difference or terminal
voltage. Potential difference is equal to the product of (vii) When the cell is being charged i.e. current is given to the
current and resistance of that given part i.e. V = iR. cell then E = V – ir and E < V.
(3) Internal resistance (r) : In case of a cell the opposition (2) Open circuit : When no current is taken from the cell it
of electrolyte to the flow of current through it is called is said to be in open circuit.
internal resistance of the cell. The internal resistance of a
cell depends on the distance between electrodes (r v d),
area of electrodes [r v (1/A)] and nature, concentration
(r v C) and temperature of electrolyte [r v (1/ temp.)].
A cell is said to be ideal, if it has zero internal resistance.

9.1 Cell in Various Positions

(i) Current through the circuit i = 0
(1) Closed circuit : Cell supplies a constant current in the
circuit. (ii) Potential difference between A and B, VAB = E
(iii) Potential difference between C and D, VCD = 0
(3) Short circuit : If two terminals of cell are join together
by a thick conducting wire

E
(i) Current given by the cell i
Rr
(ii) Potential difference across the resistance V iR
 CURRENT ELECTRICITY

(i) Maximum current (called short circuit current) flows

plates of cells are connected together their emf’s are added to
E each other while if their similar plates are connected together
momentarily isc
r their emf’s are subtractive.
(ii) Potential difference V = 0

Memory note
1. It is important to note that during charging of a cell, the
positive electrode of the cell is connected to positive
terminal of battery charger and negative electrodes of the
cell is connected to negative terminal of battery charger.
In this process, current flows from positive electrode to (1) Series grouping : In series grouping anode of one cell is
negative electrode through the cell. Refer figure connected to cathode of other cell and so on. If n identical
cells are connected in series

? V = H + Ir
Hence, the terminal potential difference becomes greater
than the emf of the cell.
(i) Equivalent emf of the combination E eq nE
2. The difference of emf and terminal voltage is called lost
voltage as it is not indicated by a voltmeter. It is equal to Ir. (ii) Equivalent internal resistance req nr

nE
9.2 Distinction between E.M.E. and Potential Difference (iii) Main current = Current from each cell i
R  nr
E.M.F. of a Cell Potential Difference (iv) Potential difference across external resistance V iR

1 The emf of a cells is the 1. The potential difference V

(v) Potential difference across each cell V '
maximum potential between the two points is n
difference between the the difference of potential
2
two electrodes of a cell between those two points § nE ·
(vi) Power dissipated in the external circuit ¨ ¸ .R
when the cell is in the in a closed circuit. © R  nr ¹
open circuit.
2. It is independent of the 2. It depends upon the resis- § E2 ·
(vii) Condition for maximum power R nr and Pmax n¨ ¸
resistance of the circuit tance between the two points © 4r ¹
and depends upon the of the circuit and current
(viii) This type of combination is used when nr << R.
nature of electrodes and flowing through the
the nature of electrolyte circuit. (2) Parallel grouping : In parallel grouping all anodes are
of the cell. connected at one point and all cathode are connected together
3. The term emf is used for 3. The potential difference is at other point. If n identical cells are connected in parallel
the source of electric measured between any two
E, r
current. points of the electric circuit.
4. It is a cause. 4. It is an effect. E, r

E, r
9.3 Grouping of Cells
i
R
In series grouping of cell’s their emf’s are additive or subtractive
while their internal resistances are always additive. If dissimilar
 CURRENT ELECTRICITY

(i) Equivalent emf Eeq = E (iii) Main current flowing through the load
(ii) Equivalent internal resistance Req r/n
nE mnE
i
E R
nr mR  nr
(iii) Main current i m
R r/n
(iv) Potential difference across external resistance = p.d. (iv) Potential difference across load V = iR
across each cell = V = iR V
(v) Potential difference across each cell V '
i n
(v) Current from each cell i '
n i
(vi) Current from each cell i '
2 n
§ E ·
(vi) Power dissipated in the circuit P ¨ ¸ .R
© R r/n ¹ nr
(vii) Condition for maximum power R and
m
§ E2 ·
(vii) Condition for max. power is R r / n and Pmax n¨ ¸ E2
© 4r ¹ Pmax (mn)
4r
(viii) This type of combination is used when nr >> R
(viii) Total number of cell = mn
Generalized Parallel Battery Memory note
Note that (i) If the wo cells connected in parallel are of the
same emf H and same internal resistance r, then

Hr  Hr
H eq H
rr

1 1 1 2 r
 or req
req r r r 2

(ii) If n identical cells are connected in parallel, then the

E1 E 2 E equivalent emf of all the cells is equal to the emf of one
  ... n
r1 r2 rn 1 1 1 1 cell.
E eq and   ... .
1 1 1 req r1 r2 rn
  ... 1 1 1 n or r = r/n
r1 r2 rn   ...  n terms eq
req r r r
(3) Mixed Grouping : If n identical cell’s are connected in a
row and such m row’s are connected in parallel as shown.
10. ELECTRIC CURRENT

(1) The time rate of flow of charge through any cross-section

ΔQ dQ
is called current. i Lim . If flow is uniform
Δt o 0 Δt dt

Q
then i . Current is a scalar quantity. It’s S.I. unit is
t
ampere (A) and C.G.S. unit is emu and is called biot (Bi),
or ab ampere. 1A = (1/10) Bi (ab amp.)
(2) Ampere of current means the flow of 6.25 u 10 18
(i) Equivalent emf of the combination Eeq = nE
electrons/sec through any cross–section of the conductor.
nr (3) The conventional direction of current is taken to be the
(ii) Equivalent internal resistance of the combination req
m direction of flow of positive charge, i.e. field and is
 CURRENT ELECTRICITY

opposite to the direction of flow of negative charge as (i) Solids : In solid conductors like metals current carriers
shown below. are free electrons.
(ii) Liquids : In liquids current carriers are positive and
negative ions.
(iii) Gases : In gases current carriers are positive ions and
free electrons.
(iv) Semi conductor : In semi conductors current carriers are
(4) The net charge in a current carrying conductor is zero. holes and free electrons.
(5) For a given conductor current does not change with (v) The amount of charge flowing through a crossection of a
change in cross-sectional area. In the following figure conductor from t = ti to t = tf is given by :
i1 = i2 = i3
tf
q ³ti I dt

From Graphs
(i) Slope of Q vs t graph gives instantaneous current.

(6) Current due to translatory motion of charge : If n

particle each having a charge q, pass through a given area
in time t then

If n particles each having a charge q pass per second per (ii) Area under the I vs t graph gives net charge flown.
unit area, the current associated with cross-sectional area A
is i = nqA
If there are n particle per unit volume each having a charge
q and moving with velocity v, the current thorough, cross
section A is i = nqvA
(7) Current due to rotatory motion of charge : If a point
charge q is moving in a circle of radius r with speed v
(frequency Q, angular speed Z and time period T) then
q qv qω
corresponding current i = qν = = = 11. KIRCHHOFF’S LAW
T 2πr 2π
11.1 Kirchhoff’s first law or Kirchhoff’s junction law
or Kirchhoff’s current law.

1. the algebraic sum of the currents meeting at a junction in a

closed electric circuit is zero, i.e., ¦ I 0

(8) Current carriers : The charged particles whose flow in 2. Consider a junction O in the electrical circuit at which
a definite direction constitutes the electric current are the five conductors are meeting. Let I1, I2, I3, I4 and I5 be
the currents in these conductors in directions, shown in
called current carriers. In different situation current
figure,
carriers are different.
 CURRENT ELECTRICITY

3. Let us adopt the following sign convention : the current

flowing in a conductor towards the junction is taken as
positive and the current flowing away from the junction is We adopt the following sign convention :
taken as negative.
Traverse a closed path of a circuit once completely in
4. According to Kirchhoff’s first law, at junction O clockwise or anticlockwise direction.
(–I1) + (–I2) + I3 + (–I4) + I5 = 0 Difference between Kirchhoff’s I and II laws
or –I1 – I2 + I3 – I4 + I5 = 0

or ¦I 0 First Law Second Law

or I 3 + I5 = I 1 + I2 + I 4 1. This law supports the 1. This law supports the law

law of conservation of of conservation of energy.
5. i.e., total current flowing towards the junction is equal to
charge.
total current flowing out of the junction.
2. According to this law 2. According to this law
6. Current cannot be stored at a junction. It means, no point/
junction in a circuit can act as a source or sink of charge. ¦I 0 ¦H ¦ IR

7. Kirchhoff’s first law supports law of conservation of 3. This law can be used in 3. This law can be used in
charge. open and closed circuits. closed circuit only.

11.2 Kirchhoff’s Second law or Kirchhoff’s loop law

or Kirchhoff’s voltage law. 12. EXPERIMENTS

The algebraic sum of changes in potential around any closed 12.1 Galvanometer
path of electric circuit (or closed loop) involving resistors
It is an instrument used to detect small current passing through it
and cells in the loop is zero, i.e., ¦ 'V 0.
by showing deflection. Galvanometers are of different types e.g.
In a closed loop, the algebraic sum of the emfs and algebraic moving coil galvanometer, moving magnet galvanometer, hot wire
sum of the products of current and resistance in the various galvanometer. In dc circuit usually moving coil galvanometer
arms of the loop is zero, i.e., ¦ H  ¦ IR 0. are used.
Kirchhoff’s second law supports the law of conservation of (i) It’s symbol : ; where G is the total
energy, i.e., the net change in the energy of a charge, after
internal resistance of the galvanometer.
the charge completes a closed path must be zero.
(ii) Full scale deflection current : The current required for
Kirchhoff’s second law follows from the fact that the
full scale deflection in a galvanometer is called full scale
electrostatic force is a conservative force and work done by
deflection current and is represented by ig.
it in any closed path is zero.
(iii) Shunt : The small resistance connected in parallel to
Consider a closed electrical circuit as shown in figure.
galvanometer coil, in order to control current flowing
containing two cells of emfs. H1 and H2 and three resistors of
through the galvanometer is known as shunt.
resistances R1, R2 and R3.
 CURRENT ELECTRICITY

Table : Merits and demerits of shunt i

(c) To pass nth part of main current (i.e. i g ) through the
n
Merits of shunt Demerits of shunt
G
To protect the galvano- Shunt resistance decreases the galvanometer, required shunt S = .
(n –1)
meter coil from burning . sensitivity of galvanometer.
12.3 Voltmeter
It can be used to convert
any galvanometer into It is a device used to measure potential difference and is always
ammeter of desired range. put in parallel with the ‘circuit element’ across which potential
difference is to be measured.
12.2 Ammeter

It is a device used to measure current and is always connected

in series with the ‘element’ through which current is to be
measured.

(i) The reading of a voltmeter is always lesser than true value.

(ii) Greater the resistance of voltmeter, more accurate will
be its reading. A voltmeter is said to be ideal if its
resistance is infinite, i.e., it draws no current from the
circuit element for its operation.
(iii) Conversion of galvanometer into voltmeter : A
(i) The reading of an ammeter is always lesser than actual galvanometer may be converted into a voltmeter by
current in the circuit. connecting a large resistance R in series with the
(ii) Smaller the resistance of an ammeter more accurate will galvanometer as shown in the figure.
be its reading. An ammeter is said to be ideal if its
resistance r is zero.
(iii) Conversion of galvanometer into ammeter : A
galvanometer may be converted into an ammeter by
connecting a low resistance (called shunt S) in parallel to
the galvanometer G as shown in figure.

(a) Equivalent resistance of the combination = G + R

(b) According to ohm’s law Maximum reading of V which
can be taken V = ig (G + R); which gives

V §V ·
Required series resistance R = –G =¨ – 1¸ G
ig ¨ ¸
© Vg ¹
(c) If nth part of applied voltage appeared across galvanometer
GS
(a) Equivalent resistance of the combination V
G S (i.e. Vg ) then required series resistance R = (n – 1) G..
n
(b) G and S are parallel to each other hence both will have
equal potential difference i.e. i g G (i  i g ) S ; which 12.4 Wheatstone Bridge Principle
gives Wheatstone Bridge Principle states that if four resistances
ig P, Q, R and S are arranged to form a bridge as shown in
Required shunt S = G figure, if galvanometer shows no deflection, the bridge is
(i – i g )
balanced.
 CURRENT ELECTRICITY

In that case through the galvanometer or in other words VB = VD. In the

P R P R
balanced condition = , on mutually changing the
Q S Q S
position of cell and galvanometer this condition will not
change.
(ii) Unbalanced bridge : If the bridge is not balanced current
will flow from D to B if VD > VB i.e. (VA  VD )  (VA  VB )
which gives PS > RQ.
(iii) Applications of wheatstone bridge : Meter bridge, post
office box and Carey Foster bridge are instruments based
on the principle of wheatstone bridge and are used to
measure unknown resistance.

12.5 Slide Wire Bridge or Meter Bridge

Proof : A slide wire bridge is a practical form of Wheatstone bridge.

Let I be the total current given out by the cell. On reaching It consists of a wire AC of constantan or manganin of 1
the point A, it is divided into two parts : metre length and of uniform area of cross-section.
1. I1 is flowing through P A meter scale is also fitted on the wooden board parallel to
the length of the wire.
2. (I – I1) through R.
At B, the current I1 is divided into two parts, Ig through the Copper strip fitted on the wooden board in order to provide
galvanometer G and (I1 – Ig) through Q. two gaps in strips.

A current (I – I1 + Ig) through S. Across one gap, a resistance box R and in another gap the
unknown resistance S are connected.
Applying Kirchhoff’s Second Law to the closed circuit
ABDA, we get The positive pole of the battery E is connected to terminal
A and the negative pole of the battery to terminal C through
I1P + Ig G – (I – I1) R = 0 ...(1)
one way key K.
where G is the resistance of galvanometer.
The circuit is now exactly the same as that of the Wheatstone
Again applying Kirchhoff’s Second Law to the closed circuit bridge figure.
BCDB, we get
(I1 – Ig) Q – (I – I1 + Ig) S – IgG = 0 ...(2)
The value of R is adjusted such that the galvanometer shows
no deflection, i.e., Ig = 0. Now, the bridge is balanced. Putting
Ig = 0 in (1) and (2) we have
I1P – (I – I1) R = 0 or I1P = (I – I1) R ...(3)
and I1Q – (I – I1) S = 0 or I1Q = (I – I1) S ...(4)

P R
Dividing (3) by (4), we get
Q S

Note that in Wheatstone bridge circuit, arms AB and BC

having resistances P and Q form ratio arm. The arm AD,
having a resistance R, is a known variable resistance arm and Adjust the position of jockey on the wire (say at B) where
arm DC, having a resistance S is unknown resistance arm. on pressing, galvanometer shows no deflection.

(i) Balanced bridge : The bridge is said to be balanced when Note the length AB ( = l say) to the wire. Find the length BC
deflection in galvanometer is zero i.e. no current flows ( = 100 – l) of the wire.
CURRENT ELECTRICITY

According to Wheatstone bridge principle If I is the current flowing through the wire, then from Ohm’s
law; V = IR; As, R = Ul/A
P R
Q S A § IU ·
? V IU KA, ¨ where K ¸
If r is the resistance per cm length of wire, then © ¹
P = resistance of the length l of the wire AB = lr or Vvl (if I and A are constant)
Q = resistance of the length (100–l) of the wire BC=(100 – l) r. i.e., potential difference across any portion of potentiometer
wire is directly proportional to length of the wire of that
Ar R § 100  A · protion.
? 100  A r
or S ¨ ¸u R
S © A ¹ Here, V/l = K = is called potential gradient, i.e., the fall of
Knowing l and R, we can calculate S. potential per unit length of wire.

12.6 Potentiometer and its principle of working 12.7 Determination of Potential Difference
using Potentiometer
Potentiometer is an apparatus used for measuring the emf
A battery of emf H is connected between the end terminals A
of a cells or potential difference between two points in an
and B of potentiometer wire with ammeter A1, resistance
electrical circuit accurately.
box R and key K in series. This circuit is called an auxillary
A potentiometer consists of a long uniform wire generally circuit. The ends of resistance R1 are connected to terminals
made of manganin or constantan, stretched on a wooden A and Jockey J through galvanometer G. A cell H1 and key
board. K1 are connected across R1 as shown in figure.
Its ends are connected to the binding screws A and B. A
meter scale is fixed on the board parallel to the length of the
wire. The potentiometer is provided with a jockey J with
the help of which, the contact can be made at any point on
the wire, figure. A battery H (called driving cell), connected
across A and B sends the current through the wire which is
kept constant by using a rheostat Rh.

Working and Theory : Close key K and take out suitable

resistance R from resistance box so that the fall of potential
across the potentiometer wire is greater than the potential
difference to be measured.
It can be checked by pressing, firstly the jockey J on
potentiometer wire near end A and later on near end B, the
deflections in galvanometer are in opposite directions.

Principle : The working of a potentiometer is based on the Close key K1. The current flows through R1. A potential
fact that the fall of potential across any portion of the wire difference is developed across R1. Adjust the position of
is directly proportional to the length of that portion provided jockey on potentiometer wire where if pressed, the
the wire is of uniform area of cross-section and a constant galvanometer shows no deflection. Let it be when jockey is
current is flowing through it. at J. Note the length AJ (= l) of potentiometer wire. This
would happen when potential difference across R1 is equal
Suppose A and U are respectively the area of cross-section to the fall of potential across the potentiometer wire of length
and specific resistance of the material of the wire. l. If K is the potential gradient of potentiometer wire, then
Let V be the potential difference across the portion of the potential difference across R1, i.e.,
wire of length l whose resistance is R. V = Kl
 CURRENT ELECTRICITY

If r is the resistance of potentiometer wire of length L, then i.e., H1 = Kl1 ...(1)

current through potentiometer wire is where K is the potential gradient across the potentiometer
H wire.
I
Rr Now remove the plug from the gap between 1 and 3 and
insert in the gap between 2 and 3 of two way key so that
Potential drop across potentiometer wire
§ H · cells of emf H2 comes into the circuit. Again find the position
Ir ¨ ¸r
©Rr¹ of jockey on potentiometer wire, where galvanometer shows
no deflection. Let it be at J2. Note the length of the wire AJ2
Potential gradient of potentiometer wire, i.e., fall of potential
( = l2 say). Then
per unit length is
H2 = Kl2 ...(2)
§ H ·r § H ·r
K ¨ ¸ . V ¨ ¸ A
©Rr¹L ©Rr ¹L H1 A1
Dividing (1) by (2), we get
H2 A2
Hence, V can be calculated.

12.8 Comparison of emfs of two cells using Potentiometer 12.9 Precautions of experiment

A battery of emf H is connected between the end terminals A 1. The current in the potentiometer wire from driving cell must
and B of potentiometer wire with rheostat Rh, ammeter A1 be kept constant during experiment.
and key K in series. 2. While adjusting the position of jockey on potentiometer wire,
The positive terminals of both the cells are connected to the edge of jockey should not be rubbed on the wire,
point A of the potentiometer. Their negative terminals are otherwise area of cross-section of wire will not be uniform
connected to two terminals 1 and 2 of two ways key, while and constant.
its common terminal 3 is connected to jockey J through a 3. The current in the potentiometer wire from driving cell
galvanometer G. should not be passed for long time as this would cause
Insert the plug in the gap between the terminals 1 and 3 of heating effect, resulting the change in resistance of wire.
two way key so that the cell of emf H1 is in the circuit. Memory note
Adjust the position of jockey on potentiometer wire, where A balance point is obtained on the potentiometer wire if
if pressed, the galvanometer shows no deflection. Let it be the fall of potential along the potentiometer wire, due to
when jockey be at J1. Note the length AJ1 (= l1 say) of the wire. driving cell is greater than the e.m.f. of the cells to be balanced.
There is no current in arm AH1J1. It means the potential of
positive terminal of cell = potential of the point A, and the 12.10 Determination of Internal Resistance
potential of negative terminal of cell = potential of the point J1. of a Cell by Potentiometer Method

To find the internal resistance r of a cell of emf H using

potentiometer, set up the circuit as shown in figure.

Therefore, the e.m.f. of the cell ( =H1) is equal to potential

difference between the points A and J1 of the potentiometer
wire.
 CURRENT ELECTRICITY

Close key K and maintain suitable constant current in the potentiometer wire circuit with the help of rheostat and using
potentiometer wire with the help of rheostat Rh. Adjust the a single cell.
position of jockey on the potentiometer wire where if Difference between Potentiometer and Voltmeter
pressed, the galvanometer show no deflection. Let it be when
jockey is as J1. Note the length AJ1 (= l1) of the potentiometer Potentiometer Voltmere
wire. Now emf of the cell, H = potential difference across
1. It measures the emf of 1. It measures the emf of a
the length l1 of the potentiometer wire.
a cell very accurately. cell approximately.
or H = Kl1 ...(1)
2. While measuring emf it 2. While measuring emf, it
where K is the potential gradient across the wire. does not draw any current drws some current from
Close key K1 and take out suitable resistance R from the from the source of the source of emf.
resistance box in the cell circuit. Again find the position of known emf.
the jockey on the potentiometer wire where galvanometer 3. While measuring emf, 3. While measuring emf the
shows no deflection. Let it be at J2. Note the length of the the resistance of poten- resistance of voltmeter is
wire AJ2 ( = l2 say). As current is being drawn from the cell, tiometer becomes infinite. high but finite.
its terminal potential difference V is balanced and not emf 4. Its sensitivity is high. 4. Its sensitivity is low.
H. Therefore, potential difference between two poles of the 5. It is based on null 5. It is based on deflection
cell, V = potential difference across the length l2 of the deflection method. method.
potentiometer wire 6. It can be used for 6. It can be used only to
i.e. V = Kl2 ...(2) various purposes. measure emf or potential
difference.
Dividing (1) by (2), we have

H A1
...(3) 13. HEATING EFFECT OF CURRENT
V A2

We know that the internal resistance r of a cell of emf H, When some potential difference V is applied across a resistance
when a resistance R is connected in its circuit is given by R then the work done by the electric field on charge q to flow
through the circuit in time t will be
HV §H ·
r uR ¨  1¸ R ...(4)
V ©V ¹ V2 t
W = qV = Vit = i2R Joule .
R
Putting the value (3) in (4), we get

§ A1 · A1  A 2
r ¨  1¸ R uR
A
© 2 ¹ A2

Thus, knowing the values of l1, l2 and R, the internal This work appears as thermal energy in the resistor.
resistance r of the cell can be determined. Heat produced by the resistance R is

12.11 Sensitiveness of Potentiometer W Vit i 2 Rt V2 t

H Cal. This relation is called joules
J 4˜ 2 4˜2 4 ˜ 2R
The sensitiveness of potentiometer means the smallest
heating.
potential difference that can be measured with its help.
Some important relations for solving objective questions are as
The sensitiveness of a potentiometer can be increased by
follow :
decreasing its potential gradient. The same can be achieved.
(i) By increasing the length of potentiometer wire.
(ii) If the potentiometer wire is of fixed length, the potential
gradient can be decreased by reducing the current in the
 CURRENT ELECTRICITY

of any electrical appliance can be calculated by rated

Condition Graph
VR2
If R and t are constant power and rated voltage i.e. by using R = e.g.
PR
H v i 2 and H v V
2

220 u 220
Resistance of 100W, 220 volt bulb is R 484 :
100
If i and t are constant (series grouping) (4) Power consumed (illumination) : An electrical appliance
HvR (Bulb, heater, …. etc.) consume rated power (PR) only if
applied voltage (VA) is equal to rated voltage (VR) i.e. If
VA2
VA = VR so Pconsumed = PR. If VA < VR then Pconsumed
If V and t are constant (Parallel grouping) R

1 VR2
Hv also we have R so
R PR

If V, i and R constant H v t § VA2 ·

Pconsumed (Brightness) ¨ 2 ¸ .PR
© VR ¹

Pconsumed v (Brightness)
e.g. If 100 W, 220 V bulb operates on 110 volt supply then
13.1 Electric Power
2
§ 110 ·
The rate at which electrical energy is dissipated into other Pconsumed ¨ ¸ u 100 25 W
forms of energy is called electrical power i.e. © 220 ¹

W V2
P= = Vi = i 2 R =
t R
(1) Units : It’s S.I. unit is Joule/sec or Watt
If VA < VR then % drop in output power
Bigger S.I. units are KW, MW and HP,
(PR  Pconsumed )
remember 1 HP = 746 Watt u100
PR
(2) Rated values : On electrical appliances
(Bulbs, Heater … etc.) For the series combination of bulbs, current through
them will be same so they will consume power in the
ratio of resistance i.e., P v R {By P = i2R) while if they
are connected in parallel i.e. V is constant so power
consumed by them is in the reverse ratio of their
1
resistance i.e. P v
R

Wattage, voltage, ……. etc. are printed called rated values (5) Thickness of filament of bulb : We know that resistance
e.g. If suppose we have a bulb of 40 W, 220 V then rated VR2 l
power (PR) = 40 W while rated voltage (VR) = 220 V. It of filament of bulb is given by R , also R U ,
PR A
means that on operating the bulb at 220 volt, the power
dissipated will be 40 W or in other words 40 J of electrical 1
hence we can say that A v PR v i.e. If rated
energy will be converted into heat and light per second. Thickness R
(3) Resistance of electrical appliance : If variation of power of a bulb is more, thickness of it’s filament is also
resistance with temperature is neglected then resistance more and it’s resistance will be less.
 CURRENT ELECTRICITY

1
If applied voltage is constant then P(consumed) v
R
VA2 If quantity of water is given n litre then
(By P ). Hence if different bulbs (electrical
R
4180(4200) n 'T
appliance) operated at same voltage supply then t
p
1
Pconsumed v PR v thickness v
R
13.2 Electric Energy

The total electric work done or energy supplied by the

source of emf in maintaining the current in an electric circuit
Different bulbs
for a given time is called electric energy consumed in the
circuit.
? Electric energy, W = VIt = P.t
? Electric energy = electric power × time
Ÿ Resistance R25 > R100 > R1000 SI unit of electric energy is joule, wherre
Ÿ Thickness of filament t1000– > t100 > t40 1 joule = 1 volt × 1 ampere × 1 second = 1 watt × 1 second
Ÿ Brightness B1000 > B100 > B25 The commercial unit of electric energy is called a kilowatt-
hour (kWh) or Board to Trade Unit (BOT) or UNIT of
(6) Long distance power transmission : When power is Electricity, in brief, where
transmitted through a power line of resistance R, power-
1 kWh = 1 kilo watt × 1 hour = 1000 watt × 1 hour
loss will be i 2 R
Thus 1 kilo watt hour is the total electric energy consumed
Now if the power P is transmitted at voltage V
when an electrical appliance of power 1 kilo-watt works for
P2 one hours.
P = Vi i.e. i = (P/V) So, Power loss uR
V2 6
1 kWh = 1000 Wh = (1000 W) × (60 × 60 s) = 3.6 × 10 J.
Now as for a given power and line, P and R are constant
Note that the number of units of electricity consumed = No.
so Power loss v (1/V2)
So if power is transmitted at high voltage, power loss watt u hour
of kWh =
will be small and vice-versa. e.g., power loss at 22 kV 1000
is 10 –4 times than at 220 V. This is why long distance
power transmission is carried out at high voltage. Electric energy VI t I 2 Rt V2t / R

(7) Time taken by heater to boil the water : We know that

13.3 Electricity Consumption
heat required to raise the temperature 'T of any
substance of mass m and specific heat S is H = m.S.'T (1) The price of electricity consumed is calculated on the
basis of electrical energy and not on the basis of electrical
Here heat produced by the heater = Heat required to raise power.
the temp. 'T of water.
(2) The unit Joule for energy is very small hence a big
J(m.S.'T) practical unit is considered known as kilowatt hour
i.e. p u t = J u m.S.'T Ÿ t
p (KWH) or board of trade unit (B.T.U.) or simple unit.
{J = 4.18 or 4.2 J/cal) (3) 1 KWH or 1 unit is the quantity of electrical energy which
4180 ( or 4200) m 'T dissipates in one hour in an electrical circuit when the
for m kg water t electrical power in the circuit is 1 KWH thus
p
{S = 1000 cal/kgoC) 1 KWH = 1000 W u 3600 sec = 3.6 u 106 J.
 CURRENT ELECTRICITY

(4) Important formulae to calculate the no. of consumed units If they are connected If they are connected

Total watt u Total hours in series in parallel

is n
1000 1 1 1
 PP = P1 + P2
PS P1 P2
13.4 Combination of Bulbs (or Electrical Appliances)
1 1 1 HP H1 H 2
Bulbs (Heater etc.) Bulbs (Heater etc.) Ÿ  Ÿ 
HS / t S H1 / t1 H2 / t 2 tp t1 t2
are in series are in parallel
' HS=H1= H2 ' Hp = H 1 = H2
(1) Total power consumed (1) Total power consumed 1 1 1
so ts = t1+ t2 so 
1 1 1 tp t1 t 2
  .... Ptotal = P1 + P2 + P3 .... + Pn
Ptotal P1 P2 i.e. time taken by i.e. time taken by parallel
combinationto boil the combination to boil the
same quantity of water same quantity of water
t1t 2
ts = t1 + t2 tp
t1  t 2

(3) If three identical bulbs are connected in series as shown

(2) In ‘n’ bulbs are identical, (2) If ‘n’ identical bulbs are in figure then on closing the switch S. Bulb C short
circuited and hence illumination of bulbs A and B
P increases
Ptotal in parallel. Ptotal = nP
N

Pconsumed Brightness Pconsumed Brightness

1 1
vV v R v v PR v i v
Prated R
i.e. in series combination i.e. in parallel combination
bulb of lesser wattage will bulb of greater wattage will Reason : Voltage on A and B increased.
give more bright light and give more bright light and (4) If three bulbs A, B and C are connected in mixed
p.d. appeared across it will more current will pass combination as shown, then illumination of bulb A
be more. through it. decreases if either B or C gets fused

Some Standard Cases for Series and Parallel Combination

P
(1) If n identical bulbs first connected in series so PS and
n

PP
then connected in parallel. So PP = nP hence n2
PS
Reason : Voltage on A decreases.
(2) An electric kettle has two coils when one coil is switched
on it takes time t1 to boil water and when the second coil (5) If two identical bulb A and B are connected in parallel
is switched on it takes time t2 to boil the same water. with ammeter A and key K as shown in figure.
 CURRENT ELECTRICITY

It should be remembered that on pressing key reading of 14. ELECTRICAL CONDUCTING

ammeter becomes twice.
MATERIALS FOR SPECIFIC USE
(1) Filament of electric bulb : Is made up of tungsten which
has high resistivity, high melting point.
(2) Element of heating devices (such as heater, geyser or
press) : Is made up of nichrome which has high resistivity
and high melting point.
(3) Resistances of resistance boxes (standard resistances) :
Reason : Total resistance becomes half.
Are made up of alloys (manganin, constantan or nichrome)
Concepts these materials have moderate resistivity which is
practically independent of temperature so that the
When a heavy current appliance such us motor, heater specified value of resistance does not alter with minor
or geyser is switched on, it will draw a heavy current changes in temperature.
from the source so that terminal voltage of source
decreases. Hence power consumed by the bulb (4) Fuse-wire : Is made up of tin-lead alloy (63% tin + 37%
decreases, so the light of bulb becomes less. lead). It should have low melting point and high resistivity.
It is used in series as a safety device in an electric circuit
and is designed so as to melt and thereby open the circuit
if the current exceeds a predetermined value due to some
fault. The function of a fuse is independent of its length.

Safe current of fuse wire relates with it’s radius as i v r 3/2

(5) Thermistors : A thermistor is a heat sensitive resistor
usually prepared from oxides of various metals such as
13.5 Some aspects of heating effects of current nickel, copper, cobalt, iron etc. These compounds are also
semi-conductor. For thermistors D is very high which
1. The wire supplying current to an electric lamp are not may be positive or negative. The resistance of thermistors
practically heated while the filament of lamp becomes white changes very rapidly with change of temperature.
hot.
We know that in series connections the heat produced due
to a current in a conductor is proportional to its resistance
(i.e. H v R). The filament of the lamp and the supply wires
are in series. The resistance of the wire supplying the current
to the lamp is very small as compared to that of the filament
of the lamp. Therefore, there is more heating effect in the
filament of the lamp than that in the supply wires. Due to it, Thermistors are used to detect small temperature change
the filament of the lamp becomes white hot whereas the and to measure very low temperature.
wires remain practically unheated.
15. SUPER CONDUCTIVITY
2. Electric Iron
Prof. K. Onnes, in 1911, discovered that certain metals and alloys
3. Electric Arc
at very low temperature lose their resistance considerably. This
4. Incandescent electric lamp phenomenon is known as super-conductivity. As the temperature
decreases, the resistance of the material also decreases, but when
the temperature reaches a certain critical value (called critical
temperature or transition temperature), the resistance of the
material completely disappears i.e., it becomes zero. Then the
material behaves as if it is a super-conductor and there will be
flow of electrons without any resistance whatsoever. The critical
temperature is different for different materials. It has been found
5. Fuse wire
 CURRENT ELECTRICITY

that mercury at critical temperature 4.2 K, lead at 7.25 K and V = potential difference across the conductor and l =
niobium at critical temperature 9.2 K become super-conductors. length of the conductor. Electric field out side the current
A team of scientists discovered that an alloy of plutonium, cobalt carrying conductor is zero.
and gallium exhibits super conductivity at temperatures below
18.5 K. Since 1987, many superconductors have been prepared
with critical temperature upto 125 K, as listed below
Bi2Ca2Sr2Cu3O10 at 105 K and Tl2Ca2Ba2Cu3O10 at 125 K.
The super-conductivity shown by materials can be verified by
simple experiment. If a current is once set up in a closed ring of
1
super-conducting material, it continues flowing for several weeks 4. For a given conductor JA = i = constant so that J v
after the source of e.m.f. has been withdrawn. A
The cause of super-conductivity is that, the free electrons in super- i.e., J1 A1 = J2 A2 ; this is called equation of continuity
conductor are no longer independent but become mutually
dependent and coherent when the critical temperature is reached.
The ionic vibrations which could deflect free electrons in metals
are unable to deflect this coherent or co-operative cloud of
electrons in super-conductors. It means the coherent cloud of
electrons makes no collisions with ions of the super-conductor
and, as such, there is no resistance offered by the super-conductor
to the flow of electrons. 5. The drift velocity of electrons is small because of the
frequent Collisions suffered by electrons.
Super-conductivity is a very interesting field of research all over
the world these days. The scientists have been working actively 6. The small value of drift velocity produces a large amount
to prepare super-conductor at room temperature and they have of electric current, due to the presence of extremely large
met with some success only. number of free electrons in a conductor. The propagation
of current is almost at the speed of light and involves
Application of super conductors
electromagnetic process. It is due to this reason that the
1. Super conductors are used for making very strong electric bulb glows immediately when switch is on.
electromagnets.
7. In the absence of electric field, the paths of electrons
2. Super conductivity is playing an important role in material
science research and high energy partical physics. between successive collisions are straight line while in
presence of electric field the paths are generally curved.
3. Super conductivity is used to produce very high speed
computers. NA x d
8. Free electron density in a metal is given by n
4. Super conductors are used for the transmission of electric A
power. where N A = Avogadro number, x = number of free
electrons per atom, d = density of metal and A = Atomic
TIPS AND TRICKS weight of metal.
9. In the absence of radiation loss, the time in which a fuse will
1. Human body, though has a large resistance of the order of
melt does not depends on it’s length but varies with radius
k: (say 10 k:), is very sensitive to minute currents even
as low as a few mA. Electrocution, excites and disorders as t v r 4
the nervous system of the body and hence one fails to 10. If length (l) and mass (m) of a conducting wire is given
control the activity of the body.
A2
2. dc flows uniformly throughout the cross-section of then R v
m
conductor while ac mainly flows through the outer surface
area of the conductor. This is known as skin effect. V
11. Macroscopic form of Ohm’s law is R , while it’ss
3. It is worth noting that electric field inside a charged i
conductor is zero, but it is non zero inside a current microscopic form is J = V E.
12. After stretching if length increases by n times then
V
carrying conductor and is given by E where
A resistance will increase by n 2 times i.e. R 2 n 2 R1
CURRENT ELECTRICITY

Similarly if radius be reduced to 1/n times then area of 24. Resistance of a conducting body is not unique but
cross-section decreases 1/n2 times so the resistance depends on it’s length and area of cross-section i.e. how
becomes n4 times i.e. R 2 n 4 R1 the potential difference is applied. See the following
figures
13. After stretching if length of a conductor increases by x%
then resistance will increases by 2x % (valid only if x < 10%)
14. Decoration of lightning in festivals is an example of series
grouping whereas all household appliances connected in
parallel grouping.
15. Using n conductors of equal resistance, the number of
possible combinations is 2n – 1.
16. If the resistance of n conductors are totally different, then
the number of possible combinations will be 2n. Length = a Length = b
17. If n identical resistances are first connected in series and Area of cross-section = b u c Area of cross-section = a u c
then in parallel, the ratio of the equivalent resistance is
Resistance R U §¨
b ·
Resistance R U §¨
a ·
Rp n2 ¸ ¸
given by © buc ¹ ©auc¹
Rs 1
25. Some standard results for equivalent resistance
18. If a wire of resistance R, cut in n equal parts and then
these parts are collected to form a bundle then equivalent
R
resistance of combination will be .
n2
19. If equivalent resistance of R1 and R2 in series and parallel
be Rs and Rp respectively then

1ª
R1 R s  R s2  4R s R p º» and
2 «¬ ¼
R1R 2 (R 3  R 4 )  (R1  R 2 )R 3R 4  R 5 (R1  R 2 ) (R 3  R 4 )
1ª R AB
R2 R s  R s2  4R s R p º» R 5 (R1  R 2  R 3  R 4 )  (R1  R 3 )(R 2  R 4 )
2 ¬« ¼
20. If a skeleton cube is made with 12 equal resistance each
having resistance R then the net resistance across

2R 1R 2  R 3 (R 1  R 2 )
R AB
2R 3  R1  R 2
5
21. The longest diagonal (EC or AG) R
6

3
22. The diagonal of face (e.g. AC, ED, ....) R
4

7
23. A side (e.g. AB, BC.....) R
12
 CURRENT ELECTRICITY

32. If n identical cells are connected in a loop in order, then

1 1 1/ 2
R AB (R 1  R 2 )  ¬ª(R 1  R 2 ) 2  4R 3 (R 1  R 2 ) ¼º emf between any two points is zero.
2 2

1 ª §R ·º
R AB R 1 «1  1  4 ¨ 2 ¸»
2 « © R1 ¹ »¼ 33. In parallel grouping of two identical cell having no internal
¬
resistance
26. It is a common misconception that “current in the circuit
will be maximum when power consumed by the load is
maximum.”
27. Actually current i = E/(R + r) is maximum (= E/r) when
R = min = 0 with PL = (E/r)2 × 0 = 0 min. while power
consumed by the load E2R/(R + r)2 is maximum (= E2/4r)
when R = r and i (E / 2r) z max ( E / r).

28. Emf is independent of the resistance of the circuit and 34. When two cell’s of different emf and no internal resistance
depends upon the nature of electrolyte of the cell while are connected in parallel then equivalent emf is
potential difference depends upon the resistance between indeterminate, note that connecting a wire with a cell with
the two points of the circuit and current flowing through no resistance is equivalent to short circuiting. Therefore
the circuit. the total current that will be flowing will be infinity.
29. Whenever a cell or battery is present in a branch there
must be some resistance (internal or external or both)
present in that branch. In practical situation it always
happen because we can never have an ideal cell or battery
with zero resistance.
30. In series grouping of identical cells. If one cell is wrongly
connected then it will cancel out the effect of two cells 35. In the parallel combination of non-identical cell’s if they
e.g. If in the combination of n identical cells (each having are connected with reversed polarity as shown then
emf E and internal resistance r) if x cell are wrongly equivalent emf

connected then equivalent emf Eeq (n  2 x ) E and E1r2  E 2 r1

E eq
equivalent internal resistance req nr r1  r2

31. Graphical view of open circuit and closed circuit of a

cell.
CURRENT ELECTRICITY

36. Wheatstone bridge is most sensitive if all the arms of 39. The measurement of resistance by Wheatstone bridge is
bridge have equal resistances i.e. P = Q = R = S not affected by the internal resistance of the cell.
37. If the temperature of the conductor placed in the right 40. In case of zero deflection in the galvanometer current
gap of metre bridge is increased, then the balancing length flows in the primary circuit of the potentiometer, not in
decreases and the jockey moves towards left. the galvanometer circuit.
38. In Wheatstone bridge to avoid inductive effects the battery 41. A potentiometer can act as an ideal voltmeter.
key should be pressed first and the galvanometer key
afterwards.
MOVING CHARGES AND MAGNETISM

MOVING CHARGES AND MAGNETISM

GENERAL KEY CONCEPT

1. Force on a moving charge:– A moving charge is a source of magnetic field.

Z


+q v cos B Y

sin
v 
X v
 
Let a positive charge q is moving in a uniform magnetic field B with velocity v .’
F  q  F  v sin  F  B
 F  qBv sin F = kq Bv sin  [k = constant]
k = 1 in S.I. system.
  
 F = qBv sin and F  q( v  B )

2. Magnetic field strength ( B ) :
In the equation F = qBv sin  , if q = 1, v = 1,
sin = 1 i.e.  = 90° then F = B.
 Magnetic field strength is defined as the force experienced by a unit charge
moving with unit velocity perpendicular to the direction of magnetic field.
Special Cases:
(1) It  = 0° or 180°, sin = 0
 F=0
A charged particle moving parallel to the magnetic field, will not experience
any force.
(2) If v = 0, F = 0
A charged particle at rest in a magnetic field will not experience any force.
(3) If  = 90°, sin = 1 then the force is maximum
Fmax. = qvB
A charged particle moving perpendicular to magnetic field will experience
maximum force.
3. S.I. unit of magnetic field intensity. It is called tesla (T).
F
B
qv sin 
If q = 1C, v = 1m/s,  = 90° i.e. sin = 1 and F = 1N
Then B = 1T.
 MOVING CHARGES AND MAGNETISM

The strength of magnetic field at a point is said to be 1T if a charge of 1C while

moving at right angle to a magnetic field, with a velocity of 1 m/s experiences a
force of 1N at that point.
4. Biot-Savart’s law:– The strength of magnetic field
Y
or magnetic flux density at a point P (dB) due to
current element dl depends on,
(i) dB  I dl  
r
(ii) dB  dl
P
(iii) dB  sin 
X I
1
(iv) dB  ,
r2
Idl sin  Idl sin 
Combining, dB  2
 dB  k [k = Proportionality constant]
r r2
0
In S.I. units, k  where µ0 is called permeability of free space.
4
 0 = 4 × 10–7 TA–1m
 
0 Idl sin   0 (dl  r )
 dB  and dB  I
4 r 2 4 r3
  
d B is perpendicular to the plane containing d and r and is directed inwards.
5. Applications of Biot-Savart’s law:–
(a) Magnetic field (B) at the Centre of a Circular Coil Carrying Current.
 nI
B 0
2r
where n is the number of turns of the coil. I is
the current in the coil and r is the radius of the
coil. I
2
(b) Magnetic field due to a straight conductor carrying current. a P
1
0 I
B (sin 2  sin 1 )
4a
where a is the perpendicular distance of the
conductor from the point where field is to the
measured.
1 and 2 are the angles made by the two ends of the conductor with the point.
(c) For an infinitely long conductor, 1  2   / 2
0 2I
 B= .
4 a
(d) Magnetic field at a point on the axis of a Circular Coil Carrying Current.
when point P lies far away from the centre of the coil.
0 2M
B . 3
4 x
where M = nIA = magnetic dipole moment of the coil.
x is the distance of the point where the field is to the measured, n is the number
of turns, I is the current and A is the area of the coil.
MOVING CHARGES AND MAGNETISM

6. Ampere’s circuital law:–


The line integral of magnetic field B around any closed path in vacuum is  0
 
times the total current through the closed path. i.e.  B.d l   0 I
7. Application of Ampere’s circuital law:–
(a) Magnetic field due to a current carrying solenoid, B = µ0nI
n is the number of turns per unit length of the solenoid.
µ0 nI
At the edge of a short solenoid, B =
2
(b) Magnetic field due to a toroid or endless solenoid
B = µ0nI
8. Motion of a charged particle in uniform electric field:–
The path of a charged particle in an electric field is a parabola.

2mv2
Equation of the parabola is x 2  y
qE
where x is the width of the electric field.
y is the displacement of the particle from its straight path.
v is the speed of the charged particle.
q is the charge of the particle
E is the electric field intensity.
m is the mass of the particle.
9. Motion of the charged particle in a magnetic field. The path of a charged particle
 
moving in a uniform magnetic field ( B ) with a velocity v making an angle  with

B is a helix.
n
v si 
v

B
O cos 
The component of velocity v cos  will not provide a force to the charged particle,
so under this velocity the particle with move forward with a constant velocity

along the direction of B . The other component v sin  will produce the force F = q
Bv sin  , which will supply the necessary centripetal force to the charged particle
in moving along a circular path of radius r.
m(vsin )2
 Centripetal force = = B qv sin 
r
Bqr
 v sin  =
m
v sin  Bq
Angular velocity of rotation = w = 
r m
 Bq
Frequency of rotation =   
2 2m
1 2m
Time period of revolution = T = 
 Bq
 MOVING CHARGES AND MAGNETISM

10. Cyclotron: It is a device used to accelerate and hence energies the positively
charged particle. This is done by placing the particle in an oscillating electric
field and a perpendicular magnetic field. The particle moves in a circular path.
 Centripetal force = magnetic Lorentz force
mv2 mv
 = Bqv  = r  radius of the circular path
r Bq
r m
Time to travel a semicircular path =  = constant.
v Bq
If v0 be the maximum velocity of the particle and r0 be the maximum radius of its
path then
mv0 2 Bqr0
 Bqv0  v0 
r0 m
1 1  Bqr0 
2 B2 q 2 r0 2
Max. K.E. of the particle = mv 0 2  m   (K.E.)max. =
2 2  m  2m

2 m
Time period of the oscillating electric field  T = .
Bq

Time period is independent of the speed and radius.

1 Bq
Cyclotron frequency =   
T 2m

Bq
Cyclotron angular frequency = 0  2 
m

11. Force on a current carrying conductor placed in a magnetic field:

  
F  I   B or F = I  B sin 

where I is the current through the conductor

B is the magnetic field intensity.
l is the length of the conductor.
 is the angle between the direction of current and magnetic field.
(i) When  = 0° or 180°, sin  = 0  F = 0
 When a conductor is placed along the magnetic field, no force will act on
the conductor.
(ii) When  = 90°, sin  = 1, F is maximum.
Fmax = I  B
when the conductor is placed perpendicular to the magnetic field, it will
experience maximum force.
12. Force between two parallel conductors carrying current:–
(a) When the current is in same direction the two conductors will attract each
other with a force
0 2I1I2
F . per unit length of the conductor
4 r
MOVING CHARGES AND MAGNETISM

(b) When the current is in opposite direction the two conductors will repel each
other with the same force.
(c) S.I. unit of current is 1 ampere. (A).
1A is the current which on flowing through each of the two parallel uniform
linear conductor placed in free space at a distance of 1 m from each other produces
a force of 2 × 10–7 N/m along their lengths.
13. Torque on a current carrying coil placed in a magnetic field:–
  
  M  B   = MB sin = nIBA sin where M is the magnetic dipole moment of
the coil.
M = nIA
where n is the number of turns of the coil.
I is the current through the coil.
B is the magnetic field intensity.
A is the area of the coil.

 is the angle between the magnetic field  B and the perpendicular to the plane
of the coil.
Special Cases:
(i) If the coil is placed parallel to magnetic field  = 0°, cos  = 1 then torque is
maximum.
max.  nIBA

(ii) If the coil is placed perpendicular to magnetic field,  = 90°, cos  = 0

  =0
14. Moving coil galvanometer:– This is based on the principle that when a current
carrying coil is placed in a magnetic field it experiences a torque. There is a
restoring torque due to the phosphor bronze strip which brings back the coil to its
normal position.
In equilibrium, Deflecting torque = Restoring torque
nIBA = k  [k = restoring torque/unit twist of the phosphor bronze strip]
k k
I   G where G  = Galvanometer constant
nBA nBA
 I
Current sensitivity of the galvanometer is the deflection produced when unit
current is passed through the galvanometer.
 nBA
Is  
I k
Voltage sensitivity is defined as the deflection produced when unit potential
difference is applied across the galvanometer.
  nBA
Vs    [R = Resistance of the galvanometer]
V IR kR
 MOVING CHARGES AND MAGNETISM
15. Condition for the maximum sensitivity of the galvanometer:-
The galvanometer is said to be sensitive if a small current produces a large
deflection.
nBA
  I
k
  will be large if (i) n is large, (ii) B is large (iii) A is large and (iv) k is small.
16. Conversion of galvanometer into voltmeter and ammeter
(a) A galvanometer is converted to voltmeter by putting a high resistance in series
with it.
Tot al r esist an ce of volt m et er = Rg + R where Rg is the galvonometer resistance.
R is the resistance added in series.
V
Current through the galvanometer = Ig = Rg  R

where V is the potential difference across the voltmeter.

Ig Rg HR
G
Voltmeter

I R I
M N
V
 R= G
Ig

Range of the voltmeter: 0 – V volt.

(b) A galvanometer is converted into an ammeter by connecting a low resistance
in parallel with it (shunt)

 I g 
Shunt = S    R g where Rg is the galvanometere resistance.
 I  Ig 
Ig Rg
G
R
I I
S
M (I - Ig) N
I is the total current through the ammeter.
Ig is the current through the ammeter. Effective resistance of the ammeter
Rg
R = R S
g

The range of the ammeter is 0 – I A. An ideal ammeter has zero resistance.

 MAGNETISM

MAGNETIC
THEORY
THEORY
G
If v and B are in the plane of paper, then according to
G
1. MAGNETIC FIELD AND FORCE
Right-Hand Rule, the direction of F on positively charged
G
G
In order to define the magnetic field B , we deduce an expression particle will be perpendicular to the plane of paper upwards
for the force on a moving charge in a magnetic field. as shown in figure (a), and on negatively charged particle will
Consider a positive charge q moving in a uniform magnetic field be perpendicular to the plane of paper downwards, figure (b).
G
B , with a velocity V . Let the angle between V and B be T.
G G G

G
(i) The magnitude of force F experienced by the moving charge
is directly proportional to the magnitude of the charge i.e.
Fvq
G
Definition of B
If v = 1, q = 1 and sin T = 1 or T= 90°, the nfrom (1),
G
(ii) The magnitude of force F is directly proportional to the
component of velocity acting perpendicular to the direction F = 1 × 1 × B × 1 = B.
of magnetic field, i.e. Thus the magnetic field induction at a point in the magnetic
F v vsin T field is equal to the force experienced by a unit charge moving
G with a unit velocity perpendicular to the direction of magnetic
(iii) The magnitude of force F is directly proportional to the field at that point.
magnitude of the magnetic field applied i.e.,
Special Cases
FvB Case (i) If T = 0° or 180°, then sin T= 0.
Combining the above factors, we get
? From (1),
F v qv sin TB or F = kqv B sin T F = qv B (0) = 0.
where k is a constant of proportionality. Its value is found It means, a charged particle moving along or opposite to the
to be one i.e. k = 1. direction of magnetic field, does not experience any force.
? F = qv B sin T ...(1) Case (ii) If v = 0, then F = qv B sin T= 0.
It means, if a charged particle is at rest in a magnetic field, it
G G G
F q vu B ...(2)
experiences no force.
G
The direction of F is the direction of cross-product of Case (iii) If T= 90°, then sin T= 1
G
velocity vG and magnetic field B , which is perpendicular to ? F = qv B (1) = qv B (Maximum).
G G
the plane containing vG and B . It is directed as given by the Unit of B . SI unit of B is tesla (T) or weber/(metre)2 i.e. (Wb/m2)
Right-handed-Screw Rule or Right-Hand Rule. or Ns C–1 m–1
 MAGNETISM

Thus, the magnetic field induction at a point is said to be speed, velocity, momentum and kinetic energy of charged
one tesla if a charge of one coulomb while moving at right particle will change.
angle to a magnetic field, with a velocity of 1 ms–1 experiences G G G
a force of 1 newton, at that point. Case II. When v, E and B are mutually perpendicular to
G G
each other. In this situation if E and B are such that
MLT 2
Dimensions of B ª MA 1T 2 º G G G
AT LT 1 ¼ F Fe  Fm 0 , then acceleration in the particle,
G
G F
a 0 . It means the particle will pass through the fields
2. LORENTZ FORCE m
without any change in its velocity. Here, Fe = Fm so qE = q v B
The force experienced by a charged particle moving in space or v = E/B.
where both electric and magnetic fields exist is called Lorentz This concept has been used in velocity-selector to get a
force. charged beam having a definite velocity.
Force due to electric field. When a charged particle carrying
G 3. MOTION OF A CHARGED PARTICLE IN A
charge +q is subjected to an electric field of strength E , it
experiences a force given by UNIFORM MAGNETIC FIELD
G G
Fe qE ...(5) Suppose a particle of mass m and charge q, entering a
G
G
whose direction is the same as that of E . uniform magnetic field induction B at O, with velocity vG ,
making an angle T with the direction of magnetic field acting
Force due to magnetic field. If the charged particle is moving
G in the plane of paper as shown in figure
in a magnetic field B , with a velocity vG it experiences a
force given by
G G G
Fm q v u B

G G
The direction of this force is in the direction of v u B i.e.
G
perpendicular to the plane contaning vG and B and is
directed as given by Right hand screw rule.
Due to both the electric and magnetic fields, the total force
experienced by the charged particle will be given by
G G G G G G G G G
F Fe  Fm qE  q v u B q E  v u B
Resolving vG into two rectangular components, we have :
G G G G v cos T (= v1) acts in the direction of the magnetic field and
F q E  vuB ...(6) v sin T (= v2) acts perpendicular to the direction of magnetic
field.
This is called Lorentz force. G
Special cases For component velocity v2 , the force acting on the charged
particle due to magnetic field is
G G G
Case I. When v, E and B , all the three are collinear.. In G G G
this situation, the charged particle is moving parallel or F q v2 u B
antiparallel to the fields, the magnetic force on the charged G
G
particle is zero. The electric force on the charged particle or F q v2 u B qv 2 Bsin 90q q vsin T B ...(1)
G
G qE G
will produce acceleration a , The direction of this force F is perpendicular to the plane
m G G
containing B and v 2 and is directed as given by Right
along the direction of electricl field. As a result of this, there
hand rule. As this force is to remain always perpendicular to
will be change in the speed of charged particle along the G
direction of the field. In this situation there will be no change v 2 it does not perform any work and hence cannot change
in the direction of motion of the charged particle but, the G
the magnitude of velocity v 2 . It changes only the direction
 MAGNETISM

of motion of the particle. Due to it, the charged particle is G G

angle between v1 and B is zero. Thus the charged particle
made to move on a circular path in the magnetic field, as
shown in figure covers the linear distance in direction of the magnetic field
with a constant speed v cos T.
Therefore, under the combined effect of the two component
velocities, the charged particle in magnetic field will cover
linear path as well as circular path i.e. the path of the charged
particle will be helical, whose axis is parallel to the direction
of magnetic field, figure

Here, magnetic field is shown perpendicular to the plane of

paper directed inwards and particle is moving in the plane
of paper. When the particle is at points A, C and D the
direction of magnetic force on the particle will be along AO,
CO and DO respectively, i.e., directed towards the centre O
of the circular path.
The force F on the charged particle due to magnetic field
2
provides the required centripetal force = mv 2 / r necessary
for motion along a circular path of radius r.

? Bq v 2 mv22 / r or v2 Bq r / m
or v sin T = B q r/m ...(2)
The angular velocity of rotation of the particle in magnetic
field will be

vsin T Bqr Bq
Z The linear distance covered by the charged particle in the
r mr m magnetic field in time equal to one revolution of its circular
The frequency of rotation of the particle in magnetic field path (known as pitch of helix) will be
will be
2Sm
d v1T v cos T
Z Bq Bq
v ...(3)
2S 2Sm Important points
The time period of revolution of the particle in the magnetic
1. If a charged particle having charge q is at rest in a magnetic
field will be G
field B , it experiences no force; as v = 0 and F = q v B sin T = 0.
1 2Sm G
T ...(4)
v Bq 2. If charged particle is moving parallel to the direction of B , it
also does not experience any force because angle T between
From (3) and (4), we note that v and T do not depend upon G G
v and B is 0° or 180° and sin 0° = sin 180° = 0. Therefore,
velocity vG of the particle. It means, all the charged particles
the charged particle in this situation will continue moving
having the same specific charge (charge/mass) but moving
along the same path with the same velocity.
with different velocities at a point, will complete their circular
paths due to component velocities perpendicular to the 3. If charged particle is moving perpendicular to the direction
magnetic fields in the same time. G
of B , it experiences a maximum force which acts
G G
For component velocity v1 vcos T , there will be no force perpendicular to the direction B as well as v . Hence this
on the charged particle in the magnetic field, because the force will provide the required centripetal force and the
 MAGNETISM

charged particle will describe a circular path in the magnetic sufficiently high energy with the help of smaller values of
oscillating electric field by making it to cross the same electric
mv 2 field time and again with the use of strong magnetic field.
field of radius r, given by Bqv .
r

4. MOTION IN COMBINED
ELECTRON AND MAGNETIC FIELDS

4.1 Velocity Filter

Velocity filter is an arrangement of cross electric and
magnetic fields in a region which helps us to select from a
beam, charged particles of the given velocity irrespective of
their charge and mass.
A velocity selector consists of two slits S1 and S2 held parallel
to each other, with common axis, some distance apart. In the
region between the slits, uniform electric and magnetic fields
are applied, perpendicular to each other as well as to the
axis of slits, as shown in figure. When a beam of charged
particles of different charges and masses after passing
G
through slit S1 enters the region of crossed electric field E
G
and magnetic field B , each particle experiences a force due
to these fields. Those particles which are moving with the Construction. It consists of two D-shaped hollow evacuated
velocity v, irrespective of their mass and charge, the force metal chambers D1 and D2 called the dees. These dees are
on each such particle due to electric field (qE) is equal and placed horizontally with their diametric edges parallel and
opposite to the force due to magnetic field (q v B), then slightly separated from each other. The dees are connected
q E = q v B or v = E/B to high frequency oscillator which can produce a potential
difference of the order of 104 volts at frequency | 107 Hz.
The two dees are enclosed in an evacuated steel box and
are well insulated from it. The box is placed in a strong
magnetic field produced by two pole pieces of strong
electromagnets N, S. The magnetic field is perpendicular to
the plane of the dees. P is a place of ionic source or positively
charged particle figure.
Working and theory. The positive ion to be accelerated is
produced at P. Suppose, at that instant, D1 is at negative
potential and D2 is at positive potential. Therefore, the ion
will be accelerated towards D1. On reaching inside D1, the
Such particles will go undeviated and filtered out of the ion will be in a field free space. Hence it moves with a
region through the slit S2. Therefore, the particles emerging constant speed in D 1 say v. But due to perpendicular
from slit S2 will have the same velocity even though their magnetic field of strength B, the ion will describe a circular
charge and mass may be different.
The velocity filter is used in mass spectrograph which helps mv 2
path of radius r (say) in D1, given by Bqv where m
to find the mass and specific charge (charge/mass) of the r
charged particle. and q are the mass and charge of the ion.
4.2 Cyclotron mv
? r
A cyclotron is a device developed by Lawrence and Bq
Livingstone by which the positively charged particles like Time taken by ion to describe a semicircular path is given
proton, deutron, alpha particle etc. can be accelerated.
Sr Sm S
Principle. The working of the cyclotron is based on the fact by, t = a constant.
that a positively charged particle can be accelerated to a v Bq B q/m
MAGNETISM

This time is independent of both the speed of the ion and in a conductor is due to motion of electrons, therefore,
radius of the circular path. In case the time during which electrons are moving from the end Q to P (along X’ axis).
the positive ion describes a semicircular path is equal to the
time during which half cycle of electric oscillator is completed,
then as the ion arrives in the gap between the two dees, the
polarity of the two dees is reversed i.e. D1 becomes positive
and D2 negative. Then, the positive ion is accelerated
towards D2 and it enters D2 with greater speed which remains
constant in D2. The ion will describe a semicircular path of
greater radius due to perpendicular magnetic field and again G
Let, vd drift velocity of electron
will arrive in a gap between the two dees exactly at the
instant, the polarity of the two dees is reversed. Thus, the – e = charge on each electron.
positive ion will go on accelerating every time it comes into Then magnetic Lorentz force on an electron is given by
the gap between the dees and will go on describing circular
G G G
path of greater and greater radius with greater and greater f  e vd u B
speed and finally acquires a sufficiently high energy. The
accelerated ion can be removed out of the dees from window If n is the number density of free electrons i.e. number of
W, by applying the electric field across the deflecting plates free electrons per unit volume of the conductor, then total
E and F. number of free electrons in the conductor will be given by
Maximum Energy of positive ion N = n (AA) = nAA
Let v0, r0 = maximum velocity and maximum radius of the ? Total force on the conductor is equal to the force acting on
circular path followed by the positive ion in cyclotron. all the free electrons inside the conductor while moving in
the magnetic field and is given by
mv 02 Bqr0 G G G G G G
Then, Bqv0 or v0 F Nf nAA ª  e v d u B º  nAAe v d u B ...(7)
r0 m ¬ ¼

2
We know that current through a conductor is related with
1 2 1 § Bqr0 · B2q 2 r02 drift velocity by the relation
? Max. K.E. mv0 m¨ ¸
2 2 © m ¹ 2m I = n A e vd
Cyclotron Frequency ? IA nAev d .A
If T is the time period of oscillating electric field then G
We represent IA as current element vector. It acts in the
T = 2t = 2S m/Bq G G
direction of flow of current i.e. along OX. Since I A and vd
1 Bq have opposite directions, hence we can write
The cyclotron frequency is given by v
T 2Sm G G
I A  nAAevd ...(8)
It is also known as magnetic resonance frequency.
From (7) and (8), we have
The cyclotron angular frequency is given by G G G
F IA u B ...(9)
Zc 2Sv Bq / m
G G G
F I Au B
5. FORCE ON A CURRENT CARRYING CONDUCTOR
PLACED IN A MAGNETIC FIELD F IABsin T ...(10)
G G
were T is the smaller angle between I A and B .
Expression for the force acting on the conductor carrying
current placed in a magnetic field Special cases
Consider a straight cylindrical conductor PQ of length A, Case I. If T = 0° or 180°, sin T= 0,
area of cross-section A, carrying current I placed in a uniform From (10), F = IAB (0) = 0 (Minimum)
G
magnetic field of induction, B . Let the conductor be placed It means a linear conductor carrying a current if placed parallel
along X-axis and magnetic field be acting in XY plane making to the direction of magnetic field, it experiences no force.
an angle T with X-axis. Suppose the current I flows through Case II. If T = 90°, sin T= q ;
the conductor from the end P to Q, figure. Since the current
From (10), F = IAB × 1 = IAB (Maximum)
 MAGNETISM

It means a linear conductor carrying current if placed G JJJG G

The force on the arm QR is given by F2 I QR u B or
perpendicular to the direction of magnetic field, it experiences
maximum force. The direction of which can be given by F2 = I (QR) B sin T = I b B sin T
Right handed screw rule. The direction of this force is in the plane of the coil directed
downwards.
6. TORQUE ON A CURRENT CARRYING COIL IN G G
A MAGNETIC FIELD Since the forces F2 and F4 are equal in magnitude and acting
in opposite directions along the same straight line, they cancel
Consider a rectangular coil PQRS suspended in a uniform out each other i.e. their resultant effect on the coil is zero.
G
magnetic field of induction B . Let PQ = RS = A and QR = SP = b. Now, the force on the arm PQ is given by
Let I be the current flowing through the coil in the direction G JJJG G JJJG G
PQRS and T be the angle which plane of the coil makes with F1 I PQ u B or F1 = I (PQ) B sin 90° = IAB ' PQ A B
the direction of magnetic field figure. The forces will be
acting on the four arms of the coil. Direction of this force is perpendicular to the plane of the
coil directed outwards (i.e. perpendicular to the plane of
paper directed towards the reader).
And, force on the arm RS is given by
G JJJG G JJJG G
F3 I RS u B or F3 = I (PQ) B sin 90° = IAB ' RS A B

The direction of this force, is perpendicular to the plane of paper

directed away from the reader i.e. into the plane of the coil.
The forces acting on the arms PQ and RS are equal, parallel
and acting in opposite directions having different lines of
action, form a couple, the effect of which is to rotate the coil
in the anticlockwise direction about the dotted line as axis.
The torque on the coil (equal to moment of couple) is given by
W = either force × arm of the couple
The forces F1 and F3 acting on the arms PQ and RS will be as
shown in figure when seen from the top.
Arm of couple = ST = PS cos T = b cos T.
? W IAB u b cos T IBA cos T (' A
× b = A = area of coil
PQRS)
If the rectangular coil has n turns, then
W nIBA cos T
Note that if the normal drawn on the plane of the coil makes
an angle D with the direction of magnetic field, then T+ D = 90°
or T= 90° – D; And cos T= cos (90° – D) = sin D
Then torque becomes,
G G G G
W nIBA sin D MBsin D MuB nIA u B
G G G G
Let F1, F2 , F3 and F4 be the forces acting on the four current
where, nIA = M = magnitude of the magnetic dipole moment
carrying arms PQ, QR, RS and SP of the coil. of the rectangular current loop
The force on arm SP is given by, G G G G G
G JJG G ? W M u B nI A u B
F4 I SP u B or F4 = I (SP) B sin (180° – T) = Ib B sin T
This torque tends to rotate the coil about its own axis. Its
JJG G value changes with angle between plane of coil and direction
The direction of this force is in the direction of SP u B i.e. of magnetic field.
in the plane of coil directed upwards.
 MAGNETISM

Special cases 1. The lower end of the coil is connected to one end of a hair
If the coil is set with its plane parallel to the direction of spring S’ of quartz or phosphor bronze. The other end of this
magnetic field B, then highly elastic spring S’ is connected to a terminal T2. L is soft
iron core which may be spherical if the coil is circular and
T 0q and cos T 1 cylindrical, if the coil is rectangular. It is so held within the
? Torque, W = nIBA (1) = nIBA (Maximum) coil, that the coil can rotate freely without touching the iron
core and pole pieces. This makes the magnetic field linked
This is the case with a radial field.
with coil to be radial field i.e. the plane of the coil in all positions
2. If the coil is set with its plane perpendicular to the direction remains parallel to the direction of magnetic field. M is concave
of magentic field B, then T = 90° and cos T = 0 mirror attached to the phosphor bronze strip. This helps us to
? Torque, W= nIBA (0) = 0 (Minimum) note the deflection of the coil using lamp and scale
arrangement. The whole arrangement is enclosed in a non-
7. MOVING COIL GALVANOMETER metallic case to avoid disturbance due to air etc. The case is
provided with levelling screws at the base.
Moving coil galvanometer is an instrument used for detection The spring S’ does three jobs for us : (i) It provides passage
and measurement of small electric currents. of current for the coil PQRS1 (ii) It keeps the coil in position
Principle. Its working is based on the fact that when a current and (iii) generates the restoring torque on the twisted coil.
carrying coil is placed in a magnetic field, it experiences a torque. The torsion head is connected to terminal T 1. The
Construction. It consists of a coil PQRS1 having large galvanometer can be connected to the circuit through
number of turns of insulated copper wire, figure. The coil is terminals T1 and T2.
wound over a non-magnetic metallic frame (usually brass) Theory. Suppose the coil PQRS1 is suspended freely in the
which may be rectangular or circular in shape. The coil is magnetic field.
suspended from a movable torsion head H by means of Let, A = length PQ or RS1 of the coil,
phosphor bronze strip in a uniform magnetic field produced
b = breadth QR or S1P of the coil,
by two strong cylindrical magnetic pole pieces N and S.
n = number of turns in the coil.
Area of each turn of the coil, A = A × b.
Let, B = strength of the magnetic field in which coil is
suspended.
I = current passing through the coil in the direction PQRS1
as shown in figure.
Let at any instant, D be the angle which the normal drawn on
the plane of the coil makes with the direction of magnetic field.
As already discussed, the rectangular coil carrying current
when placed in the magnetic field experiences a torque whose
magnitude is given by W = nIBA sin D.
If the magnetic field is radial i.e. the plane of the coil is
parallel to the direction of the magnetic field then D= 90°
and sin D= 1.
? W= nIBA
Due to this torque, the coil rotates. The phosphor bronze
strip gets twisted. As a result of it, a restoring torque comes
into play in the phosphor bronze strip, which would try to
restore the coil back to its original position.
Let T be the twist produced in the phosphor bronze strip
due to rotation of the coil and k be the restoring torque per
unit twist of the phosphor bronze strip, then total restoring
torque produced = k T.
In equilibrium position of the coil, deflecting torque
= restoring torque
 MAGNETISM

? nIBA = kT (b) The value of B can be increased by using a strong horse

shoe magnet.
k
or I T or I GT (c) The value of A can not be increased beyond a limit because
nBA
in that case the coil will not be in a uniform magnetic field.
k Moreover, it will make the galvanometer bulky and
where G a constant for a galvanometer. It is unmanageable.
nBA
known as galvanometer constant. (d) The value of k can be decreased. The value of k depends
upon the nature of the material used as suspension strip.
Hence, I v T The value of k is very small for quartz or phosphor bronze.
It means, the deflection produced is proportional to the That is why, in sensitive galvanometer, quartz or phosphor
current flowing through the galvanometer. Such a bronze strip is used as a suspension strip.
galvanometer has a linear scale.
Current sensitivity of a galvanometer is defined as the 8. AMMETER
deflection produced in the galvanometer when a unit current
flows through it. An ammeter is a low resistance galvanometer. It is used to
measure the current in a circuit in amperes.
If T is the deflection in the galvanometer when current I is
passed through it, then A galvanometer can be converted into an ammeter by using
a low resistance wire in parallel with the galvanometer. The
Current sensitivity, resistance of this wire (called the shunt wire) depends upon
the range of the ammeter and can be calculated as follows :
T nBA § k ·
Is ¨' I T¸ Let G = resistance of galvanometer, n = number of scale
I k © nBA ¹
divisions in the galvanometer,
The unit of current sensitivity is rad. A–1 or div. A–1. K = figure of merit or current for one scale deflection in the
Voltage sensitivity of a galvanometer is defined as the galvanometer.
deflection produced in the galvanometer when a unit voltage Then current which produces full scale deflection in the
is applied across the two terminals of the galvanometer. galvanometer, Ig = nK.
Let, V = voltage applied across the two terminals of the Let I be the maximum current to be measured by galvanometer.
galvanometer,
To do so, a shunt of resistance S is connected in parallel
T = deflection produced in the galvanometer. with the galvanometer so that out of the total current I, a
Then, voltage sensitivity, VS = T/V part I g should pass through the galvanometer and the
If R = resistance of the galvanometer, I = current through it. remaining part (I – Ig) flows through the shunt figure
Then V = IR
? Voltage sensitivity,

T nBA IS
VS
IR kR R
the unit of VS is rad V–1 or div. V–1.
Conditions for a sensitive galvanometer
A galvanometer is said to be very sensitive if it shows large
deflection even when a small current is passed through it.

nBA VA – VB = IgG = (I – Ig) S

From the theory of galvanometer, T I
k
§ Ig ·
For a given value of I, T will be large if nBA/k is large. It is so S ¨ ¸¸ G
or ¨ I  Ig ...(20)
if (a) n is large (b) B is large (c) A is large and (d) k is small. © ¹
(a) The value of n can not be increased beyond a certain limit Thus S can be calculated.
because it results in an increase of the resistance of the If this value of shunt resistance S is connected in parallel
galvanometer and also makes the galvanometer bulky. This with galvanometer, it works as an ammeter for the range 0 to I
tends to decrease the sensitivity. Hence n can not be ampere. Now the same scale of the galvanometer which was
increased beyond a limit. recording the maximum current Ig before conversion into ammeter
 MAGNETISM

will record the maximum current I, after conversion into ammeter. V V

It means each division of the scale in ammeter will be showing From Ohm’s law, Ig or G  R
GR Ig
higher current than that of galvanometer.

V
or R G
Ig

Initial reading of each division of galvanometer to be used as If this value of R is connected in series with galvanometer, it
ammeter is Ig/n and the reading of the same each division works as a voltmeter of the range 0 to V volt. Now the same
after conversion into ammeter is I/n. scale of the galvanometer which was recording the maximum
The effective resistance R P of ammeter (i.e. shunted potential Ig G before conversion will record and potential V
galvanometer) will be after conversion in two voltmeter. It means each division of
the scale in voltmeter will show higher potential than that of
1 1 1 S G GS the galvanometer.
 or R P
RP G S GS G S Effective resistance RS of converted galvanometer into
voltmeter is
As the shunt resistance is low, the combined resistance of
the galvanometer and the shunt is very low and hence RS = G + R
ammeter has a much lower resistance than galvanometer. An For voltmeter, a high resistance R is connected in series
ideal ammeter has zero resistance. with the galvanometer, therefore, the resistance of voltmeter
is very large as compared to that of galvanometer. The
resistance of an ideal voltmeter is infinity.
9. VOLTMETER
A voltmeter is a high resistance galvanometer. It is used to 10. BIOT­SAVART’S LAW
measure the potential difference between two points of a
circuit in volt. According to Biot-Savart’s law, the magnitude of the
magnetic field induction dB (also called magnetic flux
A galvanometer can be converted into a voltmeter by density) at a point P due to current element depends upon
connecting a high resistance in series with the galvanometer. the factors at stated below :
The value of the resistance depends upon the range of
voltmeter and can be calculated as follows : (i) dB v I (ii) dB v dA
Let, G = resistance of galvanometer, 1
(iii) dB v sin T (iv) dB v
n = number of scale divisions in the galvanometer, r2
K = figure of merit of galvanometer i.e. current for one scale Combining these factors, we get
deflection of the galvanometer.
? IdA sin T
Current which produces full scale deflection in the dB v
galvanometer, Ig = nK. r2
Let V be the potential difference to be measured by IdA sin T
galvanometer. or dB K
r2
To do so, a resistance R of such a value is connected in
series with the galvanometer so that if a potential difference
V is applied across the terminals A and B, a current Ig flows
through the galvanometer. figure

where K is a constant of proportionality. Its value depends

Now, total resistance of voltmeter = G + R on the system of units chosen for the measurement of the
various quantities and also on the medium between point P
 MAGNETISM

and the current element. When there is free space between 8. If T = 0° or 180°, then dB = 0 i.e. minimum.
current element and point, then Similarities and Dis-similarities between the Biot-Savart’s law
for the magnetic field and coulomb’s law for electrostatic field
P0
In SI units, K and In cgs system K = 1 Similarities
4S
where P0 is absolute magnetic permeability of free space (i) Both the laws for fields are long range, since in both the
laws, the field at a point varies inversely as the square of the
and P0 4Su107 Wb A1m1 4Su107 TA 1m distance from the source to point of observation.
(ii) Both the fields obey superposition principle.
(' 1 T = 1 Wb m–2)
G
(iii) The magnetic field is linear in the source Id A , just as the
P0 IdA sin T
In SI units, dB u ...(3) electric field is linear in its source, the electric charge q.
4S r2

IdA sin T 11. MAGNETIC FIELD DUE TO A STRAIGHT

In cgs system, dB 2 CONDUCTOR CARRYING CURRENT
r
In vector form, we may write Consider a straight wire conductor XY lying in the plane of
G G G G paper carrying current I in the direction X to Y, figure. Let P
G P0 I d A u r G P0 I d A u r be a point at a perpendicular distance a from the straight
dB or dB ...(4)
4S r 3
4S r3 wire conductor. Clearly, PC = a. Let the conductor be made
of small current elements. Consider a small current element
G G G G
Direction of dB . From (4), the direction of dB would Id A of the straight wire conductor at O. Let r be the
obviously be the direction of the cross product vector, position vector of P w.r.t. current element and T be the angle
G G G
d A u r . It is represented by the Right handed screw rule or G
between Id A and r. Let CO = A.
G
Right Hand Rule. Here dB is perpendicular to the plane
G
containing d A and Gr and is directed inwards. If the point P
G
is to the left of the current element, dB will be perpendicular
G
to the plane containing d A and Gr , directed outwards.
Some important features of Biot Savart’s law
1. Biot Savart’s law is valid for a symmetrical current distribution.
2. Biot Savart’s law is applicable only to very small length
conductor carrying current.
3. This law can not be easily verified experimentally as the
current carrying conductor of very small length can not be
obtained practically.
4. This law is analogous to Coulomb’s law in electrostatics.
G G
5. The direction of dB is perpendicular to both Id A and Gr .
6. If T = 0° i.e. the point P lies on the axis of the linear conductor
carrying current (or on the wire carrying current) then G
According to Biot-Savart’s law, the magnetic field dB (i.e.
P0 IdA sin 0q magnetic flux density or magnetic induction) at point P due
dB 0 G
4S r2 to current element Id A is given by
It means there is no magnetic field induction at any point on G G
the thin linear current carrying conductor.
G P 0 Id A u r
dB .
4S r 3
7. If T = 90° i.e. the point P lies at a perpendicular position w.r.t.
current element, then
P0 IdA sin T
or dB u ...(5)
P 0 IdA 4S r2
dB , which is maximum.
4S r 2
 MAGNETISM

In rt. angled 'POC, T+ I = 90° or T= 90° – I

P0 I P 0 2I P0 2I L
? sin T = sin (90° – I) = cos I ...(6) Then, B sin I  sin I sin I
4Sa 4S a 4S a 4a 2  L2
a a G
Also, cos I or r ...(7) (iv) When point P lies on the wire conductor, then d A and Gr for
r cos I
each element of the straight wire conductor are parallel.
G
A Therefore, d A u Gr 0 . So the magnetic field induction at P = 0.
And, tan I or A a tan I
a
Direction of magnetic field
Differentiating it, we get The magnetic field lines due to straight conductor carrying
dA a sec I dI 2
...(8) current are in the form of concentric circles with the
conductor as centre, lying in a plane perpendicular to the
Putting the values in (5) from (6), (7) and (8), we get straight conductor. The direction of magnetic field lines is
anticlockwise, if the current flows from A to B in the straight
2
P0 I a sec I dI cos I P0 I conductor figure (a) and is clockwise if the current flows
dB cos I dI ...(9) from B to A in the straight conductor, figure (b). The direction
4S § a2 · 4S a
¨¨ cos 2 I ¸¸ of magnetic field lines is given by Right Hand Thumb Rule
© ¹ or Maxwell’s cork screw rule.
G
The direction of dB , according to right hand thumb rule,
will be perpendicular to the plane of paper and directed
inwards. As all the current elements of the conductor will
also produce magnetic field in the same direction, therefore,
the total magnetic field at point P due to current through the
whole straight conductor XY can be obtained by integrating
Eq. (9) within the limits – I1 and + I2. Thus
I2 I2
P0 I P0 I I2
B ³ dB 4S a ³ cos I dI 4S a
sin I I1
I1 I1

P0 I P0 I
ªsin I2  sin I1 º¼ sin I1  sin I2 ...(10)
4S a ¬ 4S a Right hand thumb rule. According to this rule, if we imagine
the linear wire conductor to be held in the grip of the right
Special cases. (i) When the conductor XY is of infinite length
and the point P lies near the centre of the conductor then hand so that the thumb points in the direction of current,
then the curvature of the fingers around the conductor
I1 I2 90q will represent the direction of magnetic field lines, figure
(a) and (b).
P0 I P0 2I
So, B sin 90q  sin 90q ...(11)
4S a 4S a
(ii) When the conductor XY is of infinite length but the point P
lies near the end Y (or X) then I1 = 90° and I2 = 0°.

P0 I P0 I
So, B sin 90q  sin 0q ...(11 a)
4S a 4S a
Thus we note that the magnetic field due to an infinite long
linear conductor carrying current near its centre is twice
than that near one of its ends.
(iii) If length of conductor is finite, say L and point P lies on
right bisector of conductor, then

L/ 2 L
I1 I2 I and sin I
2
a2  L / 2 4a 2  L2
 MAGNETISM

12. MAGNETIC FIELD AT THE CENTRE OF THE P0 I P 0 2 SI

? B .2Sr
CIRCULAR COIL CARRYING CURRENT 4S r 2 4S r
If the circular coil consists of n turns, then
Consider a circular coil of radius r with centre O, lying with
its plane in the plane of paper. Let I be the current flowing in P 0 2SnI P0 I
B u 2 Sn ...(13)
the circular coil in the direction shown, figure (a). Suppose 4S r 4S r
the circular coil is made of a large number of current elements
each of length dA. P0 I
i.e. B × angle subtended by coil at the centre.
4S r
G
Direction of B
The direction of magnetic field at the centre of circular current
loop is given by Right hand rule.
Right Hand rule. According to this rule, if we hold the thumb
of right hand mutually perpendicular to the grip of the fingers
such that the curvature of the fingers represent the direction
of current in the wire loop, then the thumb of the right hand
will point in the direction of magnetic field near the centre of
the current loop.

According to Biot-Savart’s law, the magnetic field at the

G
centre of the circular coil due to the current element Id A is
given by
G G
G P0 § d A u r ·
dB I¨ ¸
4S © r 3 ¹

P0 IdAr sin T P 0 IdA sin T

or dB
4S r3 4S r 2
where Gr is the position vector of point O from the current
G
element. Since the angle between d A and Gr is 90° (i.e., T = 90°),
therefore, 13. AMPERE’S CIRCUITAL LAW
P0 IdA sin 90q P 0 IdA
dB or dB ...(12) Consider an open surface with a boundary C, and the current
4S r2 4S r 2 I is passing through the surface. Let the boundary C be
G made of large number of small line elements, each of length
In this case, the direction of dB is perpendicular to the G
plane of the current loop and is directed inwards. Since the dA. The direction of d A of small line element under study is
current through all the elements of the circular coil will acting tangentially to its length dA. Let Bt be the tangential
contribute to the magnetic feild in the same direction, component of the magnetic field induction at this element
therefore, the total magnetic field at point O due to current G G
then Bt and d A are acting in the same direction, angle
in the whole circular coil can be obtained by integrating eq.
between them is zero. We take the product of Bt and dA for
(12). Thus
that element. Then
P IdA P0 I G G
B ³ dB ³ 4S0 r 2 4S r 2 ³
dA Bt dA B.d A

But ³ dA = total length of the circular coil = circumference of the

current loop = 2Sr
MAGNETISM

The relation (19) is independent of the size and shape of the

closed path or loop enclosing the current.

14. MAGNETIC FIELD DUE TO INFINITE LONG

STRAIGHT WIRE CARRYING CURRENT
Consider an infinite long straight wire lying in the plane of
paper. Let I be the current flowing through it from X to Y. A
magnetic field is produced which has the same magnitude
at all points that are at the same distance from the wire, i.e.
If length dA is very small and products for all elements of
the magnetic field has cylindrical symmetry around the wire.
closed boundary are added together, then sum tends to be

an integral around the closed path or loop (i.e., ) . v³

G G
Therefore, 6 of B.d A over all elements on a closed path
G G G
v³
B.d A = Line integral of B around the closed path or
loop whose boundary coincides with the closed path.
According to Ampere’s circuital law,
G G
v³B.d A P0 I ...(19)

where I is the total current threading the closed path or loop

and P0 is the absolute permeability of the space. Thus, Let P be a point at a perpendicular distance r from the straight
G
Ampere’s circuital law states that the line integral of magnetic wire and B be the magnetic field at P. It will be acting
G tangentially to the magnetic field line passing through P.
field induction B around a closed path in vacuum is equal to
Consider an amperian loop as a circle of radius r, perpendicular
P0 times the total current I threading the closed path. to the plane of paper with centre on wire such that point P
The relation (19) involves a sign convention, for the sense lies on the loop, figure. The magnitude of magnetic field is
of closed path to be traversed while taking the line integral G
same at all points on this loop. The magnetic field B at P
of magnetic field (i.e., direction of integration) and current
will be tangential to the circumference of the circular loop.
threading it, which is given by Right Hand Rule. According
We shall integrate the amperian path anticlockwise. Then
to it, if curvature of the fingers is perpendicular to the thumb G G
of right hand such that the curvature of the fingers represents B and d A are acting in the same direction. The line integral
the sense, the boundary is traversed in the closed path or G
G G of B around the closed loop is
v³
loop for B.d A , then the direction of thumb gives the sense G G
in which the current I is regarded as positive. v³ B.d A v³ v³
BdA cos 0q B dA B2Sr

According to sign convention, for the closed path as shown As per sign convention, here I is positive,
in figure, I1 is positive and I2 is negative. Then, according to Using Ampere’s circuital law
Ampere’s circuital law G G
G G v³B.d A P 0I or B2 Sr P 0 I
v³ B.d A P 0 I1  I2 P0 Ie
P0 I P 0 2I
where Ie is the total current enclosed by the loop or closed path. or B ...(21)
2Sr 4S r

15. MAGNETIC FIELD DUE TO CURRENT THROUGH

A VERY LONG CIRCULAR CYLINDER
Consider an infinite long cylinder of radius R with axis XY.
Let I be the current passing through the cylinder. A magnetic
field is set up due to current through the cylinder in the form
of circular magnetic lines of force, with their centres lying
 MAGNETISM

on the axis of cylinder. These lines of force are perpendicular

P0P r Ir
to the length of cylinder. or B i.e., B v r
2 SR 2
If we plot a graph between magnetic field induction B and
distance from the axis of cylinder for a current flowing through
a solid cylinder, we get a curve of the type as shown figure

Case I. Point P is lying outside the cylinder. Let r be the

perpendicular distance of point P from the axis of cylinder, Here we note that the magnetic field induction is maximum
G for a point on the surface of solid cylinder carrying current
where r > R. Let B be the magnetic field induction at P. It is
acting tangential to the magnetic line of force at P directed and is zero for a point on the axis of cylinder.
G G
into the paper. Here B and d A are acting in the same direction.
16. FORCE BETWEEN TWO PARALLEL CONDUCTORS
Applying Ampere circuital law we have CARRYING CURRENT
G G
v³B.d A P0 I or v³
BdA cos 0q P 0 I Consider C 1D 1 and C 2 D 2 , two infinite long straight
conductors carrying currents I1 and I2 in the same direction.
or v³ BdA P 0 I or B2Sr P0 I They are held parallel to each other at a distance r apart, in
the plane of paper. The magnetic field is produced due to
current through each conductor shown separately in figure.
P0 I
or B , i.e., B v 1/ r Since each conductor is in the magnetic field produced by
2 Sr the other, therefore, each conductor experiences a force.
Case II. Point P is lying inside cylinder. Here r < R. we may
have two possibilities.
(i) If the current is only along the surface of cylinder which is
so if the conductor is a cylindrical sheet of metal, then current
through the closed path L is zero. Using Ampere circutal
law, we have B = 0.
(ii) If the current is uniformly distributed throughout the cross-
section of the conductor, then the current through closed
path L is given by

I Ir 2
I' u Sr 2
SR 2 R2
Magnetic field induction at a point P on conductor C2D2
Applying Ampere’s circuital law, we have
due to current I1 passing through C1D1 is given by
G G
v³B.d A P0P r I '
B1
P0 2I1
...(12)
4S r
2
P0P r Ir According to right hand rule, the direction of magnetic field
or 2SrB P 0P r I '
R2 G
B1 is perpendicular to the plane of paper, directed inwards.
 MAGNETISM

As the current carrying conductor C2D2 lies in the magnetic Q

G G G G G RG G SG G PG G
field B1 (produced by the current through C1D1), therefore, v³ B.d A ³ ³ ³ ³
B.d A  B.d A  B.d A  B.d A
PQRS P Q R S
the unit length of C2D2 will experience a force given by
F2 = B1I2 × 1 = B1I2 Q Q
G G
Putting the value of B1, we have ³
Here, B.d A ³ BdA cos 0q BL
P P
P0 2I1I2
F2 . ...(13)
4S r R
G G R P
G G
It means the two linear parallel conductors carrying
and ³ B.d A ³ BdA cos90q 0 ³ B.d A
Q Q S
currents in the same direction attract each other.
Thus one ampere is that much current which when flowing S
G G
through each of the two parallel uniform long linear
conductors placed in free space at a distance of one metre
³
Also, B.d A 0 (' outside the solenoid, B = 0)
R
from each other will attract or repel each other with a force
G G
of 2 × 10–7 N per metre of their length.
v³ B.d A BL  0  0  0 BL
...(21)
PQRS
17. THE SOLENOID
From Ampere’s circuital law
A solenoid consists of an insulating long wire closely wound G G
in the form of a helix. Its length is very large as compared to v³ B.d A P0 × total current through the rectangle PQRS
its diameter. PQRS

Magnetic field due to a solenoid = P0 × no. of turns in rectangle × current

Consider a long straight solenoid of circular cross-section. = P0 n LI ...(22)
Each two turns of the solenoid are insulated from each other. From (21) and (22), we have
When current is passed through the solenoid, then each
turn of the solenoid can be regarded as a circular loop BL = P0 n LI or B = P0 n I
carrying current and thus will be producing a magnetic field. This relation gives the magnetic field induction at a point
At a point outside the solenoid, the magnetic fields due to well inside the solenoid. At a point near the end of a solenoid,
neighbouring loops oppose each other and at a point inside the magnetic field induction is found to be P0 n I/2.
the solenoid, the magnetic fields are in the same direction.
As a result of it, the effective magnetic field outside the 18. TOROID
solenoid becomes weak, whereas the magnetic field in the
interior of solenoid becomes strong and uniform, acting The toroid is a hollow circular ring on which a large number of
along the axis of the solenoid. insulated turns of a metallic wire are closely wound. In fact, a
toroid is an endless solenoid in the form of a ring, figure.
Let us now apply Ampere’s circuital law.
Let n be the number of turns per unit length of solenoid and
I be the current flowing through the solenoid and the turns
of the solenoid be closely packed.
Consider a rectangular amperian loop PQRS near the middle
of solenoid as shown in figure

Magnetic field due to current in ideal toroid

G Let n be the number of turns per unit length of toroid and I
The line integral of magnetic field induction B over the be the current flowing through it. In case of ideal toroid, the
closed path PQRS is coil turns are circular and closely wound. A magnetic field
 MAGNETISM

of constant magnitude is set up inside the turns of toroid in

the form of concentric circular magnetic field lines. The 19. MAGNETISM & MATTER
direction of the magnetic field at a point is given by the
tangent to the magnetic field line at that point. We draw 19.1 The Bar Magnet
three circular amperian loops, 1, 2 and 3 of radii r1, r2 and r3 to
be traversed in clockwise direction as shown by dashed It is the most commonly used form of an artificial magnet.
circles in figure, so that the points P, S and Q may lie on
When we hold a sheet of glass over a short bar magnet and
them. The circular area bounded by loops 2 and 3, both cut
sprinkle some iron filings on the sheet, the iron filings
the toroid. Each turn of current carrying wire is cut once by
rearrange themselves as shown in figure. The pattern
the loop 2 and twice by the loop 3. Let B1 be the magnitude
suggests that attraction is maximum at the two ends of the
of magnetic field along loop 1. Line integral of magnetic
bar magnet. These ends are called poles of the magnet.
field B1 along the loop 1 is
G G
v³ B1 .d A v³ B1dA cos 0q B1 2Sr1 ...(i)
loop 1 loop 1

Loop 1 encloses no current.

According to Ampere’s circuital law
G G
v³ B1 .d A P 0 u current enclosed by loop 1 = P0 × 0 = 0
loop 1

or B12 S r1 = 0 or B1 = 0
Let B3 be the magnitude of magnetic field along the loop 3.
The line integral of magnetic field B3 along the loop 3 is
G G
v³ B3 .d A v³ B3dA cos 0q B3 2 Sr3
loop 3 loop 3

From the sectional cut as shown in figure, we note that the

current coming out of the plane of paper is cancelled exactly
by the current going into it. Therefore, the total current
enclosed by loop 3 is zero.
According to Ampere’s circuital law
G G
v³ B3 .d A P0 × total current through loop 3
loop 3

or B3 2Sr3 P 0 u 0 0 or B3 0
Let B the magnitude of magnetic field along the loop 2. Line
integral of magnetic field along the loop 2 is 1. The earth behaves as a magnet.
G G 2. Every magnet attracts small pieces of magnetic substances
v³ B.d A B2Sr2
like iron, cobalt, nickel and steel towards it.
loop 2
3. When a magnet is suspended freely with the help of an
Current enclosed by the loop 2 = number of turns × current unspun thread, it comes to rest along the north south
in each turn = 2 S r2 n × I direction.
According to Ampere’s circuital law 4. Like poles repel each other and unlike poles attract each
G G other.
v³ B.d A P0 u total current
5. The force of attraction or repulsion F between two magnetic
loop 2
poles of strengths m1 and m2 separated by a distance r is
or B2 S r2 P 0 u 2Sr2 nI or B P0 nI directly proportional to the product of pole strengths and
inversely proportional to the square of the distance between
their centres, i.e.,
 MAGNETISM

m1m2 mm the field of the magnet. The torque acting on a compass

Fv or F K 1 2 2 , where K is magnetic force needle aligns it in the direction of the magnetic field.
r2 r
constant. The path along which the compass needles are aligned is
known as magnetic field line.
P0
In SI units, K 107 Wb A 1m 1
4S
where P0 is absolute magnetic permeability of free space
(air/vacuum).

P0 m1m 2
? F ...(1)
4S r 2
This is called Coulomb’s law of magnetic force. However, in
cgs system, the value of K = 1.

This corresponds to Coulomb’s law in electrostatics.

SI Unit of magnetic pole strength

Suppose m1 = m2 = m (say),
r = 1 m and F = 10–7 N
From equation (1),

m m
107 107 u or m 2 1 or m = +1 ampere-metre
12
(Am). Therefore, strength of a magnetic pole is said to be
one ampere-metre, if it repels an equal and similar pole, when
placed in vacuum (or air) at a distance of one metre from it,
with a force of 10–7 N.
6. The magnetic poles always exist in pairs. The poles of a
magnet can never be separated i.e. magnetic monopoles do
not exist.

20. MAGNETIC FIELD LINES

Magnetic field line is an imaginary curve, the tangent to
which at any point gives us the direction of magnetic field
G
B at that point.
If we imagine a number of small compass needless around a
magnet, each compass needle experiences a torque due to
 MAGNETISM

Properteis of magnetic field lines

1. The magnetic field lines of a magnet (or of a solenoid
carrying current) form closed continuous loops.
2. Outside the body of the magnet, the direction of magnetic
field lines is from north pole to south pole.
We shall show that the SI unit of M is joule/tesla or ampere
3. At any given point, tangent to the magnetic field line
G metre2.
represents the direction of net magnetic field ( B ) at that ? SI unit of pole strength is Am.
point.
Bar magnet as an equivalent solenoid
4. The magnitude of magnetic field at any point is represented
We know that a current loop acts as a magnetic dipole.
by the number of magnetic field lines passing normally
According to Ampere’s hypothesis, all magnetic phenomena
through unit area around that point. Therefore, crowded
can be explained in terms of circulating currents.
lines represent a strong magnetic field and lines which are
not so crowded represent a weak magnetic field. In figure magnetic field lines for a bar magnet and a current
carrying solenoid resemble very closely. Therefore, a bar
5. No two magnetic field lines can intersect each other.
magnet can be thought of as a large number of circulating
currents in analogy with a solenoid. Cutting a bar magnet is
like cutting a solenoid. We get two smaller solenoids with
weaker magnetic properties. The magnetic field lines remain
continuous, emerging from one face of one solenoid and
entering into other face of other solenoid. If we were to
move a small compass needle in the neighbourhood of a bar
magnet and a current carrying solenoid, we would find that
21. MAGNETIC DIPOLE the deflections of the needle are similar in both cases.
To demonstrate the similarity of a current carrying solenoid
A magnetic dipole consists of two unlike poles of equal
to a bar magnet, let us calculate axial field of a finite solenoid
strength and separated by a small distance.
carrying current.
For example, a bar magnet, a compass needle etc. are
magnetic dipoles. We shall show that a current loop behaves
as a magnetic dipole. An atom of a magnetic material behaves
as a dipole due to electrons revolving around the nucleus.
The two poles of a magnetic dipole (or a magnet), called
north pole and south pole are always of equal strength, and
of opposite nature. Further such two magnetic poles exist
always in pairs and cannot be separated from each other.
The distance between the two poles of a bar magnet is called
the magnetic length of the magnet. It is a vector directed from In figure, suppose
G
S-pole of magnet to its N-pole, and is represented by 2 A . a = radius of solenoid,
Magnetic dipole moment is the product of strength of either 2A = length of solenoid with centre O
G
pole (m) and the magnetic length ( 2 A ) of the magnet. n = number of turns per unit length of solenoid,
G i = strength of current passed through the solenoid
It is represented by M .
Magnetic dipole moment = strength of either pole × magnetic We have to calculate magnetic field at any point P on the
length axis of solenoid, where OP = r. Consider a small element of
thickness dx of the solenoid, at a distance x from O.
G G
M m 2A Number of turns in the element = n dx.
Using equation, magnitude of magnetic field at P due to this
Magnetic dipole moment is a vector quantity directed from current element is
South to North pole of the magnet, as shown in figure
P0ia 2 n dx
dB 3/ 2
2ª r  x  a2 º ...(10)
2
¬ ¼
 MAGNETISM

If P lies at a very large distance from O, i.e., r >> a and r >> x,

U W  MB cos T2  cos T1 ...(17)
then [(r – x)2 + a2]3/2 | r3
When T1 = 90°, and T2 = T, then
P0ia 2 ndx
dB ...(11) U = W = – MB (cos T – cos 90°)
2r 3
W = – MB cos T ...(18)
As range of variation of x is from x = – A to x = +
A, therefore
the magnitude of total magnetic field at P due to current In vector notation, we may rewrie (18) as
carrying solenoid G G
U  M.B ...(19)
2 x A 2
P 0nia P 0nia x A
B
2r 3 ³ dx
2r 3
x x A
Particular Cases
x A 1. When T = 90°
U = – MB cos T = – MB cos 90° = 0
P0 2n 2A iSa
2
P0 ni a 2
B 2A ...(12) i.e., when the dipole is perpendicular to magnetic field its potential
2 r3 4S r3 energy is zero.
If M is magnetic moment of the solenoid, then Hence to calculate potential energy of diole at any position
M = total no. of turns × current × area of cross section making angle T with B, we use
M = n (2A) × i × (Sa2) U = – MB (cos T2 – cos T1) and take T1 = 90° and T2 = T.
Therefore,
P0 2M U = – MB (cos T – cos 90°) = – MB cos T
? B ...(13)
4S r 3 2. When T= 0°
This is the expression for magnetic field on the axial line of U = – MB cos 0° = – MB
a short bar magnet. which is minimum. This is the position of stable equilibrium,
Thus, the axial field of a finite solenoid carrying current is i.e., when the magnetic dipole is aligned along the magnetic
same as that of a bar magnet. Hence, for all practical purposes, field, it is in stable equilibrium having minimum P.E.
a finite solenoid carrying current is equivalent to a bar magnet. 3. When T= 180°
Potential energy of a magnetic dipole in a magnetic field U = – MB cos 180° = MB, which is maximum. This is the
Potential energy of a magnetic dipole in a magnetic field is position of unstable equilibrium.
the energy possessed by the dipole due to its particular
position in the field.
G
When a magnetic dipole of moment M is held at an angle T
G
with the direction of a uniform magnetic field B , the
magnitude of the torque acting on the dipole is
W MBsin T ...(16)
This torque tends to align the dipole in the direction of the
field. Work has to be done in rotating the dipole against the
action of the torque. This work done is stored in the
magnetic dipole as potential energy of the dipole.
Now, small amount of work done in rotating the dipole
through a small angle dT against the restoring torque is 22. MAGNETISM AND GAUSS’S LAW
dW = WdT= MB sin TdT
According to Gauss’s law for magnetism, the net magnetic
Total work done in rotating the dipole from T= T1 to T = T2 is
flux (IB) through any closed surface is always zero.
T2
T2
W ³ MBsin T dT MB  cos T T1
 MB cos T2  cos T1 23. EARTH’S MAGNETISM
T1
Magnetic elements of earth at a place are the quantities
? Potential energy of the dipole is which describe completely in magnitude as well as direction,
the magnetic field of earth at that place.
 MAGNETISM

Square (23) and (24), and add

23.1 Magnetic declination
H2 + V2 = R2 (cos2 G + sin2 G) = R2
Magnetic declination at a place is the angle between
magnetic meridian and geographic meridian at that place. ? R H2  V 2 ...(25)

Dividing (24) by (23), we get

R sin V V
or tan ...(26)
R cos H H
The value of horizontal component H = R cos G is different
at different places. At the magnetic poles, G = 90°
? H = R cos 90° = zero
At the magnetic equator, G = 0°
? H = R cos 0° = R
Horizontal component (H) can be measured using both, a
vibration magnetometer and a deflection magnetometer.
The value of H at a place on the surface of earth is of the
order of 3.2 × 10–5 tesla.
Retain in Memory
Memory note
1. The earth’s magnetic poles are not at directly opposite positions
on globe. Current magnetic south is farther from geographic Note that the direction of horizontal component H of earth’s
south than magnetic north is from geographic north. magnetic field is from geographic south to geographic north
above the surface of earth. (if we ignore declination).
2. Infact, the magnetic field of earth varies with position and
also with time. For example, in a span of 240 years from 1580 24. MAGNETIC PROPERTIES OF MATTER
to 1820 A.D., the magnetic declination at London has been
found to change by 3.5° – suggesting that magnetic poles To describe the magnetic properties of materials, we define
of earth change their position with time. the following few terms, which should be clearly understood
3. The magnetic declination in India is rather small. At Delhi,
declination is only 0° 41’ East and at Mumbai, the declination 24.1 Magnetic Permeability
is 0° 58’ West. Thus at both these places, the direction of
geographic north is given quite accurately by the compass It is the ability of a material to permit the passage of magnetic
needle (within 1° of the actual direction). lines of force through it i.e. the degree or extent to which magnetic
field can penetrate or permeate a material is called relative
23.2 Magnetic Dip or Magnetic Inclination magnetic permeability of the material. It is represented by Pr.
Relative magnetic permeability of a mterial is defined as the
Magnetic dip or magnetic inclination at a place is defined as ratio of the number of magnetic field lines per unit area (i.e.
the angle which the direction of total strength of earth’s flux density B) in that material to the number of magnetic
magnetic field makes with a horizontal line in magnetic meridian. field lines per unit area that would be present, if the medium
were replaced by vacuum. (i.e. flux density B0).
23.3 Horizontal Component
B
It is the component of total intensity of earth’s magnetic i.e., Pr
field in the horizontal direction in magnetic meridian. It is B0
represented by H.
Relative magnetic permeability of a material may also be
In figure, AK represents the total intensity of earth’s magnetic defined as the ratio of magnetic permeability of the material
field, ‘BAK = G. The resultant intensity R along AK is (P) and magnetic permeability of free space (P0)
resolved into two rectangular components :
Horizontal component along AB is P
? Pr or P P rP0
AL = H = R cos G ...(23) P0
Vertical component along AD is We know that P0 = 4S × 10–7 weber/amp-metre (Wb A–1 m–1)
AM = V = R sin G ...(24) or henry/metre (Hm–1)
 MAGNETISM

? SI units of permeability (P) are But B = PH

Hm–1 = Wb A–1 m–1 = (T m2) A–1 m–1 = T m A–1 P
? PH P0 H 1  F m or 1  Fm
P0
G
24.2 Magnetic Intensity ( H )
or Pr 1  Fm
The degree to which a magnetic field can magnetise a material
This is the relation between relative magnetic permeability
is represented in terms of magnetising force or magnetise
G and magnetic susceptibility of the material.
intensity ( H ).
25. CLASSIFICATION OF MAGNETIC MATERIALS
24.3 Magnetisation or Intensity of Magnetisation ‘I’
There is a large variety of elements and compounds on earth.
It represents the extent to which a specimen is magnetised, Some new elements, alloys and compounds have been
when placed in a magnetising field. Quantitatively, synthesized in the laboratory. Faraday classified these
The magnetisation of a magnetic material is defined as the substances on the basis of their magnetic properties, into
magnetic moment per unit volume of the material. the following three categories :
(i) Diamagnetic substances,
Magnetic moment m
M (ii) Paramagnetic substances, and
volume V
(iii) Ferromagnetic substances
There are SI unit of I, which are the same as SI units of H.
Their main characteristics are discussed below :
Magnetic susceptibility ( F m ) of a magnetic material is
25.1 Diamagnetic Substances
defined as the ratio of the intensity of magnetisation (I)
induced in the material to the magnetising force (H) applied The diamagnetic substances are those in which the
individual atoms/molecules/ions do not possess any net
on it. Magnetic susceptibility is represented by F m .
magnetic moment on their own. When such substances are
placed in an external magnetising field, they get feebly
I magnetised in a direction opposite to the magnetising field.
Thus Fm
H when placed in a non-uniform magnetic field, these
substances have a tendency to move from stronger parts of
Relation between magnetic permeability and magnetic
the field to the weaker parts.
susceptibility
When a specimen of a diamagnetic material is placed in a
When a magnetic material is placed in a magnetising field of
magnetising field, the magnetic field lines prefer not to pass
magnetising intensity H, the material gets magnetised. The through the specimen.
total magnetic induction B in the material is the sum of the
Relative magnetic permeability of diamagnetic substances
magnetic induction B0 in vacuum produced by the magnetic
is always less than unity.
intensity and magnetic induction Bm, due to magnetisation
of the material. Therefore, From the relation Pr 1  Fm , a P r  1, Fm is negative.
B = B0 + Bm Hence susceptibility of diamagnetic substances has a small
But B0 = P0 H and Bm = m0 I, where I is the intensity of negative value.
magnetisation induced in the magnetic material. Therefore, A superconductor repels a magnet and in turn, is repelled
from above by the magnet.
The phenomenon of perfect diamagnetism in
B P 0 H  P0 I P 0 H  I , superconductors is called Meissner effect. Superconducting
magnets have been used for running magnetically leviated
i.e., B P0 H  I superfast trains.

25.2 Paramagnetic substances

I
Now as Fm ? I FmH
H Paramagnetic substacnes are those in which each individual
atom/molecule/ion has a net non zero magnetic moment of
From above, B P0 H  Fm H P0 H 1  F m its own. When such substances are placed in an external
 MAGNETISM

magnetic field, they get feebly magnetised in the direction inversely proportional to the temperature (T) of the material.
of the magnetising field.
1
When placed in a non-uniform magnetic field, they tend to i.e., I v B, and I v
T
move from weaker parts of the field to the stronger parts.
When a specimen of a paramagnetic substance is placed in B
a magnetising field, the magnetic field lines prefer to pass Combining these factors, we get I v
T
through the specimen rather than through air.
As B v H , magnetising intensity
From the SI relation, P r 1  F m , as P r ! 1 , therefore, F m
I 1
must be positive. Hence, susceptibility of paramagnetic ? Iv or v
substances is positive, though small. T T
Susceptibility of paramagnetic substances varies inversely I
But Fm
1
as the temperature of the substance i.e. F m v i.e. they
T
lose their magnetic character with rise in temperature. 1 C
? Fm v or Fm
T T
25.3 Ferromagnetic substances where C is a constant of proportionality and is called Curie
Ferromagnetic substances are those in which each individual constant.
atom/molecule/ion has a non zero magnetic moment, as in a
paramagnetic substance. 26. HYSTERISIS CURVE
When such substances are placed in an external magnetising
The hysterisis curve represents the relation between
field, they get strongly magnetised in the direction of the field. G G
magnetic induction B (or intensity of magnetization I ) of
The ferromagnetic materials show all the properties of
paramagnetic substances, but to a much greater degree. For a ferromagnetic material with magnetiziing force or magnetic
example, G
intensity H . The shape of the hysterisis curve is shown in
(i) They are strongly magnetised in the direction of external figure. It represents the behaviour of the material as it is
field in which they are placed. taken through a cycle of magnetization.
(ii) Relative magnetic permeability of ferromagnetic materials is G
very large ( | 103 to 105) Suppose the material is unmagnetised initially i.e., B 0
G
(iii) The susceptibility of ferromagnetic materials is also very and H 0 . This state is represented by the origin O. Wee
large. ' Fm P r  1 place the material in a solenoid and increase the current
G
That is why they can be magnetised easily and strongly. through the solenoid gradually. The magnetising force H
(iv) With rise in temperature, susceptibility of ferromagnetics G
increases. The magnetic induction B in the material
decreases. At a certain temperature, ferromagnetics change
increases and saturates as depicted in the curve oa. This
over to paramagnetics. This transition temperature is called
curie temperature. For example, curie temperature of iron is behaviour represents alignment and merger of the domains
G
about 1000 K. of ferromagnetic material until no further enhancement in B
is possible. Therefore, there is no use of inreasing solenoid
current and hence magnetic intensity beyond this.

25.4 Curie Law in Magnetism

According to Curie law,
Intensity of magnetisation (I) of a magnetic material is (i)
directly proportional to magnetic induction (B), and (ii)
MAGNETISM

This phenomenon of lagging of I or B behind H when a

specimen of a magnetic material is subjected to a cycle of
magnetisation is called hysteresis.
For example, hysteresis loop for soft iron is narrow and
large, whereas the hysteresis loop for steel is wide and short,
figure

Next, we decrease the solenoid current and hence magnetic The hysterisis loops of soft iron and steel reveal that
G (i) The retentivity of soft iron is greater than the retentivity of
intensity H till it reduces to zero. The curve follows the
G G steel,
path ab showing that when H 0 , B z 0 . Thus, some
(ii) Soft iron is more strongly magnetised than steel,
magnetism is left in the specimen.
G (iii) Coercivity of soft iron is less than coercivity of steel. It
The value of magnetic induction B left in the specimen means soft iron loses its magnetism more rapidly than steel
when the magnetising force is reduced to zero is called does.
Retentivity or Remanence or Residual magnetism of the (iv) As area of I-H loop for soft iron is smaller than the area of
material. I-H loop for steel, therefore, hysterisis loss in case of soft
It shows that the domains are not completely randomised iron is smaller than the hysterisis loss in case of steel.
even when the magnetising force is removed. Next, the (a) Permanent Magnets
current in the solenoid is reversed and increased slowly.
Permanent magnets are the materials which retain at room
Certain domains are flipped until the net magnetic induction
G temperature, their ferromagnetic properties for a long time.
B inside is reduced to zero. This is represented by the The material chosen should have
curve bc. It means to reduce the residual magnetism or (i) high retentivity so that the magnet is strong,
retentivity to zero, we have to apply a magnetising force =
OC in opposite direction. This value of magnetising force is (ii) high coercivity so that the magnetisation is not erased by
called coercivity of the material. stray magnetic fields, temperature changes or mechanical
damage due to rough handling etc.
As the reverse current in solenoid is increased in magnitude,
we once again obtain saturation in the reverse direction at (iii) high permeability so that it can be magnetised easily.
d. The variation is represented by the curve cd. Next, the Steel is preferred for making permanent magnets.
solenoid current is reduced (curve de), reversed and (b) Electromagnets
increased (curve ea). The cycle repeats itself. From figure,
The core of electromagnets are made of ferromagnetic
we find that saturated magnetic induction BS is of the order
materials, which have high permeability and low retentivity.
of 1.5 T and coercivity is of the order of –90 Am–1.
Soft iron is a suitable material for this purpose. When a soft
From the above discussion, it is clear that when a specimen iron rod is placed in a solenoid and current is passed through
of a magnetic material is taken through a cycle of the solenoid, magnetism of the solenoid is increased by a
magnetisation, the intensity of magnetisation (I) and thousand fold. When the solenoid current is switched off,
magnetic induction (B) lag behind the magnetising force the magnetism is removed instantly as retentivity of soft
(H). Thus, even if the magnetising force H is made zero, the iron is very low. Electromagnets are used in electric bells,
values of I and B do not reduce to zero i.e., the specimen loudspeakers and telephone diaphragms. Giant
tends to retain the magnetic properties. electromagnets are used in cranes to lift machinery etc.
MAGNETISM
MAGNETISM

Specific example
27. HALL EFFECT
In the above circular loop tension in part A and B.
The Phenomenon of producing a transverse emf in a current
In balanced condition of small part AB of the loop is shown below
carrying conductor on applying a magnetic field perpendicular
to the direction of the current is called Hall effect.
Hall effect helps us to know the nature and number of charge
carriers in a conductor.
Consider a conductor having electrons as current carriers.
The electrons move with drift velocity vG opposite to the
direction of flow of current

dT dT
2Tsin dF BidA Ÿ 2T sin BiRdT
2 2

dT dT dT
If dT is small so, sin | Ÿ 2T. BiRdT
G G 2 2 2
Force acting on electron Fm  e v u B . This force acts
along x-axis and hence electrons will move towards face (2) BiL
T BiR, if 2SR L so T
and it becomes negatively charged. 2S

28. STANDARD CASES FOR FORCE ON

CURRENT CARRYING CONDUCTORS
Case 1 : When an arbitrary current carrying loop placed in
If no magnetic field is present, the loop will still open into
a magnetic field ( A to the plane of loop), each element of
a circle as in it’s adjacent parts current will be in opposite
loop experiences a magnetic force due to which loop
direction and opposite currents repel each other.
stretches and open into circular loop and tension developed
in it’s each part.

Case 2 : Equilibrium of a current carrying conductor :

When a finite length current carrying wire is kept parallel to
another infinite length current carrying wire, it can suspend
freely in air as shown below
 MAGNETISM

Wire is placed along the axis of coil so magnetic field

produced by the coil is parallel to the wire. Hence it will not
experience any force.
Case 4 : Current carrying spring : If current is passed
through a spring, then it will contract because current will
flow through all the turns in the same direction.

In both the situations for equilibrium of XY it’s downward

P 0 2i1i 2
weight = upward magnetic force i.e. mg . .A
4S h

In the first case if wire XY is slightly displaced from its

equilibrium position, it executes SHM and it’s time period
If current makes to flow through spring, then spring will
h contract and weight lift up.
is given by T 2S .
g

If direction of current in movable wire is reversed then

it’s instantaneous acceleration produced is 2gp.

Case 3 : Current carrying wire and circular loop : If a

current carrying straight wire is placed in the magnetic field
of current carrying circular loop.

If switch is closed then current start flowing, spring will

execute oscillation in vertical plane.
Case 5 : Tension less strings : In the following figure the
value and direction of current through the conductor XY so
that strings becomes tensionless ?
Strings becomes tensionless if weight of conductor XY
balanced by magnetic force (Fm).
Wire is placed in the perpendicular magnetic field due to
coil at it’s centre, so it will experience a maximum force
P0i1
F BiA u i2A
2r
MAGNETISM

In the following situation conducting rod (X, Y) slides at

constant velocity if

mg
Fcos T mgsin T Ÿ BiA cos T mgsin T Ÿ B tan T
iA

TIPS & TRICKS

1. The device whose working principle based on Halmholtz
Hence direction of current is from X o Y and in balanced coils and in which uniform magnetic field is used called as
mg “Halmholtz galvanometer”.
condition Fm = mg Ÿ BiA = mg Ÿ i =
BA 2. The value of magnetic field induction at a point, on the
centre of separation of two linear parallel conductors
Case 6 : A current carrying conductor floating in air such
carrying equal currents in the same direction is zero.
that it is making an angle T with the direction of magnetic
field, while magnetic field and conductor both lies in a 3. If a current carrying circular loop (n = 1) is turned into a
horizontal plane. coil having n identical turns then magnetic field at the
centre of the coil becomes n2 times the previous field i.e.
B(n turn) = n2 B(single turn)
4. When a current carrying coil is suspended freely in earth’s
magnetic field, it’s plane stays in East-West direction.
G
5. Magnetic field B produced by a moving charge q is given
G G G
G P0 q v u r P0 q v u rˆ
mg by B ; where v = velocity of
In equilibrium mg = BiA sinT Ÿ i 4S r3 4S r 2
BA sin T
charge and v < < c (speed of light).
Case 7 : Sliding of conducting rod on inclined rails : When
a conducting rod slides on conducting rails.

6. If an electron is revolving in a circular path of radius r with

speed v then magnetic field produced at the centre of circular

P 0 ev v
path B Ÿrv
4S r 2 B
G
7. The line integral of magnetising field H for any closed
path called magnetomotive force (MMF). It’s S.I. unit is amp.
8. Ratio of dimension of e.m.f. to MMF is equal to the dimension
of resistance.
9. The positive ions are produced in the gap between the two
dees by the ionisation of the gas. To produce proton,
hydrogen gas is used; while for producing alpha-particles,
helium gas is used.
 MAGNETISM

10. Cyclotron frequency is also known as magnetic resonance

frequency.
11. Cyclotron can not accelerate electrons because they have
very small mass.
12. The energy of a charged particle moving in a uniform magnetic
field does not change because it experiences a force in a
direction, perpendicular to it’s direction of motion. Due to
which the speed of charged particle remains unchanged and
hence it’s K.E. remains same.
17. If no magnetic field is present, the loop will still open into a
13. Magnetic force does no work when the charged particle is circle as in it’s adjacent parts current will be in opposite
displaced while electric force does work in displacing the
direction and opposite currents repel each other.
charged particle.
14. Magnetic force is velocity dependent, while electric force
is independent of the state of rest or motion of the charged
particle.
15. If a particle enters a magnetic field normally to the
magnetic field, then it starts moving in a circular orbit.
The point at which it enters the magnetic field lies on the
circumference. (Most of us confuse it with the centre of the
orbit)
16. Deviation of charged particle in magnetic field : If a
G 18. In the following case if wire XY is slightly displaced from its
charged particle (q, m) enters a uniform magnetic field B equilibrium position, it executes SHM and it’s time period is
(extends upto a length x) at right angles with speed v as
shown in figure. The speed of the particle in magnetic h
given by T 2S .
field does not change. But it gets deviated in the magnetic g
field.

§ Bq ·
Deviation in terms of time t ; T Zt ¨ ¸t
© m¹

Deviation in terms of length of the magnetic field ;

§x·
T sin 1 ¨ ¸ . This relation can be used only when x d r .
©r¹

For x > r, the deviation will be 180° as shown in the following figure 19. In the previous case if direction of currnet in movable wire
is reversed then it’s instantaneous acceleration produced is
2gp.
20. Electric force is an absolute concept while magnetic force is
a relative concept for an observer.
21. The nature of force between two parallel charge beams
decided by electric force, as it is dominator. The nature of
force between two parallel current carrying wires decided
by magnetic force.
MAGNETISM

24. If a current carrying conductor AB is placed transverse to a

22. If a straight current carrying wire is placed along the axis of long current carrying conductor as shown then force.
a current carrying coil then it will not experience magnetic
Experienced by wire AB
force because magnetic field produced by the coil is parallel
to the wire. P0i1i 2 §xA·
F loge ¨ ¸
23. The force acting on a curved wire joining points a and b as 2S © x ¹
shown in the figure is the same as that on a straight wire
G G G
joining these points. It is given by the expression F iL u B
 EMI & AC

EMI & AC
THEORY

1. MAGNETIC FLUX
Various Methods of Producing induced E.M.F.
(1) The total number of magnetic lines of force passing
normally through an area placed in a magnetic field is equal We have learnt that e.m.f. is induced in a circuit, whenever
to the magnetic flux linked with that area. the amount of magnetic flux linked with the circuit is
changed. As I = BA cos T, the magnetic flux I can be
changed by changing B, A or T. Hence there are three
methods of producing induced e.m.f.
1. By changing the magnitude of magnetic field B,
2. By changing the area A, i.e., by shrinking or stretching or
changing the shape of the coil.
3. By changing angle T between the direction of B and normal
to the surface area A, i.e., changing the relative orientation
of the surface area and the magnetic field.
(2) Net flux through the surface I = ³ B. dA = BA cos T
3. LENZ’S LAW
(T is the angle between area vector and magnetic field
vector) If T = 0o then I= BA, If T = 90o then I = 0 This law gives the direction of induced emf/induced current.
According to this law, the direction of induced emf or current in a
(3) Unit and Dimension : Magnetic flux is a scalar quantity. It’s
circuit is such as to oppose the cause that produces it. This law is
S.I. unit is weber (wb), CGS unit is Maxwell or Gauss × cm2;
based upon law of conservation of energy.
(1wb = 108 Maxwell).
(1) When N-pole of a bar magnet moves towards the coil, the
Num Joule Volt u Coulomb
(4) Other units : Tesla × m 2 flux associated with loop increases and an emf is induced
Amp Amp Amp
in it. Since the circuit of loop is closed, induced current
= Volt × sec = Ohm × Coulomb = Henry × Amp. It’s also flows in it.
dimensional formula [I] = [ML2T–2A–1]
(2) Cause of this induced current, is approach of north pole
2. FARADAY’S LAWS OF EMI and therefore to oppose the cause, i.e., to repel the
approaching north pole, the induced current in loop is in
(1) First law : Whenever the number of magnetic lines of such a direction so that the front face of loop behaves as
force (magnetic flux) passing through a circuit changes an north pole. Therefore induced current as seen by observer
emf is produced in the circuit called induced emf. The O is in anticlockwise direction. (figure)
induced emf persists only as long as there is change or
cutting of flux.
(2) Second law : The induced emf is given by rate of change
dI
of magnetic flux linked with the circuit i.e. e  . . For
dt
NdI
N turns e  ; Negative sign indicates that induced
dt
emf (e) opposes the change of flux. (3) If the loop is free to move the cause of induced emf in the
coil can also be termed as relative motion. Therefore to
Induced current (i) Induced charge (q) Induced power (P) oppose the cause, the relative motion between the
e N dI e2 N 2 § dI ·
2 approaching magnet and the loop should be opposed.
N
i  . dq idt  .dI P ¨ ¸ For this, the loop will itself start moving in the direction of
R R dt R R R © dt ¹
motion of the magnet.
Induced charge It depends on (4) It is important to remember that whenever cause of induced
is time indepen- time and resistance emf is relative motion, the new motion is always in the
dent. direction of motion of the cause.
 EMI & AC

Table : The various positions of relative motion between the magnet and the coil

Position of magnet

Direction of Anticlockwise direction Clockwise direction Clockwise direction Anticlockwise direction

induced current

Behaviour of face As a north pole As a south pole As a south pole As a north pole
of the coil

Type of magnetic Repulsive force Attractive force Repulsive force Attractive force
force opposed

Magnetic field linked Cross (×), Increases Cross (×), Decreases Dots (˜) Increases Dots (˜) Decreases
with the coil and it’s
progress as viewed
from left

4. EDDY CURRENT
(i) Dead-beat galvanometer : A dead beat galvanometer
When a changing magnetic flux is applied to a bulk piece of means one whose pointer comes to rest in the final
conducting material then circulating currents called eddy currents equilibrium position immediately without any oscillation
are induced in the material. Because the resistance of the bulk about the equilibrium position when a current is passed
conductor is usually low, eddy currents often have large in its coil.
magnitudes and heat up the conductor.
This is achieved by winding the coil on a metallic
(1) These are circulating currents like eddies in water.
frame the large eddy currents induced in the frame provide
(2) Experimental concept given by Focault hence also named electromagnetic damping.
as “Focault current”.
(ii) Electric-brakes : When the train is running its wheel is
(3) The production of eddy currents in a metallic block leads
moving in air and when the train is to be stopped by
to the loss of electric energy in the form of heat.
electric breaks the wheel is made to move in a field created
(4) By Lamination, slotting processes the resistance path for
by electromagnet. Eddy currents induced in the wheels
circulation of eddy current increases, resulting in to
due to the changing flux oppose the cause and stop
weakening them and also reducing losses causes by them
the train.
(iii) Induction furnace : Joule’s heat causes the melting of a
metal piece placed in a rapidly changing magnetic field.
(iv) Speedometer : In the speedometer of an automobile, a
magnet is geared to the main shaft of the vehicle and it
rotates according to the speed of the vehicle. The magnet
is mounted in an aluminium cylinder with the help of
hair springs. When the magnet rotates, it produces eddy
currents in the drum and drags it through an angle, which
indicates the speed of the vehicle on a calibrated scale.
(v) Energy meter : In energy meters, the armature coil carries
a metallic aluminium disc which rotates between the poles
of a pair of permanent horse shoe magnets. As the
armature rotates, the current induced in the disc tends
(5) Application of eddy currents : Though most of the times to oppose the motion of the armature coil. Due to this
eddy currents are undesirable but they find some useful braking effect, deflection is proportional to the energy
applications as enumerated below consumed.
 EMI & AC

5. INDUCED CHARGE FLOW

When a current is induced in the circuit due to the flux change,
charge flows through the circuit and the net amount of charge
which flows along the circuit is given as :

1 dI 1
q ³ i dt ³ R dt
dt
R
dI ³ when r < a; E =
r dB
2 dt
; En v r

'I 'I 7. DYNAMIC (MOTIONAL) EMI DUE

Ÿ q and q N for N turns.
R R TO TRANSLATORY MOTION

6. INDUCED ELECTRIC FIELD (1) Consider a conducting rod of length l moving with a

It is non-conservative and non-electrostatic in nature. Its field uniform velocity v perpendicular to a uniform magnetic
lines are concentric circular closed curves.
field B , directed into the plane of the paper. Let the rod be
dB moving to the right as shown in figure. The conducting
A time varying magnetic field always produced induced electrons also move to the right as they are trapped within
dt
the rod.
electric field in all space surrounding it.
Induced electric field (E in) is directly proportional to

induced emf so e = ³E in .d A ..…(i)

dI
From Faraday’s second laws e  ..…(ii)
dt

dI Conducting electrons experiences a magnetic force

From (i) and (ii) e ³ E in .dA 
dt
This is known as Fm = evB. So they move from P to Q within the rod. The
end P of the rod becomes positively charged while end Q
integral form of Faraday’s laws of EMI.
becomes negatively charged, hence an electric field is set
up within the rod which opposes the further downward
movement of electrons i.e. an equilibrium is reached and
in equilibrium Fe = Fm i.e. eE = evB or E = vB

ª Vº
Ÿ Induced emf e = El = Bvl «E
¬ A »¼

(2) If rod is moving by making an angle T with the direction of

magnetic field or length. Induced emf e = Bvl sinT

A uniform but time varying magnetic field B(t) exists in a

circular region of radius ‘a’ and is directed into the plane
of the paper as shown, the magnitude of the induced
electric field (Ein) at point P lies at a distance r from the
centre of the circular region is calculated as follows.

dI dB dB
So ³E in .dA e
dt
A
dt
i.e. E 2 Sr Sa 2
dt

a 2 dB 1
where r t a or E ; E in v (3) Motion of conducting rod on an inclined plane : When
2r dt r conductor start sliding from the top of an inclined plane
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 EMI & AC

as shown, it moves perpendicular to it’s length but at an (2) Magnetic force : Conductor PQ experiences a magnetic
angle (90  T ) with the direction of magnetic field. force in opposite direction of it’s motion and

§ BvA · B 2 vA 2
Fm BiA B¨ ¸A
© R ¹ R

(3) Power dissipated in moving the conductor : For uniform

motion of rod PQ, the rate of doing mechanical work by
external agent or mech. Power delivered by external source
is given as

dW B 2 vA 2 B2 v 2 A 2
Pmech Pext Fext .v uv
Hence induced emf across the ends of conductor dt R R
e = Bv sin(90 – T)l = Bvl cosT (4) Electrical power : Also electrical power dissipated in
BvA cos T resistance or rate of heat dissipation across resistance is
So induced current i (Directed from Q to P). given as
R
2
The forces acting on the bar are shown in following figure. H § BvA · B2v 2A 2
Pthermal i 2R ¨ ¸ .R ; Pthermal
The rod will move down with constant velocity only if t © R ¹ R
Fm cos T = mg cos (90 – T) = mg sin T (It is clear that Pmech. = Pthermal which is consistent with the
Ÿ Bil cos T = mg sin T principle of conservation of energy.)
(5) Motion of conductor rod in a vertical plane : If conducting
§ Bv A cos T · mgR sin T
B¨ T ¸A cos T mg sin T Ÿ vT rod released from rest (at t = 0) as shown in figure then
© R ¹ B 2 A 2 cos 2 T with rise in it’s speed (v), induces emf (e), induced current
(i), magnetic force (Fm) increases but it’s weight remains
8. MOTIONAL EMI IN LOOP BY GENERATED AREA constant.
Rod will achieve a constant maximum (terminal) velocity
If conducting rod moves on two parallel conducting rails
vT if Fm = mg
as shown in following figure then phenomenon of induced
emf can also be understand by the concept of generated B 2 v T2 A 2 mgR
area (The area swept of conductor in magnetic field, during So mg Ÿ vT
R B2 A 2
it’s motion)

As shown in figure in time t distance travelled by conductor = vt

SPECIAL CASES
Area generated A = lvt. Flux linked with this area I = BA =
Motion of train and aeroplane in earth’s magnetic field
dI
Blvt. Hence induced emf e BvA
dt

(1) Induced current :

e BvA
i
R R
 EMI & AC

Induced emf across the axle of the wheels of the train and it is
across the tips of the wing of the aeroplane is given by e = Bvlv
where l = length of the axle or distance between the tips of the
wings of plane, B v = vertical component of earth’s magnetic field
and v = speed of train or plane.

9. MOTIONAL EMI DUE TO ROTATIONAL MOTION

(1) Conducting rod : A conducting rod of length l whose one
end is fixed, is rotated about the axis passing through it’s (4) Semicircular conducting loop : If a semi-circular
fixed end and perpendicular to it’s length with constant conducting loop (ACD) of radius ‘r’ with centre at O, the
angular velocity Z. Magnetic field (B) is perpendicular to plane of loop being in the plane of paper. The loop is now
the plane of the paper. made to rotate with a constant angular velocity Z, about
an axis passing through O and perpendicular to the plane
emf induces across the ends of the rod
of paper. The effective resistance of the loop is R.
where Q = frequency (revolution per sec) and T = Time
period.

In time t the area swept by the loop in the field i.e. region II

(2) Cycle wheel : A conducting wheel each spoke of length l 1 1 2 dA r 2Z

A r rT r Zt ;
is rotating with angular velocity Z in a given magnetic 2 2 dt 2

field as shown below in fig. Flux link with the rotating loop at time t I = BA. Hence induced

Due to flux cutting each metal spoke becomes identical dI dA BZr 2

emf in the loop in magnitude e B and
cell of emf e (say), all such identical cells connected in dt dt 2
parallel fashion enet = e (emf of single cell). Let N be the e BZr 2
induced current i
R 2R
1
number of spokes hence e net BZA 2 ; Z 2 Sv
2 10. PERIODIC EMI
Suppose a rectangular coil having N turns placed initially in a
magnetic field such that magnetic field is perpendicular to it’s
plane as shown.
Z–Angular speed
v–Frequency of rotation of coil
R–Resistance of coil
Here e net v N 0 i.e. total emf does not depends on number
of spokes ‘N’.
(3) Faraday copper disc generator : A metal disc can be
assumed to made of uncountable radial conductors when
metal disc rotates in transverse magnetic field these
radial conductors cuts away magnetic field lines and
because of this flux cutting all becomes identical cells each

1
of emf ‘e’ where e BZr 2
2
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 EMI & AC

For uniform rotational motion with Z, the flux linked with coil at (1) Coefficient of self-induction : Number of flux linkages with
any time t the coil is proportional to the current i. i.e. NI v i or

I = NBA cos T = NBA cos Zt NI Li (N is the number of turns in coil and NI – total

I = I0 cosZt where I0 = NBA = maximum flux flux linkage). Hence L

NI
= coefficient of self-induction.
i
(1) Induced emf in coil : Induced emf also changes in periodic
manner that’s why this phenomenon called periodic EMI (2) If i = 1amp, N = 1 then, L = I i.e. the coefficient of self
dI induction of a coil is equal to the flux linked with the coil
e  = NBA Z sinZt Ÿ e = e0 sinZt where e0 = emf when the current in it is 1 amp.
dt
amplitude or max. emf = NBA Z = I0Z dI
(3) By Faraday’s second law induced emf e N . Which
dt
e e0
(2) Induced current : At any time t, i sinZt = i0
R R di di
gives e L ; If = amp/sec then | e |= L.
sinZt where i 0 = current amplitude or max. current dt dt

e0 NBAZ I0 Z Hence coefficient of self induction is equal to the emf

i0
R R R induced in the coil when the rate of change of current in
the coil is unity.
11. INDUCTANCE
(4) Units and dimensional formula of ‘L’ : It’s S.I. unit
(1) Inductance is that property of electrical circuits which
opposes any change in the current in the circuit. weber Tesla u m 2 Num Joule Coulomb u volt
2 2
Amp Amp Amp Amp Amp 2
(2) Inductance is inherent property of electrical circuits. It will
always be found in an electrical circuit whether we want it
or not. volt u sec
ohm u sec . But practical unit is henry (H).
amp
(3) A straight wire carrying current with no iron part in the
circuit will have lesser value of inductance. It’s dimensional formula [L] = [ML2T–2A–2]
(4) Inductance is analogous to inertia in mechanics, because (5) Dependence of self inductance (L) : ‘L’ does not depend
inductance of an electrical circuit opposes any change of upon current flowing or change in current flowing but it
current in the circuit. depends upon number of turns (N), Area of cross section
(A) and permeability of medium (P).
11.1 Self Induction
‘L’ does not play any role till there is a constant current
Whenever the electric current passing through a coil or circuit
flowing in the circuit. ‘L’ comes in to the picture only when
changes, the magnetic flux linked with it will also change. As a
there is a change in current.
result of this, in accordance with Faraday’s laws of electromagnetic
induction, an emf is induced in the coil or the circuit which opposes (6) Magnetic potential energy of inductor : In building a
the change that causes it. This phenomenon is called ‘self steady current in the circuit, the source emf has to do
induction’ and the emf induced is called back emf, current so work against of self inductance of coil and whatever
produced in the coil is called induced current. energy consumed for this work stored in magnetic field
of coil this energy called as magnetic potential energy (U)
of coil

i 1 2
U ³0
Lidi
2
Li ;

1 NIi
Also U Li i
2 2

(7) The various formulae for L

 EMI & AC

Condition Figure 11.2 Mutual Induction

Whenever the current passing through a coil or circuit changes,

the magnetic flux linked with a neighbouring coil or circuit will
Circular coil also change. Hence an emf will be induced in the neighbouring
coil or circuit. This phenomenon is called ‘mutual induction’.
P 0 SN 2 r
L
2

Solenoid

P0 N 2r
L P 0 n 2 AA
A

(1) Coefficient of mutual induction : Total flux linked with the

secondary due to current in the primary is N2I2 and N2I2
P 0N 2r
Toroid L
2 v i1 Ÿ N2I2= Mi1 where N1 - Number of turns in primary;
N2 - Number of turns in secondary; I2 - Flux linked with
each turn of secondary; i1 - Current flowing through
primary; M-Coefficient of mutual induction or mutual
inductance.
Square coil (2) According to Faraday’s second law emf induces in

2 2P 0 N 2 a dI 2 di1
L secondary e 2 N 2 ; e2 M
S dt dt

Triangular coil di1 1Amp

(3) If then |e2| = M. Hence coefficient of mutual
dt sec
P 0 18 Ni
B . induction is equal to the emf induced in the secondary
4S A
coil when rate of change of current in primary coil is unity.

§ P 18 Ni · §¨ 3 2 ·¸ (4) Units and dimensional formula of M : Similar to self-

N¨ 0 . ¸u A
© 4S A ¹ ¨© 4 ¸
¹
inductance (L)
L
i (5) Dependence of mutual inductance
(i) Number of turns (N1, N2) of both coils
9 3 P0 N 2A
L Ÿ L v N2 (ii) Coefficient of self inductances (L1, L2) of both the coils
8S
(iii) Area of cross-section of coils

Coaxial cylinders (iv) Magnetic permeability of medium between the coils (Pr)
P0 r or nature of material on which two coils are wound
L log e 2
2 Sr r1
(v) Distance between two coils (As d increases so M
2.303 r decreases)
P 0 log10 2
2 Sr r1
(vi) Orientation between primary and secondary coil (for 90o
orientation no flux relation M = 0)
(vii) Coupling factor ‘K’ between primary and secondary
coil
 EMI & AC

(6) Relation between M, L1 and L2 : For two magnetically Condition Figure

coupled coils M K L1L 2 ; where k – coefficient of
Two concentric coplaner
coupling or coupling factor which is defined as
circular coils

K
Magnetic flux linked in sec ondary
; SP 0 N1N 2 r 2
M
Magnetic flux linked in primary 2R
0dKd1

Two Solenoids

P 0 N1 N 2 A
M
A

Two concentric
(7) The various formulae for M : coplaner square coils

P 0 2 2 N1 N 2 A 2
M
SL

12. COMBINATION OF INDUCTANCE

(1) Series : If two coils of self-inductances L1 and L2 having mutual inductance are in series and are far from each other, so that the
mutual induction between them is negligible, then net self inductance LS = L1 + L2

When they are situated close to each other, then net inductance LS = L1 + L2 ± 2M

Mutual induction is absent (k = 0) Mutual induction is present and Mutual induction is present and
favours self inductance of coils opposes self inductance of coils

Leq = L1 + L2

Current in same direction Current in opposite direction

Winding nature same Opposite winding nature
Their flux assist each other Their flux opposes each other
Leq = L1 + L2 + 2M Leq = L1 + L2 – 2M

(2) Parallel : If two coils of self-inductances L1 and L2 having When they are situated close to each other, then
mutual inductance are connected in parallel and are far
1 1 1 L1L 2  M 2
from each other, then net inductance L is  LP
LP L1 L 2 L1  L 2 r 2M

L1L 2
Ÿ LP
L1  L 2
 EMI & AC

Mutual induction is absent (k = 0) Mutual induction is present and Mutual induction is present and
favours self inductance of coils opposes self inductance of coil

L 1L 2 L1L 2  M 2 L1L 2  M 2
L eq L eq L eq
L1  L 2 L1  L 2  2M L1  L 2  2M

13. GROWTH AND DECAY OF CURRENT IN LR­ CIRCUIT

If a circuit containing a pure inductor L and a resistor R in series

with a battery and a key then on closing the circuit current
through the circuit rises exponentially and reaches up to a certain
maximum value (steady state). If circuit is opened from it’s steady
state condition then current through the circuit decreases
exponentially.
(4) Behaviour of inductor : The current in the circuit grows
exponentially with time from 0 to the maximum value

§ E·
i¨ ¸ . Just after closing the switch as i = 0, inductor act
© R¹

as open circuit i.e. broken wires and long after the switch
has been closed as i = i0, the inductor act as a short circuit
i.e. a simple connecting wire.

(1) The value of current at any instant of time t after closing

the circuit (i.e. during the rising of current) is given by

ª  tº
R
E
i i 0 «1  e L » ; where i 0 i max = steady state
«¬ »¼ R

current.
(2) The value of current at any instant of time t after opening
from the steady state condition (i.e. during the decaying
R
 t
of current) is given by i i 0e L

L
(3) Time constant (W) : It is given as W ; It’s unit is second.
R
In other words the time interval, during which the current
in an inductive circuit rises to 63% of its maximum value at 14. LC­ OSCILLATION
make, is defined as time constant or it is the time interval,
during which the current after opening an inductive circuit When a charged capacitor C having an initial charge q0 is
falls to 37% of its maximum value. discharged through an inductance L, the charge and current in the
Lakshya Educare
 EMI & AC

circuit start oscillating simple harmonically. If the resistance of

the circuit is zero, no energy is dissipated as heat. We also assume
an idealized situation in which energy is not radiated away from
the circuit. The total energy associated with the circuit is constant.

1 rad
Frequency of oscillation is given by Z
LC sec

1 The oscillation of the LC circuit are an electromagnetic analog to

or Hz
2S LC the mechanical oscillation of a block-spring system.
 EMI & AC

15. DC MOTOR Ee E  kZ ; When motor

(5) Current in the motor : i
R R
It is an electrical machine which converts electrical energy into
mechanical energy. E
is just switched on i.e. Z = 0 so e = 0 hence i =
(1) Principle : It is based on the fact that a current carrying R
coil placed in the magnetic field experiences a torque. This
maximum and at full speed, Z is maximum so back emf e is
torque rotates the coil.
maximum and i is minimum. Thus, maximum current is drawn
when the motor is just switched on which decreases when
motor attains the speed.
(6) Motor starter : At the time of start a large current flows
through the motor which may burn out it. Hence a starter
is used for starting a dc motor safely. Its function is to
introduce a suitable resistance in the circuit at the time of
(2) Construction : It consists of the following components starting of the motor. This resistance decreases gradually
figure. and reduces to zero when the motor runs at full speed.

The value of starting resistance is maximum at time t = 0

and its value is controlled by spring and electromagnetic
system and is made to zero when the motor attains its safe
ABCD = Armature coil, S1, S2 = split ring comutators speed.
B1, B2 = Carbon brushes, N, S = Strong magnetic poles (7) Mechanical power and Efficiency of dc motor :

(3) Working : Force on any arm of the coil is given by Pmechanical Pout e Back e.m.f .
Efficiency K
Psup plied Pin E Supply voltage
F i A u B in fig., force on AB will be perpendicular to
plane of the paper and pointing inwards. Force on CD will (8) Uses of dc motors : They are used in electric locomotives,
be equal and opposite. So coil rotates in clockwise sense electric ears, rolling mills, electric cranes, electric lifts, dc
when viewed from top in fig. The current in AB reverses drills, fans and blowers, centrifugal pumps and air
due to commutation keeping the force on AB and CD in compressors, etc.
such a direction that the coil continues to rotate in the
16. DC GENERATOR
same direction.
If the current produced by the generator is direct current, then the
(4) Back emf in motor : Due to the rotation of armature coil in
generator is called dc generator.
magnetic field a back emf is induced in the circuit. Which
is given by e = E – iR. dc generator consists of (i) Armature (coil) (ii) Magnet (iii)
Commutator (iv) Brushes
Back emf directly depends upon the angular velocity Z of
In dc generator commutator is used in place of slip rings. The
armature and magnetic field B. But for constant magnetic commutator rotates along with the coil so that in every cycle
field B, value of back emf e is given by e v Z or e = kZ when direction of ‘e’ reverses, the commutator also reverses or
(e = NBAZ sinZt) makes contact with the other brush so that in the external load the
current remains in the some direction giving dc
 EMI & AC

7. If an aeroplane is landing down or taking off and its wings

are in the east-west direction, then the potential difference
or emf will be induced across the wings. If an aeroplane is
landing down or taking off and its wings are in the north-
south direction, then no potential difference or emf will be
induced.

8. When a conducting rod moving horizontally on equator

of earth no emf induces because there is no vertical
component of earth’s magnetic field. But at poles BV is
maximum so maximum flux cutting hence emf induces.

9. When a conducting rod falling freely in earth’s magnetic

field such that it’s length lies along East - West direction
TIPS AND TRICKS then induced emf continuously increases w.r.t. time and
induced current flows from West - East.
1. If a bar magnet moves towards a fixed conducting coil,
then due to the flux changes an emf, current and charge 10. 1 henry = 109 emu of inductance or 109 ab-henry.
induces in the coil. If speed of magnet increases then
induced emf and induced current increases but induced 11. Inductance at the ends of a solenoid is half of it’s the
charge remains same
§ 1 ·
inductance at the centre. ¨ L end L centre ¸ .
© 2 ¹

12. A thin long wire made up of material of high resistivity

behaves predominantly as a resistance. But it has some
amount of inductance as well as capacitance in it. It is
thus difficult to obtain pure resistor. Similarly it is difficult
to obtain pure capacitor as well as pure inductor.
Induced parameter : e1, i1, q1
e2 ( > e1), i2( > i1), q2 (= q1 13. Due to inherent presence of self inductance in all electrical
circuits, a resistive circuit with no capacitive or inductive
2. Can ever electric lines of force be closed curve ? Yes,
element in it, also has some inductance associated with
when produced by a changing magnetic field.
it.
3. No flux cutting No EMI
G JG G
The effect of self-inductance can be eliminated as in the
4. Vector form of motional emf : e vuB A coils of a resistance box by doubling back the coil on itself.

5. In motional emf B, v and A are three vectors. If any two

vector are parallel – No flux cutting.

14. It is not possible to have mutual inductance without self

inductance but it may or may not be possible self
inductance without mutual inductance.

di
6. A piece of metal and a piece of non-metal are dropped from 15. If main current through a coil increases (in) so will be
dt
the same height near the surface of the earth. The non-
metallic piece will reach the ground first because there will positive (+ve), hence induced emf e will be negative (i.e.
be no induced current in it. opposite emf) Ÿ Enet = E – e
 EMI & AC

20. In RL-circuit with dc source the time taken by the current

to reach half of the maximum value is called half life time

L
and it is given by T = 0.693 .
R

21. dc motor is a highly versatile energy conversion device. It

16. Sometimes at sudden opening of key, because of high can meet the demand of loads requiring high starting
inductance of circuit a high momentarily induced emf torque, high accelerating and decelerating torque.
produced and a sparking occurs at key position. To 22. When a source of emf is connected across the two ends of
avoid sparking a capacitor is connected across the the primary winding alone or across the two ends of
key. secondary winding alone, ohm’s law can be applied. But
17. Sometimes at sudden opening of key, because of high in the transformer as a whole, ohm’s law should not be
inductance of circuit a high momentarily induced emf applied because primary winding and secondary winding
produced and a sparking occurs at key position. To are not connected electrically.
avoid sparking a capacitor is connected across the 23. Even when secondary circuit of the transformer is open it
key.
also draws some current called no load primary current for
18. One can have resistance with or without inductance but supplying no load Cu and iron loses.
one can’t have inductance without having resistance.
24. Transformer has highest possible efficiency out of all the
19. The circuit behaviour of an inductor is quite different from electrical machines.
that of a resistor. while a resistor opposes the current i, an
di
inductor opposes the change in the circuit.
dt
 EMI & AC

ALTERNATING CURRENT From (3) and (4), we get I m u

T
2
I0 2 I0 .T
...(5)
2 Z 2S

1. THE ALTERNATING CURRENT 2

or Im I0 0.637 I0
S
The magnitude of alternating current changes
Hence, mean or average value of alternating current over
continuously with time and its direction is reversed
positive half cycle is 0.637 times the peak value of
periodically. It is represented by
alternating current, i.e., 63.7% of the peak value.
I I 0 sin Z t or I I 0 cos Z t
3. A.C. CIRCUIT CONTAINING RESISTANCE ONLY
2S
Z 2Sv Let a source of alternating e.m.f. be connected to a pure
T resistance R, Figure. Suppose the alternating e.m.f.
supplied is represented by
2. AVERAGE VALUE OF ALTERNATING CURRENT E = E0 sin Zt ...(1)
Let I be the current in the circuit at any instant t. The
The mean or average value of alternating current over any
potential difference developed across R will be IR. This
half cycle is defined as that value of steady current which
must be equal to e.m.f. applied at that instant, i.e.,
would send the same amount of charge through a circuit in
the time of half cycle (i.e. T/2) as is sent by the alternating IR = E = E0 sin Zt
current through the same circuit, in the same time.
To calculate the mean or average value, let an alternating
current be represented by
I = I0 sin Z t ...(1)
If the strength of current is assumed to remain constant
for a small time, dt, then small amount of charge sent in a
small time dt is
dq = I dt ...(2)
Let q be the total charge sent by alternating current in the
first half cycle (i.e. 0 o T/2).
T/2

? q ³ I dt
0

T/2
ª cos Zt º
T/2

Using (1), we get, q ³I

0
0 sin Zt.dt I0 « 
¬ Z »¼ 0
E0
I ª or I sin Zt I 0 sin Zt ...(2)
T º R
 0 « cos Z  cos 0q »
Z¬ 2 ¼
where I0 = E0/R, maximum value of current.
10 This is the form of alternating current developed.
 cos S  cos 0q 'Z T 2S
Z Comparing I0 = E0/R with Ohm’s law equation, viz. current
= voltage/resistance, we find that resistance to a.c. is
I0 2I 0 represented by R–which is the value of resistance to d.c.
q  1  1 ...(3)
Z Z
Hence behaviour of R in d.c. and a.c. circuit is the same, R
If Im represents the mean or average value of alternating can reduce a.c. as well as d.c. equally effectively.
current over the 1st half cycle, then
Comparing (2) and (1), we find that E and I are in phase.
T Therefore, in an a.c. circuit containing R only, the voltage
q Im u ...(4) and current are in the same phase, as shown in figure.
2
EMI & AC

3.1 Phasor Diagram 5. A.C. CIRCUIT CONTAINING CAPACITANCE ONLY

In the a.c. circuit containing R only, current and voltage Let a source of alternating e.m.f. be connected to a capacitor
are in the same phase. Therefore, in figure, both phasors only of capacitance C, figure. Suppose the alternating e.m.f.
G G supplied is
I0 and E 0 are in the same direction making an angle (Zt)
E = E0 sin Zt ...(1)
with OX. This is so for all times. It means that the phase The current flowing in the circuit transfers charge to the
angle between alternating voltage and alternating current plates of the capacitor. This produces a potential difference
through R is zero. between the plates. The capacitor is alternately charged
and discharged as the current reverses each half cycle. At
I = I0 sin Zt and E = E0 sin Zt.
any instant t, suppose q is the charge on the capacitor.
Therefore, potential difference across the plates of
4. A.C. CIRCUIT CONTAINING INDUCTANCE ONLY capacitor V = q/C.
At every instant, the potential difference V must be equal
In an a.c. circuit containing L only alternating current I to the e.m.f. applied i.e.
lags behind alternating voltage E by a phase angle of 90°,
i.e., by one fourth of a period. Conversely, voltage across q
V E E 0 sin Zt
L leads the current by a phase angle of 90°. This is shown C
in figure. or q = CH0 sin Zt
If I is instantaneous value of current in the circuit at instant
t, then

dq d
I (CH0 sin Zt)
dt dt
I=CE0 (cos Zt) Z

E0
I sin Zt  S / 2 ...(2)
1/ ZC
The current will be maximum i.e.
I = I0, when sin (Zt + S/2) = maximum = 1

E0
? From (2), I 0 u1 ...(3)
1/ ZC
Put in (2), I = I0 sin (Zt + S/2) ...(4)
This is the form of alternating current developed.
Figure (b) represents the vector diagram or the phasor Comparing (4) with (1), we find that in an a.c. circuit
diagram of a.c. circuit containing L only. The vector containing C only, alternating current I leads the alternating
G e.m.f. by a phase angle of 90°. This is shown in figure (b)
representing E 0 makes an angle (Zt) with OX. As current
and (c).
lags behing the e.m.f. by 90°, therefore, phasor representing
G The phasor diagram or vector diagram of a.c. circuit containing
I0 is turned clockwise through 90° from the direction of G
C only in shown in figure (b). The phasor I0 is turned
G § S· v0 G
E0 . I I0 sin ¨ Zt  ¸ , I0 , XL = Z L anticlockwise through 90° from the direction of phasor E 0 .
© 2¹ xL
Their projections on YOY’ give the instantaneous values E
A pure inductance offer zero resistance to dc. It means a and I as shown in figure (b). When E0 and I0 rotate with
pure inductor cannot reduce dc. The units of inductive frequency Z, curves in figure (c). are generated.
reactance

1 1 1
XL = Z L Ÿ (henry) = ohm
sec sec amp / sec
The dimensions of inductive reactance are the same as
those of resistance.
 EMI & AC

(i) The maximum voltage across R is

G G
VR I0 R
G
In figure, current phasor I0 is represented along OX.

Comparing (3) with Ohm’s law equation, viz current =

voltage/resistance, we find that (1/Z C) represents
effective resistance offered by the capacitor. This is called
capacitative reactance and is denoted by XC.

1 1
Thus X C
ZC 2SvC
The capacitative reactance limits the amplitude of current in a
purely capacitative circuit in the same way as the resistance
limits the current in a purely resistive circuit. Clearly, G
capacitative reactance varies inversely as the frequency of As VR is in phase with current, it is represented by the
a.c. and also inversely as the capacitance of the condenser.
vector OA , along OX.
In a d.c. circuit, v = 0, ? XC = f
G G
1 1 sec (ii) The maximum voltage across L is VL I0 X L
Xc sec
ZC farad coulomb / volt As voltage across the inductor leads the current by 90°, it
G
volt sec . is represented by OB along OY, 90° ahead of I0 .
ohm
amp. sec G G
(iii) The maximum voltage across C is VC I0 X C
6. A.C. CIRCUIT CONTAINING RESISTANCE, As voltage across the capacitor lags behind the alternating
INDUCTANCE AND CAPACITANCE AND SERIES current by 90°, it is represented by OC rotated clockwise
G
6.1 Phasor Treatment through 90° from the direction of I0 . OC is along OY’.

Let a pure resistance R, a pure inductance L and an ideal 6.2 Analytical Treatment of RLC series circuit
capacitor of capacitance C be connected in series to a source
of alternating e.m.f., figure. As R, L, C are in series, therefore, Let a pure resistance R, a pure inductance L and an ideal
current at any instant through the three elements has the condenser of capacity C be connected in series to a source
same amplitude and phase. Let it be represented by of alternating e.m.f. Suppose the alterning e.m.f. supplied
I = I0 sin Zt is
E = E0 sin Zt ...(1)
At any instant of time t, suppose
q = charge on capacitor
I = current in the circuit
dI
= rate of change of current in the circuit
dt

q
? potential difference across the condenser
C
However, voltage across each element bears a different
dI
phase relationship with the current. Now, potential difference across inductor L
dt
 EMI & AC

potential difference across resistance = RI or q0 Z Z cos (Zt + T – I) = E0 sin Zt = E0 cos (Zt – S/2) ...(7)
? The voltage equation of the circuit is Comparing the two sides of this equation, we find that
E0 = q0 Z Z = I0 Z, where I0 q0Z ...(8)
dI q
L  RI  = E = E sin Zt ...(2) and Zt + TI = Zt – S/2
dt C 0

S
dq dI d 2q ? TI
As I , therefore, 2
dt dt dt 2
S
? The voltage equation becomes or T I ...(9)
2
d 2q dq q ? Current in the circuit is
L 2
R  E 0 sin Zt ...(3)
dt dt C
dq d
I q 0 sin Zt  T = q Z cos (Zt + T)
This is like the equation of a forced, damped oscillator. Let dt dt 0

the solution of equation (3) be

I = I0 cos (Zt + T) {using (8)}
q = q0 sin (Zt + T)
Using (9), we get, I = I0 cos (Zt + I– S/2)
dq I = I0 sin (Zt + I) ...(10)
? q 0 Z cos Zt  T
dt
XC  XL
From (6), I tan 1 ...(11)
2 R
d q
q 0 Z sin Zt  T
2

dt 2 2
As cos I + sin I = 1
2

Substituting these values in equation (3), we get 2 2

§ R · § XC  XL ·
¨ ¸ ¨ ¸
2
L [–q0 Z sin (Zt + T)] + R q0 Z cos (Zt + T) ? 1
©Z¹ © Z ¹
q0
 sin (Zt  T) E 0 sin Zt
2 2
or R2 + (XC – XL) = Z
C
or Z R 2  (X C  X L ) 2 ...(12)
q 0 Z[R cos Zt  T  ZL sin Zt  T

1 7. A.C. CIRCUIT CONTING RESISTANCE & INDUCTANCE

 sin Zt  T ] E 0 sin Zt
ZC
Let a source of alternating e.m.f. be connected to an ohmic
resistance R and a coil of inductance L, in series as shown
1
A s ZL = XL and XC , therefore in figure.
ZC
q0 Z [R cos (Zt + T) + (XC – XL) sin (Zt + T)] = E0 sin Zt
Multiplying and dividing by

2
Z R 2  XC  X L , we get

ªR X  XL º
q 0 ZZ « cos Zt  T  C sin Zt  T » = E sin Zt
¬Z Z ¼ 0

...(4)

R XC  XL
Let cos I and sin I ...(5)
Z Z

XC  XL
so that tan I ...(6)
R
? q0 Z Z[cos (Zt + T) cos I + sin (Zt + T) sin I] = E0 sin Zt
 EMI & AC

Z R 2  X 2L E L
dI
...(1)
dt
We find that in RL circuit, voltage leads the current by a
phase angle I, where The self induced e.m.f. is also called the back e.m.f., as it
opposes any change in the current in the circuit.
AK OL VL I0 X L
tan I Physically, the self inductance plays the role of inertia. It
OA OA VR I0R
is the electromagnetic analogue of mass in mechanics.
Therefore, work needs to be done against the back e.m.f. E
XL
tan I in establishing the current. This work done is stored in the
R
inductor as magnetic potential energy.
8. A.C. CIRCUIT CONTAINING RESISTANCE For the current I at an instant t, the rate of doing work is
AND CAPACITANCE
dW
EI
Let a source of alternating e.m.f. be connected to an ohmic dt
resistance R and a condenser of capacity C, in series as
If we ignore the resistive losses, and consider only
shown in figure.
inductive effect, then
Z R 2  X 2C
dW dI
Using (1), EI L u I or dW = LI dI
dt dt
Total amount of work done in establishing the current I is

I
1 2
W ³ dW ³ LIdI
0
2
LI

Thus energy required to build up current in an inductor =

energy stored in inductor

1 2
UB W LI
2

10. ELECTRIC RESONANCE

10.1 Series Resonance Circuit

A circuit in which inductance L, capacitance C and

resistance R are connected in series, and the circuit admits
Figure represents phasor diagram of RC circuit. We find maximum current corresponding to a given frequency of
that in RC circuit, voltage lags behind the current by a a.c., is called series resonance circuit.
phase angle I, where The impedance (Z) of an RLC circuit is given by

AK OC VC I0X C
tan I § 1 ·
2
OA OA VR I0R Z R 2  ¨ ZL  ¸ ...(1)
© Z C¹
XC
tan I
R At very low frequencies, inductive reactance XL = ZL is
negligible, but capacitative reactance (XC = 1/ZC) is very
9. ENERGY STORED IN AN INDUCTOR high.
As frequency of alternating e.m.f. applied to the circuit is
When a.c. is applied to an inductor of inductance L, the increased, X L goes on increasing and X C goes on
current in it grows from zero to maximum steady value I0. If
decreasing. For a particular value of Z ( = Zr, say)
I is the current at any instant t, then the magnitude of
induced e.m.f. developed in the inductor at that instant is XL = XC
 EMI & AC

i.e., Zr L
1
or Zr
1 § Z ·
The quantity ¨ r ¸ is regarded as a measure of
Zr C LC © 2'Z ¹
sharpness of resonance, i.e., Q factor of resonance circuit
1 1
2S v r or v r is the ratio of resonance angular frequency to band width
LC 2 S LC of the circuit (which is difference in angular frequencies at
At this particular frequency vr, as XL = XC, therefore, from which power is half the maximum power or current is
(1)
I0 / 2 .
Z R 2  0 = R = minimum 10.2 Average Power in RLC circuit or Inductive Circuit
i.e. impedance of RLC circuit is minimum and hence the
Let the alternating e.m.f. applied to an RLC circuit be
E0 E0
current I 0 becomes maximum. This frequency E = E0 sin Zt ...(1)
Z R
If alternating current developed lags behind the applied
is called series resonance frequency.
e.m.f. by a phase angle I, then
I = I0 sin (Zt – I) ...(2)

dW
Power at instant t, EI
dt

dW
E 0 sin Zt u I 0 sin Zt  I
dt
= E0 I0 sin Zt (sin Zt cos I– cos Zt sin I)
2
= E0I0 sin Zt cos I– E0I0 sin Zt cos Zt sin I

2 E 0I0
= E0I0 sin Zt cos I  sin 2 Zt sin I
2
The Q factor of series resonant circuit is defined as the If this instantaneous power is assumed to remain constant
ratio of the voltage developed across the inductance or for a small time dt, then small amount of work done in this
capacitance at resonance to the impressed voltage, which time is
is the voltage applied across R.
§ E I ·
dW ¨ E 0 I 0 sin 2 Zt cos I  0 0 sin 2 Zt sin I ¸ dt
i.e. Q
voltage across L or C © 2 ¹
applied voltage ( voltage across R )
Total work done over a complete cycle is
Zr L I Zr L T T
Q E0 I0
RI R W ³
0
E 0 I 0 sin 2 Zt cos I dt  ³
0
2
sin 2Zt sin I dt
1 / Zr C I I
or Q
RI RC Zr T T
E0 I0
1
W ³
E 0 I 0 cos I sin 2 Zt dt 
2 ³
sin I sin 2 Zt dt
Using Zr , we get 0 0
LC
T T
T
Q
L
R
1 1
R
L
C
As ³ sin 2 Zt dt
2 ³
and sin Zt dt 0
LC 0 0

1 LC 1 L T
or Q ? W E 0 I 0 cos Iu
RC R C 2
? Average power in the inductive circuit over a complete
1 L
Thus Q ...(1) cycle
R C
 EMI & AC

W E 0 I 0 cos I T E 0 I0 R
P . cos I
T T 2 2 2 2 [from impedance triangle]
R  XL  XC
2

P = Ev Iv cos I ...(3)
Hence average power over a complete cycle in an inductive Resistance
? Power factor = cos I =
circuit is the product of virtual e.m.f., virtual current and Impedance
cosine of the phase angle between the voltage and current.
In a non-inductance circuit, XL = XC

R R
? Power factor = cos I = 1, I 0q ...(4)
R 2 R
The relation (3) is applicable to all a.c. circuits. cos I and
Z will have appropriate values for difference circuits. This is the maximum value of power factor. In a pure

For example : inductor or an ideal capacitor, I = 90°

Power factor = cos I = cos 90° = 0
R
(i) In RL circuit, Z R X 2 2
L
and cos I Average power consumed in a pure inductor or ideal a
Z
capacitor, P = Ev Iv cos 90° = Zero. Therefore,
R current through pure L or pure C, which consumes no
(ii) In RC circuit, Z R 2  X 2C and cos I
Z power for its maintenance in the circuit is called Idle current
or Wattless current.
(iii) In LC circuit, Z = XL – XC and I = 90°
In actual practice, we do not have ideal inductor or ideal
R capacitor. Therefore, there does occur some dissipation
(iv) In RLC circuit, Z R 2  XL  XC
2
and cos I
Z of energy. However, inductance and capacitance continue
to be most suitable for controlling current in a.c. circuits
Ev with minimum loss of power.
In all a.c. circuits, I v
Z

10.3 Power Factor of an A.C. Circuit

We have proved that average power/cycle in an inductive

circuit is
P = EvIv cos I ...(1)
Here, P is called true power, (EvIv) is called apparent power
or virtual power and cos I is called power factor of the
circuit.

true power (P )
Thus, Power factor = cos I
apparent power E v I v

...(2)
EMI & AC

Theory and Working : As the armature coil is rotated in the

11. A.C. GENERATOR OR A.C. DYNAMO magnetic field, angle T between the field and normal to the
coil changes continuously. Therefore, magnetic flux linked
with the coil changes. An e.m.f. is induced in the coil.
An a.c. generator/dynamo is a machine which produces
alternating current energy from mechanical energy. It is To start with, suppose the plane of the coil is perpendicular
one of the most important applications of the phenomenon to the plane of the paper in which magnetic field is applied,
of electromagnetic induction. The generator was designed with AB at front and CD at the back, figure (a). The amount
originally by a Yugoslav scientist, Nikola Tesla. The word of magnetic flux linked with the coil in this position is
generator is a misnomer, because nothing is generated by maximum. As the coil is rotated anticlockwise (or
the machine. Infact, it is an alternator converting one form clockwise), AB moves inwards and CD moves outwards.
of energy into another. The amount of magnetic flux linked with the coil changes.
According to Fleming’s right hand rule, current induced
in AB is from A to B and in CD, it is from C to D. In the
11.1 Principle
external circuit, current flows from B2 to B1, figure (a)

An a.c. generator/dynamo is based on the phenomenon

of electromagnetic induction, i.e., whenever amount
of magnetic flux linked with a coil changes, an e.m.f. is
induced in the coil. It lasts so long as the change in
magnetic flux through the coil continues. The direction of
current induced is given by Fleming’s right hand rule.

11.2 Construction

The essential parts of an a.c. dynamo are shown in figure.

1. Armature : ABCD is a rectangular armatrue coil. It
consists of a large number of turns of insulated copper
wire wound over a laminated soft iron core, I. The coil can
be rotated about the central axis.
2. Field Magnets : N and S are the pole pieces of a strong
electromagnet in which the armature coil is rotated. Axis
of rotation is perpendicular to the magnetic field lines.
The magnetic field is of the order of 1 to 2 tesla.
3. Slip Rings : R1 and R2 are two hollow metallic rings, to
which two ends of armature coil are connected. These
rings rotate with the rotation of the coil.
4. Brushes : B1 and B2 are two flexible metal plates or
carbon rods. They are fixed and are kept in light contact
with R1 and R2 respectively. The purpose of brushes is to
pass on current from the armature coil to the external load
resistance R.
 EMI & AC

After half the rotation of the coil, AB is at the back and CD The current supplied by the a.c. generator is also
is at the front, figure. Therefore, on rotating further, AB sinusoidal. It is given by
moves outwards and CD moves outwards and CD moves
e e0
inwards. The current induced in AB is from B to A and in i sin Zt i 0 win Zt
CD, it is from D to C. Through external circuit, current R R
flows from B1 to B2; figure (b). This is repeated. Induced e0
current in the external circuit changes direction after every where i 0 maximum value of current.
R
half rotation of the coil. Hence the current induced is
alternating in nature.
To calculate the magnitude of e.m.f. induced, suppose
N = number of turns in the coil, Suppose to start with, the plane of the coil is not
A = area enclosed by each turn of the coil perpendicular to the magnetic field. Therefore, at t = 0,
G T z 0. Let T G, the phase angle. This is the angle which
B = strength of magnetic field G
normal to the coil makes with the direction of B. The
G
T = angle which normal to the coil makes with B at any equation (4) of e.m.f. induced in that case can be rewritten
instant t, figure. as e = e0 sin (Zt + G).

12. TRANSFORMER
A transformer which increases the a.c. voltage is called a
step up transformer, A transformer which decreases the
a.c. voltages is called a step down transformer.
? Magnetic flux linked with the coil in this position
G G 12.1 Principle
I N B . A NBA cos T NBA cos Zt ...(1)
A transformer is based on the principle of mutual
where Z is angular velocity of the coil.
induction, i.e., whenever the amount of magnetic flux linked
As the coil is rotated, T changes; therefore, magnetic flux with a coil changes, an e.m.f. is induced in the neighbouring
I linked with the coil changes and hence an e.m.f. is
coil.
induced in the coil.
12.2 Construction
At the instant t, if e is the e.m.f. induced in the coil, then
A transformer consists of a rectangular soft iron core made
dI d
e  NAB cos Zt of laminated sheets, well insulated from one another, figure.
dt dt
Two coils P1P2 (the primary coil) and S1S2 (the secondary
d coil) are wound on the same core, but are well insulated
 NAB cos Zt  NAB  sin Zt Z
dt from each other. Note that both the coils are also insulated
from the core. The source of alternating e.m.f. (to be
E = NAB Z sin Zt ...(2)
transformed) is connected to the primary coil P1P2 and a
The induced e.m.f. will be maximum, when load resistance R is connected to the secondary coil S1S2
sin Zt = maximum = 1 through an open switch S. Thus, there can be no current
? emax = e0 = NAB Z × 1 ...(3) through the secondary coil so long as the switch is open.
Put in (2), e = e0 sin Zt ...(4)
The variation of induced e.m.f. with time (i.e. with position
of the coil) is shown in figure.

For an ideal transformer, we assume that the resistances

of the primary and secondary windings are negligible.
 EMI & AC

Further, the energy losses due to magnetic hysterisis in

Ep np
the iron core is also negligible. Well designed high capacity From (2),
Es ns
transformers may have energy losses as low as 1%.

12.3 Theory and working np Ip

? Is Ip . ...(3)
Let the alternating e.m.f. supplied by the a.c. source ns K
connected to primary be For a step up transformer, Es > Ep ; K > 1 ? Is < Ip
Ep = E0 sin Zt ...(1) i.e. secondary current is weaker when secondary voltage
As we have assumed the primary to be a pure inductance is higher, i.e., whatever we gain in voltage, we lose in
with zero resistance, the sinusoidal primary current Ip lags current in the same ratio.
the primary voltage Ep by 90°. The primary’s power factor, The reverse is true for a step down transformer.
cos I = 90° = 0. Therefore, no power is dissipated in primary.
The alternating primary current induces an alternating §n · Es § ns ·
From eqn. (3) I p Is ¨ s ¸ ¨ ¸
magnetic flux IB in the iron core. Because the core extends ¨ np ¸ R ¨ np ¸
© ¹ © ¹
through the secondary winding, the induced flux also
extends through the turns of secondary.
1 §n · § ns ·
According to Faraday’s law of electromagnetic induction, Using equation (2), we get I p .Ep ¨ s ¸¨ ¸
R ¨ np ¸¨ np ¸
the induced e.m.f. per turn (Eturn) is same for both, the © ¹© ¹
primary and secondary. Also, the voltage Ep across the 2
primary is equal to the e.m.f. induced in the primary, and 1 §¨ n s ·
¸ Ep
Ip ...(4)
the voltage Es across the secondary is equal to the e.m.f. R ¨© n p ¸
¹
induced in the secondary. Thus,
Ep
dI B Ep Es This equation, has the form I p , where the
E turn R eq
dt np ns
2
§ np ·
Here, np ; ns represent total number of turns in primary and equivalent resistance Req is R eq ¨ ¸ R
¨n ¸ ...(5)
secondary coils respectively. © s ¹
Thus Req is the value of load resistance as seen by the
n source/generator, i.e., the source/generator produces
? Es Ep s ...(2)
np current Ip and voltage Ep as if it were connected to a
resistance Req.
If ns > np ; Es > Ep, the transformer is a step up transformer.
Similarly, when ns < np ; Es < Ep. The device is called a step Efficiency of a transformer is defined as the ratio of output
to the input power.
ns
down transformer. = K represents transformation ratio. E s Is
np Output power
i.e., K
Input power Ep Ip
Note that this relation (2) is based on three assumptions
In an ideal transformer, where there is no power loss, K = 1
(i) the primary resistance and current are small,
(i.e. 100%). However, practically there are many energy
(ii) there is no leakage of magnetic flux. The same magnetic losses. Hence efficiency of a transformer in practice is
flux links both, the primary and secondary coil, less than one (i.e. less than 100%).
(iii) the secondary current is small.
12.4 Energy Losses in a Transformer
Now, the rate at which the generator/source transfer energy
to the primary = IpEp. The rate at which the primary then Following are the major sources of energy loss in a
transfers energy to the secondary (via the alternating transformer :
magnetic field linking the two coils) is IsEs. 1. Copper loss is the energy loss in the form of heat in the
As we assume that no energy is lost along the way, copper coils of a transformer. This is due to Joule heating
conservation of energy requires that of conducting wires. These are minimised using thick wires.
2. Iron loss is the energy loss in the form of heat in the iron core
Ep of the transformer. This is due to formation of eddy currents
IpEp = IsEs ? Is Ip
Es in iron core. It is minimised by taking laminated cores.
 EMI & AC

3. Leakage of magnetic flux occurs inspite of best insulations.

Therefore, rate of change of magnetic flux linked with each
turn of S1S2 is less than the rate of change of magnetic flux
linked with each turn of P1P2. It can be reduced by winding
the primary and secondary coils one over the other.
4. Hysteresis loss. This is the loss of energy due to repeated
magnetisation and demagnetisation of the iron core when
a.c. is fed to it. The loss is kept to a minimum by using a
magnetic material which has a low hysteresis loss.
5. Magnetostriction, i.e., humming noise of a transformer. Now, we consider a different surface, i.e., a tiffin box shaped
Therefore, output power in the best transformer may be surface without lid with its circular rim, which has the same
roughly 90% of the input power. boundary as that of loop C1. The box does not touch to
the connecting wire and plate P of capacitor. The flat
13. DISPLACEMENT CURRENT circular bottom S of the tiffin box lies in between the
capacitor plates. Figure (b). No conduction current is
According to Ampere circuital law :
passing through the tiffin box surface S, therefore I = 0.
G On applying Ampere’s circuital law to loop C1 of this tiffin
the line integral of magentic field B around any closed
box surface, we have
path is equal to P0 times the total current threading the
closed path, i.e.,
G G
³
C
B. d A P 0I
...(1)

Consider a parallel plate capacitor having plates P and Q

connected to a battery B, through a tapping key K. When
key K is pressed, the conduction current flows through
the connecting wires. The capacitor starts storing charge.
As the charge on the capacitor grows, the conduction
current in the wires decreases. When the capacitor is G G
fully charged, the conduction current stops flowing in ³ B.d A = B 2Sr = P × 0 = 0
0
or B = 0 ...(3)
the wires. During charging of capacitor, there is no C

conduction current between the plates of capacitor. From (2) and (3), we note that there is a magnetic field at R
During charging, let at an instant, I be the conduction calculated through one way and no magnetic field at R,
current in the wires. This current will produce magnetic calculated through another way. Since this contradition
field around the wires which can be detected by using a arises from the use of Ampere’s circuital law, hence
compass needle. Ampere’s circuital law is logically inconsisten.
Let us find the magnetic field at point R which is at a If at the given instant of time, q is the charge on the plate
perpendicular distance r from connecting wire in a region of capacitor and A is the plate area of capacitor, the
outside the parallel plate capacitor. For this we consider a magnitude of the electric field between the plates of
capacitor is
plane circular loop C1, of radius r, whose centre lies on
wire and its plane is perpendicular to the direction of current q
carrying wire (figure a). The magnitude of the magnetic E
0 A
field is same at all points on the loop and is acting
tangentially along the circumference of the loop. If B is This field is perpendicular to surface S. It has the same
magnitude over the area A of the capacitor plates and
the magnitude of magnetic field at R, then using Ampere’s
becomes zero outside the capacitor.
circuital law, for loop C1, we have
The electric flux through surface S is,
G G P0I
³
C1
B. dA ³ B dA cos 0q = B 2 Sr = P I or B
C1
0
2 Sr
...(2)
IE
G G
E.A EA cos 0q
1 q
uA
q
...(4)
0 A 0
 EMI & AC

dq
If is the rate of change of charge with time on the plate
dt
of the capacitor, then

dI E d§ q · 1 dq
¨ ¸
dt dt ¨© 0 ¸
¹ 0 dt

dq dI E
or 0
dt dt
Due to battery B, let the conduction current I be flowing
dq through the lead wires at any instant, but there is no
Here, = current through surface S corresponding to
dt conduction current across the capacitor gap, as no charge
changing electric field = ID, called Maxwell’s displacement is transported across this gap.
current. Thus, For loop C1, there is no electric flux, i.e., IE = 0 and
displacement current is that current which comes into play
dI E
in the region in which the electric field and the electric flux 0
is changing with time. dt

dI E dI E
I D 0 ? I  ID I 0 I 0 0 I ...(7)
...(5) dt
dt
For loop C2, there is no conduction current, i.e., I = 0
Maxwell modified Ampere’s circuital law in order to make
the same logically consistent. He stated Ampere’s circuital dI E
law to the form, ? I + ID = 0 + ID = ID = 0 ...(8)
dt
G G § dI · At the given instant if q is the magnitude of charge on the
³ B.dA P 0 I  ID P0 ¨ I  H0 E ¸
© dt ¹
...(6) plates of the capacitor of area A, then electric field E in the
gap between the two plates of this capacitor is given by
This is called Ampere Maxwell’s Law.
q § V q ·
14. CONTINUITY OF CURRENT E ¨'E ¸
0 A ¨ 0 0 A ¸¹
©
Maxwell’s modification of Ampere’s circuital law gives that
G G q q
? Electric flux, I E EA A
³
C
B.dA P 0 I  ID 0 A 0

d dq
dI E Thus from (8), we have I + ID = 0 q / 0 I
where I D 0 , called displacement current, I is the dt dt
dt
conduction current and IE is the electric flux across the ...(9)
loop C. From (7) and (9), we conclude that the sum (I + ID) has the
The sum of the conduction current and displacement same value on the left and right side of plate P of the
current (i.e., I + ID) has the important property of continuity parallel plate capacitor. Hence (I + ID) has the property of
along any closed path although individually they may not continuity although individually they may not be
be continuous. continuous.
To prove it, consider a parallel plate capacitor having plates 15. CONSEQUENCES OF DISPLACEMENT CURRENT
P and Q, being charged with battery B. During the time,
charging is taking place, let at an instant, I be the The discovery of displacement current is of great
conduction current flowing through the wires. Let C1 and importance as it has established a symmetry between the
C2 be the two loops, which have exactly the same boundary laws of electricity and magnetism. Faraday’s law of
as that of the plates of capacitor. C1 is little towards left electromagnetic induction states that the magnitude of
and C2 is a little towards right of the plate P of parallel plate the emf induced in a coil is equal to the rate of change of
capacitor, figure. magnetic flux linked with it. Since, the emf between two
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 EMI & AC

points A and B is the measure of maximum workdone in

taking a unit charge from point A to B, therefore, the
existence of an emf shows the existence of an electric field.
It is due to this fact, Faraday concluded that a changing
magnetic field with time gives rise to an electric field.
The Maxwell’s concept that a changing electric field with
time gives rise to displacement current which also produces
a magnetic field similar to that of conduction current. It is
infact, a symmetrical counterpart of the Faraday’s concept,
which led Maxwell to conclude that the displacement
current is also a source of magnetic field. It means the
time varying electric and magnetic fields give rise to each
other. From these concepts, Maxwell concluded the where P0 and 0 are permeability and permittivity of the
existence of electromagnetic wave in a region where free space respectively.
electric and magnetic fields were changing with time. –7
We know, P0 = 4S× 10 Wb A m ;
–1 –1

–2 2 –1 –2
16. MAXWELL’S EQUATIONS AND LORENTZ FORCE 0 = 8.85 × 10 C N m
8 –1
In the absence of any dielectric or magnetic material, the Putting these values in (10), we have c = 3.00 × 10 ms
four Maxwell’s equations are given below ? where P  are the absolute permeability and absolute
G G permittivity of the medium. We also know that P = P0Pr and
(i) ³
S
E . ds q / 0 . This equation is Gauss’s Law in
 0 r where P 0 , r are the relative permeability and
relative permittivity of the medium.
electrostatics.
The electric lines of force do not form continuous closed 1 c
Therefore, v
path. P 0 P r 0r P r r
G G
(ii)
³
S
B . ds 0 . This equation is Gauss’s Law in
ª 1 º
«' c »
magnetostatics. «¬ P 0 0 »¼
The magnetic lines of force always form closed paths.
Maxwell also concluded that electromagnetic wave is
G G d G G transverse in nature and light is electromagnetic wave.
(iii) ³ E .d A 
dt ³
s
B . d s . This equation is Faraday’s law of
17. VELOCITY OF ELECTROMAGNETIC WAVES
electromagnetic induction.
Consider a plane electromagnetic wave propagating along
The line integral of electric field around any closed path
positive direction of X–axis in space with speed c. Since
(i.e., the emf) is equal to the time rate of change of magnetic
in electromagnetic wave, the electric and magnetic fields
flux through the surface bounded by the closed path.
are transverse to the direction of wave propagation,
G G therefore, the electric and magnetic fields are in Y–Z plane.
d G G
(iv) ³ B. dA P 0 I  P 0 0 ³
dt s
E .ds . This equation is G
Let the electric field E be acting along Y–axis and
G
magnetic field B along Z–axis.
generalised form of Ampere’s law as Modified by Maxwell
and is also known as Ampere-Maxwell law. At any instant, the electric and magnetic fields varying
sinusoidally with x and t can be represented by the
The electromagnetic waves are those wave in which there
equations.
are sinusoidal variation of electric and magnetic field
vectors at right angles to each other as well as at right E = Ey = E0 sin Z (t – x/c) ...(1)
angles to the direction of wave propagation. B = Bz = B0 sin Z (t – x/c) ...(2)
Here E0 and B0 are the amplitudes of electric and magnetic
1
c fields along Y–axis and Z–axis respectively. Consider a
P 0 0 ...(10)
rectangular path PQRS in X–Y plane as shown in figure.
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EMI & AC

G Consider a rectangular path PUTQ in the X–Z plane as

The line integral of E over the closed path PQRS will be G
shown in figure. The line integral of B over the closed
Q path PUTQ, we have
G G G G RG G SG G PG G
³
PQRS
E . dA ³
P
³
Q
³
R
³
E , d A  E , dA  E , dA  E , d A
S G G U
G G TG G QG G PG G
³
PUTQ
B.dA ³
P
³
U
³
T
³
B .d A  B .d A  B .d A  B .dA
Q
0 E x2 A0E x1 A
B x1 A  0  B x 2 A  0
ª § x · § x ·º
E 0 A «sin Z ¨ t  2 ¸  sin Z ¨ t  1 ¸» ...(3)
¬ © c ¹ © c ¹¼ ª § x · § x ·º
B 0 A «sin Z¨ t  1 ¸  sin Z¨ t  2 ¸» ...(6)
¬ © c ¹ © c ¹¼
Magnetic flux linked with surface surrounded by
rectangular path PQRS will be The electric flux linked with the surface surrounded by
rectangular path PUTQ is
x2 x2
ª § x ·º
IB ³ B x A dx ³ B A«¬sin Z¨© t  c ¸¹»¼ dx
0 x2
G G
x2 x2
§ x·
x1 x1 IE ³
x1
E .ds ³
x1
E x Adx ³
E 0 A sin Z¨ t  ¸ dx
x1
© c¹

B 0 Ac ª § x2 · § x ·º
«cos Z ¨ t  ¸  cos Z ¨ t  1 ¸»
ª § x · § x ·º
Z ¬ © c ¹ © c ¹¼ c
 E 0 A « cos Z¨ t  2 ¸  cos Z¨ t  1 ¸»
Z ¬ © c ¹ © c ¹¼

dI B B 0 Ac ª § x2 · § x ·º
? « Z sin Z¨ t  ¸  Z sin Z¨ t  1 ¸» dI E ª § x · § x ·º
dt Z ¬ © c ¹ © c ¹¼ cE 0 A «sin Z¨ t  2 ¸  sin Z¨ t  1 ¸»
or dt ¬ © c ¹ © c ¹¼

ª § x · § x ·º
B 0 Ac «sin Z¨ t  2 ¸  sin Z¨ t  1 ¸» ...(4) ª § x · § x ·º
¬ © c ¹ © c ¹¼ c E 0 A «sin Z¨ t  1 ¸  sin Z¨ t  2 ¸» ...(7)
¬ © c ¹ © c ¹¼
Using Faraday’s law of electromagnetic induction, we have
In space, there is no conduction current. According to
G G dI B Ampere Maxwell law in space
³ E . dA 
dt
G G dI E
Putting the values from (3) and (4), we get ³
PUTQ
B. dA P 0 0
dt
E0 = cB0 ...(5)
Putting values from (6) and (7), we get
Since E and B are in phase, we can write.
B0 P 0 0 cE 0 P 0 0 c cB 0
E = c B at any point in space.
 EMI & AC

1 1 B02
or 1 P 0 0 c 2 or c ...(8) In terms of maximum magnetic field, u av ,
P 0 0 2 P0

Which is the speed of electromagnetic waves in vacuum. 1 B 20 1 2

so I c B rms c
–7 –1 2 P0 P0
For vacuum, P0 = 4S × 10 T mA

1 19. ELECTROMAGNETIC SPECTRUM

and 9 u 10 9 Nm 2 C  2
S 0
After the experimental discovery of electromagnetic waves
1 by Hertz, many other electromagntic waves were
or 0 N 1m  2 C 2
S u 9 u 10 9 discovered by different ways of excitation.

Putting the value in (8), we get The orderly distribution of electromagnetic radiations
according to their wavelength or frequency is called the
1 electromagnetic spectrum.
c 3 u10 8 m / s
7
S u10 u 1 / S u 9 u10 9
The electromagnetic spectrum has much wider range with
–14 2
wavelength variation ~ 10 m to 6 × 10 m. The whole
which is exactly the speed of light in vacuum.
electromagnetic spectrum has been classified into different
This shows that light is an electromagnetic wave. parts and subparts in order of increasing wavelength,
18. INTENSITY OF ELECTROMAGNETIC WAVE according to their type of excitation. There is overlapping
in certain parts of the spectrum, showing that the
Intensity of electromagnetic wave at a point is defined as corresponding radiations can be produced by two
the energy crossing per second per unit area normally methods. It may be noted that the physical properties of
around that point during the propagation of electromagnetic electromagnetic waves are decided by their wavelengths
wave. and not by the method of their excitation.
Consider the propagation of electromagnetic wave with A table given below shows the various parts of the
speed c along the X–axis. Take an imaginary cylinder of electromagnetic spectrum with approximate wavelength
area of cross-section A and length c ' t, so that the wave range, frequency range, their sources of production and
crosses the area A normally. Figure. Let uav be the average detections.
energy density of electromagnetic wave.
20. MAIN PARTS OF ELECTROMAGNETIC SPECTRUM
The electromagnetic spectrum has been broadly classified
into following main parts; mentioned below in the order of
increasing frequency.

20.1 Radiowaves

Theses are the electromagnetic wave of frequency range

5 9
from 5 × 10 Hz to 10 Hz. These waves are produced by
The energy of electromagnetic wave (U) crossing the area
oscillating electric circuits having an inductor and
of cross-section at P normally in time ' t is the energy of
capacitor.
wave contained in a cylinder of length c ' t and area of
cross-section A. It is given by U = uav (c ' t) A Uses : The various frequency ranges are used for different
types of wireless communication systems as mentioned below
The intensity of electromagnetic wave at P is,
(i) The electromagnetic waves of frequency range from 530
U u av c ' t A kHz to 1710 kHz form amplitude modulated (AM) band. It is
I u av c
A 't A 't used in ground wave propagation.

1 (ii) The electromagnetic waves of frequency range 1710 kHz

In terms of maximum electric field, u av 0 E 20 , to 54 Mhz are used for short wave bands. It is used in sky
2
wave propagation.
1
so I 0 E 02 c 0 E 2rms c (iii) The electromagnetic waves of frequency range 54 Mhz to
2 890 MHz are used in television waves.
 EMI & AC

(iv) The electromagnetic waves of frequency range 88 MHz to The visible light emitted or reflected from objects around
108 MHz from frequency modulated (FM) radio band. It is us provides the information about the world surrounding
used for commercial FM radio. us.
(v) The electromagnetic waves of frequency range 300 MHz
20.5 Ultraviolet rays
to 3000 MHz form ultra high frequency (UHF) band. It is
used in cellular phones communication. The ultraviolet rays were discovered by Ritter in 1801. The
14 16
frequency range of ultraviolet rays is 8 × 10 Hz to 5 × 10
20.2 Microwaves Hz. The ultraviolet rays are produced by sun, special lamps
Microwaves are the electromagnetic waves of frequency and very hot bodies. Most of the ultraviolet rays coming
range 1 GHz to 300 GHz. They are produced by special from sun are absorbed by the ozone layer in the earth’s
vacuum tubes. namely ; klystrons, magnetrons and Gunn atmosphere. The ultraviolet rays in large quantity produce
diodes etc. harmful effect on human eyes.
Uses : Uses : Ultraviolet rays are used :
(i) Microwaves are used in Radar systems for air craft (i) for checking the mineral samples through the property of
navigation. ultraviolet rays causing flourescence.
(ii) A radar using microwave can help in detecting the speed (ii) in the study of molecular structure and arrangement of
of tennis ball, cricket ball, automobile while in motion. electrons in the external shell through ultraviolet
(iii) Microwave ovens are used for cooking purposes. absorption spectra.
(iv) Microwaves are used for observing the movement of trains (iii) to destroy the bacteria and for sterilizing the surgical
on rails while sitting in microwave operated control rooms. instruments.
(iv) in burglar alarm.
20.3 Infrared waves
(v) in the detection of forged documents, finger prints in
Infrared waves were discovered by Herschell. These are forensic laboratory.
11
the electromagnetic waves of frequency range 3 × 10 Hz
14 (vi) to preserve the food stuff.
to 4 × 10 Hz. Infrared waves sometimes are called as
heat waves. Infrared waves are produced by hot bodies 20. 6 X–rays
and molecules. These wave are not detected by human
eye but snake can detect them. The X–rays were discovered by German Physicst W.
16 21
Roentgen. Their frequency range is 10 Hz to 3 × 10 Hz.
Uses :
These are produced when high energy electrons are
Infrared waves are used : stopped suddenly on a metal of high atomic number.
(i) in physical therapy, i.e., to treat muscular strain. X–rays have high penetrating power.
(ii) to provide electrical energy to satellite by using solar cells Uses : X–rays are used :
(iii) for producing dehydrated fruits (i) In surgery for the detection of fractures, foreign bodies
(iv) for taking photographs during the condition of fog, smoke like bullets, diseased organs and stones in the human body.
etc. (ii) In Engineering (i) for detecting faults, cracks, flaws and
(v) in green houses to keep the plants warm holes in final metal products (ii) for the testing of weldings,
casting and moulds.
(vi) in revealing the secret writings on the ancient walls
(iii) In Radio therapy, to cure untracable skin diseases and
(vii) in solar water heaters and cookers
malignant growth.
(viii) in weather forecasting through infra red photography
(iv) In detective departments (i) for detection of explosives,
(ix) in checking the purity of chemcials and in the study of opium, gold and silver in the body of smugglers.
molecular structure by taking infrared absorption spectrum.
(v) In Industry (i) for the detection of pearls in oysters and
20.4 Visible light defects in rubber tyres, gold and tennis balls etc. (ii) for
testing the uniformity of insulating material.
It is the narrow region of electromagnetic spectrum, which
is detected by the human eye. Its frequency is ranging (vi) In Scientific Research (i) for the investigation of structure
14 14
from 4×10 Hz to 8×10 Hz. It is produced due to atomic of crystal, arrangement of atoms and molecules in the
excitation. complex substances.
 EMI & AC

(i) in the treatment of cancer and tumours.

20.7 J-rays
(ii) to preserve the food stuffs for a long time as the soft
J–rays are the electromagnetic waves of frequency range
18 22 J–rays can kill microorganisms easily.
3 × 10 Hz to 5 × 10 Hz. J–rays have nuclear origin.
These rays are highly energetic and are produced by the (iii) to produce nuclear reactions.
nucleus of the radioactive substances. (iv) to provide valuable information about the structure
Uses : J–rays are used : of atomic nucleus.
 Chapter 8
Electromagnetic Waves
8.1 Introduction
• We will discuss about electromagnetic waves, their properties and characteristics, and also their practical
uses in our day-to-day life.
• One of the most important applications of electromagnetic waves is in communication.
• Some of the important applications of electromagnetic waves are:-
1. We are able to see everything around us because of electromagnetic waves.
2. It helps in aircraft navigation and helps the pilot for the smooth take-off and landing of aeroplanes. It
also helps to calculate the speed of the aeroplane.
3. In the medical field it has got very important applications. For example: - In laser eye surgery, in x-rays.
4. In radio and television broadcasting signals. These signals are transmitted by electromagnetic waves.
5. Electromagnetic waves helps in determining the speed of the passing vehicles.
6. They are used in electronic appliances like T.V. remotes, remote cars, LED TV, microwave ovens etc.
7. Voice transmission in mobile phones is possible because of electromagnetic waves.

What are Electromagnetic Waves?

• Electromagnetic (EM) waves are the waves which are related to both electricity and magnetism.
• Electromagnetic (EM) waves are the waves which are coupled time varying electric and magnetic fields
that propagate in space.
• Waves associated with electricity and magnetism and as they are waves so they will propagate in the
space.
• When the electric and magnetic fields combine together and when they are varying with time they both
will give rise to electromagnetic waves.
• Electromagnetic equations emerged from Maxwell’s equations.
• Maxwell found these EM waves have so many special properties which can be used for many practical
purposes.
• Time varying electric field + Time varying magnetic field = Electromagnetic waves.

Maxwell’s Experiments
• Maxwell proposed that the time varying electric field can generate magnetic field.
• Time varying magnetic field generates electric field (Faraday-Lenz law).
1. According to Faraday Lenz law an EMF is induced in the circuit whenever the amount of magnetic flux
linked with a circuit changes.
2. As a result electric current gets generated in the circuit which has an electric field associated with it.
According to Maxwell if Faraday’s law is true then the vice-versa should also be true, i.e. a time varying
electric field should also be able to generate a magnetic field.

Ampere’s Circuit Law:

• According to Ampere’s Circuital law, the line integral of magnetic field over the length element is equal to
μ0 times the total current passing through the surface ∫dl = μ0 l
• According to Maxwell there was some inconsistency in the Ampere’s circuital law.
• This means Ampere’s circuital law was correct for some cases but not correct for some.
• Maxwell took different scenarios i.e. he took a capacitor and tried to calculate magnetic field at a specific
point in a piece of a capacitor.
• Point P as shown in the figure is where he determined the value of B, assuming some current I is flowing
through the circuit.
• He considered 3 different amperial loops as shown in the figs.
• Ampere’s circuital law should be same for all the 3 setups.
Case 1: Considered a surface of radius r & dl is the circumference of the surface, then from Ampere’s
circuital law

∫ 𝐵. 𝑑𝑙 = 𝜇0 𝑙
𝑜𝑟 𝐵(2𝜋𝑟) = 𝜇0 𝑙
𝜇0 𝑙
𝑜𝑟 𝐵 =
2𝜋𝑟

Case 2 : Considering a surface like a box & its lid is open and applying the Ampere’s circuital law

∫ 𝐵. 𝑑𝑙 = 𝜇0 𝑙
 As there is no current flowing inside the capacitor, therefore I = 0

Or
∫ 𝐵. 𝑑𝑙 = 0

Case 3: Considering the surface between 2 plates of the capacitor, in this case also I=0, so B=0

• At the same point but with different amperial surfaces the value of magnetic field is not same. They are
different for the same point.
Maxwell suggested that there are some gaps in the Ampere’s circuital law. He corrected the Ampere’s
circuital law. And he made Ampere’s circuital law consistent in all the scenarios.
Maxwell’s correction to Ampere’s law
• Ampere’s law states that “the line integral of resultant magnetic field along a closed plane curve is equal
to μ0 time the total current crossing the area bounded by the closed curve provided the electric field
inside the loop remains constant".
• Ampere’s law is true only for steady currents.
• Maxwell found the shortcoming in Ampere’s law and he modified Ampere’s law to include time-varying
electric fields.
• For Ampere’s circuital law to be correct Maxwell assumed that there has to be some current existing
between the plates of the capacitor.
• Outside the capacitor current was due to the flow of electrons.
• There was no conduction of charges between the plates of the capacitor.
• According to Maxwell between the plates of the capacitor there is an electric field which is directed from
positive plate to the negative plate.
o Magnitude of the electric field E =(V/d)
Where V=potential difference between the plates, d = distance between the plates.
E = (Q/Cd)
where Q=charge on the plates of the capacitor, Capacitance of the capacitor=C
=>E = (Q/ (Aε0d/d)), where A =area of the capacitor.
E=Q/(Aε0)
Direction of the electric field will be perpendicular to the selected surface i.e. if considering plate of the
capacitor as surface.
 o As E =0 outside the plates and E=(Q/(Aε0)) between the plates. There may be some electric field
between the plates because of which some current is present between the plates of the capacitor.
o Electric Flux through the surface = ΦE = (EA) =(QA)/ (Aε0) =(Q/ ε0)
• Assuming Q (charge on capacitor i.e. charging or discharging of the capacitor) changes with time current
will be get generated.
o Therefore current Id =(dQ/dt)
Where Id =displacement current
o =>Differentiating ΦE =(Q/ ε0) on both sides w.r.t time,
(dΦE/dt) =(1/ ε0) (dQ/dt)
where (dQ/dt) =current
Therefore (dQ/dt) = ε0 (d ΦE/dt)
=>Current was generated because of change of electric flux with time.
o Electric flux arose because of presence of electric field in the plates of the capacitor.
Id = (dQ/dt) = Displacement current
Therefore Change in electric field gave rise to Displacement current.
▪ Current won’t be 0 it will be Id.
▪ There is some current between the plates of the capacitor and there is some current at the
surface.
▪ At certain points there is no displacement current there is only conduction current and vice-versa.
➢ Maxwell corrected the Ampere’s circuital law by including displacement current.
➢ He said that there is not only the current existed outside the capacitor but also current known as
displacement current existed between the plates of the capacitor.
➢ Displacement current exists due to the change in the electric field between the plates of the capacitor.
➢ Conclusion:-Magnetic fields are produced both by conduction currents and by time varying fields.

Ampere-Maxwell Law
• As Maxwell was able to correct the shortcomings of the Ampere’s circuital law therefore the law came to
known as Ampere-Maxwell law.
• Current which is arising due to the flow of charges is known as conduction current.
It is denoted by IC.
• Current which is arising due to change in electric field is known as displacement current.
It is denoted by Id.
• Therefore I = Ic + Id, where I = total current
• Ampere-Maxwell Law stated that
o ∫dl = μ0 (Ic + Id)
o ∫dl = μ0 Ic + μ0 ε0 (d ɸE/dt)
o The above expression is known as Modified Maxwell Law
8.2 Displacement Current
• Consider a capacitor and outside the plates of the capacitor there is conduction current IC.
• Area between the plates i.e. inside the capacitor there is displacement current I d.
• Physical behaviour of displacement current is same as that of induction current.
• Difference between Conduction current and Displacement current:-
Conduction Current Displacement Current
It arises due to the fixed charges. It arises due to the change in electric field.
• For Static electric fields:-
Id=0.
• For time varying electric fields:-
Id ≠0.
• There can be some scenarios where there will be only conduction current and in some case there will be
only displacement current.
• Outside the capacitor there is only conduction current and no displacement current.
• Inside the capacitor there is only displacement current and no conduction current.
• But there can be some scenario where both conduction as well as displacement current is present i.e. I=
IC + Id.
• Applying modified Ampere-Maxwell law to calculate magnetic field at the same point of the capacitor
considering different amperial loop, the result will be same.

Ampere – Maxwell law: Consequences

Case 1 : Magnetic field is given as

o∫dl = μ0 Ic
o∫dl = μ0 Ic / 2πr
Case 2 : Magnetic field is given as

o ∫dl = μ0 Id
o ∫dl = μ0 Id / 2πr
Conclusion: -
1. The value of B is same in both cases.
2. Total current should be the same.
o Time varying electric field generates magnetic field given by (Ampere-Maxwell law)
o Consider 1st step up there is electric field between the plates and this electric field is varying with time.
o As a result there is displacement current and this displacement current gives rise to magnetic field.
o Time varying magnetic field generates electric field given by (Faraday-Lenz law)
o Therefore if there is electric field changing with time it generates magnetic field and if there is magnetic
field changing with time it generates electric field.
o Electromagnetic waves are based on the above conclusion.

Maxwell’s Equations
o Maxwell's equations describe how an electric field can generate a magnetic field and vice-versa. These
equations describe the relationship and behaviour of electric and magnetic fields.
o Maxwell gave a set of 4 equations which are known as Maxwell’s equations.
o According to Maxwell equations:-
o A flow of electric current will generate magnetic field and if the current varies with time magnetic field
will also give rise to an electric filed.
o First equation (1) describes the surface integral of electric field.
o Second equation (2) describes the surface integral of magnetic field.
o Third equation (3) describes the line integral of electric field.
o Fourth equation (4) describes line integral of magnetic field.

o Maxwell was the first to determine the speed of propagation of EM waves is same as the speed of light.
Experimentally it was found that:-
𝟏
𝒄 =
√𝝁𝟎 𝝐𝟎
Where μ0(permeability) and ε0(permittivity) and c= velocity of light.
o Maxwell’s equations show that the electricity, magnetism and ray optics are all inter-related to each other.
8.3 Electromagnetic Waves

• Electromagnetic waves are coupled time varying electric and magnetic fields that propagate in space.
• Electric field is varying with time, and it will give rise to magnetic field, this magnetic field is varying with
time and it gives rise to electric field and the process continues so on.
• These electric and magnetic fields are time varying and coupled with each other when propagating
together in space gives rise to electromagnetic waves.
• In the fig, red line represents the electric field and it varies in the form of a sine wave.
• The magnetic field as shown in the fig. represented by blue line.
• The magnetic field will be a sine wave but in a perpendicular direction to the electric field.
• These both give rise to electromagnetic field.
• If the electric field is along x-axis, magnetic field along y-axis, the wave will then propagate in the z-axis.
• Electric and magnetic field are perpendicular to each other and to the direction of wave propagation.
• Electric and magnetic fields which is time varying and coupled to each other they give rise to
electromagnetic waves.

8.3.1 Sources of Electromagnetic Waves (EM)

o EM waves are generated by electrically charged particle oscillates (accelerating charges).
o The electric field associated with the accelerating charge vibrates which generates the vibrating magnetic
field.
o These both vibrating electric and magnetic fields give rise to EM waves.
o If the charge is at rest, electric field associated with the charge will also be static. There will be no
generation of EM waves as electric field is not varying with time.
o When the charge is moving with uniform velocity, then the acceleration is 0. The change in electric field
with time is also constant as a result again there will be no electromagnetic waves generated.
o This shows that only the accelerated charges alone can generate EM waves.
o For example:
o Consider an oscillating charge particle, it will have oscillating electric field and which give rise to
oscillating magnetic field.
o This oscillating magnetic field in turn give rise to oscillating electric field and so on process continues.
o The regeneration of electric and magnetic fields are same as propagation of the wave.
o This wave is known as electromagnetic wave.
o The frequency of EM waves= the frequency of the oscillating particle.
8.3.2 Nature of EM waves
o EM waves are transverse waves.
o The transverse waves are those in which direction of disturbance or displacement in the medium is
perpendicular to that of the propagation of wave.
o The particles of the medium are moving in a direction perpendicular to the direction of propagation of
wave.

o In case of EM waves the propagation of wave takes place along x-axis, electric and magnetic fields are
perpendicular to the wave propagation.
o This means wave propagation  x-axis , electric field  y-axis, magnetic field  z-axis.
o Because of this EM waves are transverse waves in nature.
o Electric field of EM wave is represented as:
𝑬𝒚 = 𝑬𝟎 𝒔𝒊𝒏(𝒌𝒙– 𝝎𝒕)
Where Ey= electric field along y-axis and x=direction of propagation of wave.
o Wave number 𝒌 = (𝟐𝝅/𝝀)
o Magnetic field of EM wave is represented as:
𝑩𝒛 = 𝑩𝟎 𝒔𝒊𝒏(𝒌𝒙 − 𝝎𝒕)
Where BZ = electric field along z-axis and x=direction of propagation of wave.
8.3.3 Energy of EM wave
o As the EM waves propagate, they carry energy. Because of this property they have so many practical uses
in our day-to-day life.
o Energy in EM wave is partly carried by electric field and partly by magnetic field.
o Mathematically:
o Total energy stored per unit volume in EM wave, ET =Energy stored per unit volume by electric field +
Energy stored per unit volume stored in magnetic field.
1 1
𝐸𝑇 = ( ) (𝐸 2 𝜖0 ) + ( ) (𝐵 2 𝜇0 )
2 2
𝐸
o Experimentally it has been found that the; Speed of the EM wave =Speed of the light 𝑐 = 𝐵
=> B=(E/c)
1 1
∴ 𝐸𝑇 = ( ) (𝐸 2 𝜖0 ) + ( ) (𝐸 2 /𝑐 2 𝜇0 )
2 2
From Maxwell’s equations :-
1
𝑐 =
√ 𝜇0 𝜖 0
1 1 𝐸 2 𝜇0 𝜖 0
∴ 𝐸𝑇 = ( ) (𝐸 2 𝜖0 ) + ( ) ( )
2 2 𝜇0
1 1
𝐸𝑇 = ( ) (𝐸 2 𝜖0 ) + ( ) (𝐸 2 𝜇0 𝜖0 )
2 2
∴ 𝐸𝑇 = 𝐸 2 𝜖0

This is the amount of energy carried per unit volume by the EM wave.

8.3.4 Properties of EM waves

1. Velocity of EM waves in free space or vacuum is a fundamental constant.
o Experimentally it was found that the velocity of EM wave is same as speed of light(c=3x10 8m/s).
o The value of c is fundamental constant.
1
𝑐 =
√ 𝜇0 𝜖 0
2. No material medium is necessary for EM waves. But they can propagate within a medium as well.
o EM waves require time varying electric and magnetic fields to propagate.
1
o If the medium is present then velocity 𝑣 =
√𝜇𝜖
Where μ =permeability of the medium and ε=permittivity of the medium.
For example: -Spectacles. When light falls on glass of the spectacle, light rays pass through glass .i.e. Light
waves propagate through medium which is glass here.
3. EM waves carry energy and momentum.
o Total energy stored per unit volume in EM wave, 𝐸𝑇 = 𝐸 2 𝜖0 (partly carried by electric field and partly by
magnetic field).
o As EM waves carry energy and momentum, it becomes an important property for its practical purposes.
o EM waves are used for communication purposes, voice communication in mobile phones,
telecommunication used in radio.
4. EM waves exert pressure. As they carry energy and momentum, they exert pressure. The pressure exerted
by EM waves is known as Radiation pressure.
For example: -
The sunlight which we get from sun is in the form of visible light rays. These light rays are also part of EM
waves. If we keep our palm in sun, after some time, palm becomes warm and starts sweating. This
happens because sunlight is getting transferred in the form of EM waves and these EM waves carry
energy.
Suppose total energy transferred to the hand =E.
Momentum = (E/c) as c is extremely high, therefore momentum is very small. As momentum is very less,
pressure experienced is also very less. This is the reason due to which the pressure exerted by the sun is
not experienced by the hand.
8.4 Electromagnetic Spectrum
o Electromagnetic spectrum is the classification of EM waves according to their frequency or wavelength.
o Based on the wavelength EM waves are classified into different categories. This classification is known
as electromagnetic spectrum.
o Different categories of EM waves in decreasing order of their wavelength:-
o Radio waves > 0.1m
o Microwaves 0.1 m - 1mm
o Infra-Red 1mm – 700 nm
o Visible light 700nm – 400 nm
o Ultraviolet 400nm- 1nm
o X-rays 1nm – 10-3nm
o Gamma rays <10-3nm
o These 7 waves together constitute the electromagnetic spectrum.
Tip:-
o To remember the order of wavelength of each wave, we can just write the initial letter of all the waves
and they are in the order of decreasing wavelength.
o R (max wavelength), M, I, V, U, X and G (minimum wavelength).
o It can be remembered like this Red Man In Violet Uniform X Gun.

The electromagnetic spectrum, with common names for various part of it. The various regions do not have
sharply defined boundaries.

Electromagnetic energy of each wave in Electromagnetic Spectrum

• Electromagnetic waves energy can be described by frequency, wavelength or energy.
1. Frequency- Both micro and radio waves are described in terms of frequencies.
o Frequency is number of crests that pass a given point within one second.
o Consider a wave which has 3 crests which pass a point in 1 second. Therefore frequency=3Hz.
 2. Wavelength-Infrared and visible waves are generally described in terms of wavelength.
o Wavelength is the distance between consecutive crests or troughs.
o Wavelength can vary from small value to a large value.
o SI unit: - meter.

3. Energy- X-rays and Gamma rays are described in terms of energies.

o An EM wave can be described in terms of energy –in units of eV.
o eV is the amount of kinetic energy needed to move 1 electron through a potential of 1 volt.

• Moving along the EM spectrum energy increases as the wavelength decreases.

• Relation between Wavelength and Frequency:
c=νλ
Where λ =wavelength and ν= frequency.
=>λ = (c/ν)
𝐸 = ℎ𝜈 = (ℎ𝑐/𝜆)
=> 𝐸 ∝ 𝜈 𝑎𝑛𝑑 𝐸 ∝ (1/𝜆)
∴ from EM spectrum
o Decreasing order of wavelength  R, M, I, V, U, X and G
o In terms of increasing order of frequency  G, X, U, V, I, M, and R
8.4.1 Radio Waves
o Radio waves are produced by the accelerated motion of charges in conducting wires.
o Important application of radio waves is in:-
i. Radio and television communication systems.
ii. Mobile phones for voice communication.
o In electromagnetic spectrum the wavelength (λ) of radio waves is >0.1m.
o Radio waves are further classified into different bands:-
i. (Amplitude Modulated)AM band 530 kHz to 1710 kHz (lowest frequency band).They are similar to FM
channels.
ii. Short wave band – up to 54MHz
iii. TV waves band – 54MHz to 890MHz
iv. (Frequency Modulated)FM band – 88MHz to 108MHz
v. UHF band- Ultra high frequency(used for voice communication over cell phones)
8.4.2 Micro Waves
o Micro waves are short wavelength radio waves.
o They are produced by special vacuum tubes (klystrons/magnetrons/Gunn diodes).
o They are used in microwave ovens, and radar system in aircraft navigation.
RADAR Technology:
RADAR- Radio detection and ranging.
Different applications of RADAR:
a) Air traffic control:
For example: - To manage air traffic. The pilot should know any other aeroplane is present nearby or not.
The pilot should know the climatic conditions during take-off and landing.
Radar plays very important role in aircraft navigation.
b) Speed detection:
The instruments which are used to detect the speed of the vehicles which move on the roads uses radar
technology.
c) Military purposes
It helps to detect enemies and weapons.
d) Satellite tracking
In order to track satellites, radar technology is used.
Why Radio waves use micro waves:
o As they use short wavelength waves which are same as micro waves.
o They are invisible to humans. If we are able to see the waves which get transmitted it will be very
irritating.
o Even the smallest presence of microwaves is easy to detect.
Working of Radar Set:-
It consists of:
1. Transmitter: It transmits the microwaves.
2. Receiver: It receives the echo produced by the microwaves when they strike any object. When the receiver
receives the reflected ray then it is possible to track the presence of other object in the vicinity.
Microwave ovens
o The following are the properties because of which microwaves are very useful :-
o They have smaller wavelength.
o They get absorbed by water, fats and sugar.
Working of microwave oven:-
o In order to heat anything uniformly microwave ovens are used.
o Any food material will have water, sugar and fats in it.
o When we heat any food material inside the microwave, the microwaves penetrate inside the food.
o So the microwaves get absorbed by the water and the fat molecules.
o The molecules of the food material will start moving randomly with some frequency.
o This is same as providing some wave to the food material with the same frequency with which the
molecules start vibrating.
o This shows that the frequency of microwave matches with the frequency of the molecules.
o As all the molecules are set in random motion, temperature increases and food material gets heated
uniformly throughout.
o Object can be heated by 2 ways:
 a) Conduction of heat: It happens when anything is heated over gas burner.
b) Exciting the molecules: This technique is used in microwave oven.

8.4.3 Infrared waves

o Infrared waves often known as heat waves as they are produced by hot bodies.
o Their wavelength is lesser than both radio and micro waves.
o They readily get absorbed by water.
Applications: Infrared lamps/Infrared detector/LED in remote switches of electronic devices/Greenhouse
effect.
For example:
a) Fire gives out both visible light waves and infrared waves. The light rays are visible to us but the infrared
waves cannot be seen by us.
b) Humans also generate some infrared waves.
o There are some special glasses which have infrared detector to view infrared waves.
o The infrared lamps are used to heat food materials and sometimes washrooms.
o When we switch on the TV with the help of remote, there is an LED both on TV and on remote.
o The signal gets transferred from remote to TV via infrared waves.
Greenhouse Effect: Green house effect is an atmospheric heating phenomenon that allows incoming solar
radiation to pass through but blocks the heat radiated back from the Earth’s surface.
o Consider that the sun gives radiation in the form of visible light to the earth.
o When the visible light reaches the earth’s surface all the objects on the earth becomes hot.
o The visible light carries energy from sun and that energy gets transferred to all the objects present on the
earth.
o As a result of heat transfer all the objects gets heated up.
o These hot objects transmit infrared waves.
o The earth will reradiate the infrared waves.
o When these infrared waves try to go out of the atmosphere they get trapped by the greenhouse gases
(CO2, CH4, water vapour).
o As a result heat gets trapped inside the earth which results in an increase in temperature.
o The greenhouse effect makes earth warm because of which the temperature of the earth is suitable for
the survival of life on earth.
o Global warming is due to an increase in temperature of the environment, due to pollution.

8.4.4 Visible or Light rays

o Light waves are the most common form of EM wave.
o Their wavelength range is 4x1014 Hz-7x1014
o We are able to see everything because of light rays.
o The radiation which we get from sun is in the form of visible light.
o Most of the insects have compound eyes due to which they see not only the visible light but also the
ultraviolet rays.
o Snakes can even see the infrared rays.

8.4.5 Ultraviolet rays(UV rays)

It covers wavelengths ranging from about 4 × 10–7 m (400 nm) down to 6 × 10–10m (0.6 nm).
o The UVrays are produced by special lamps and very hot bodies (sun).
o UV rays have harmful effects on humans.
o UV lamps are used to kill germs in water purifiers.
o For example:-
o When UV rays fall on the skin of humans then it leads to the production of a pigment called melamine
which causes tanning of the skin.
o In order to protect from UV rays glasses are used, as they get absorbed by the glasses.
o UV rays help in LASER assisted eye surgery. As UV rays have very short wavelength so they can be focused
into narrow beam of light.
o The ozone layer which is present outside the atmosphere protects us from the harmful UV rays.
o Ozone has a property of reflecting the harmful UV rays. But due to the use of CFC (chlorofluorocarbon)
ozone layer is depleting. So if ozone layer gets depleted humans will get exposed to harmful UV rays
coming from the sun.
8.4.6 X-Rays
o X-Rays are produced by bombarding a metal target by high energy electrons.
o It is very important diagnostic tool.
o X-Rays have lesser wavelengths as compared to all other waves.
o Because of this X-Rays can easily penetrate inside the skin (low density material). It either gets
reflected or absorbed by the high density material (like bone).
o In any X-Ray, bones look darker and lighter area is skin.
o It is also used for cancer treatment.
o In cancer there is unwanted growth of the cells.
o In order to treat cancer the abnormal growth of cells should be stopped.
o The X-Rays have the ability to damage the living tissue.
This is how it helps in the treatment of cancer

8.4.7 Gamma Rays

o Gamma rays are produced in the nuclear reactions and also emitted by radioactive nuclei.
o It is also used in the treatment of the cancer.
o Gamma rays also have very small wavelength. So they help to kill the growth of unwanted living cells
which grow when the body is suffering from cancer.
 RAY OPTICS

RAY OPTICS

1. RECTILINEAR PROPAGATION OF LIGHT As shown in the figure, the angle between reflected ray and
incident ray is180 – 2i where i is the angle of incidence. Maximum
It is a well established fact that light is a wave. Although, a light deviation is 180°, when angle of incident i is zero.
wave spreads as it moves away from its source, we can approximate
its path as a straight line. Under this approximation, we show light 2.3 Law of Reflection in Vector Form
as a ray and the study of light as a ray is called ray optics or
geometrical optics. Say unit vector along incident ray = û .

1.1 Ray Unit vector along normal = n̂

The straight line path along which light travels in a homogeneous Unit vector along reflected ray = r̂
medium is called a ray. Then r̂ û  2 û . n̂ n̂

2. REFLECTION OF LIGHT
The phenomenon in which a light ray is sent back into the same
medium from which it is coming, on interaction with a boundary,
is called reflection. The boundary can be a rigid surface or just an
interface between two media.
2.1 Law of Reflection Laws of reflection remain the same whether the reflected surface is
plane or curved.
We have few angles to define before considering law of reflection
(i) Angle of incidence : The angle which the incident ray
makes with normal at the point of incidence.
(ii) Angle of reflection : The angle which the reflected ray
makes with normal at the point of incidence.
A reflected ray lies in the plane of incidence and has an angle of
reflection equal to the angle of incidence. ‘i = ‘r.
2.2 Deviation
2.4 Reflection by a plane surface
When a ray of light suffers reflection, its path is changed. The
angle between its direction after reflection and the direction before Suppose a reflecting surface is rotated by an angle T (say
reflection is called the deviation. anticlockwise), keeping the incident ray fixed then the reflect ray
rotates by 2T along the same sense, i.e., anticlockwise.
 RAY OPTICS

Magnification of a plane mirror is unity.

The image is formed behind the mirror. It is erect. Virtual and
2.5 Reflection from plane mirror laterally inverted.

When an object is placed in front of a plane mirror, its image can Image formation by two inclined mirrors, inclined at angle
be seen behind the mirror. The distance of the object from the = T [0, 180°]
mirror is equal to the distance of the image from the mirror. The object and all its images will always lie on a circle, having
center at the point of intersection of the two inclined mirrors, in a
two dimensional view.

3. OTHER IMPORTANT INFORMATIONS

(i) When the object moves with speed u towards (or away) from
the plane mirror then image also moves toward (or away) with
speed u. But relative speed of image w.r.t. object is 2u.
(ii) When mirror moves towards the stationary object with speed
u, the image will move with speed 2u.
 RAY OPTICS

4. SPHERICAL MIRRORS Paraxial rays : Rays which are close to principal axis and make
small angles with it, i.e., they are nearly parallel to the axis, are
A spherical mirror is a part of sphere. If one of the surfaces is
called paraxial rays. Our treatment of spherical mirrors will be
silvered, the other surface acts as the reflecting surface. When
restricted to such rays which means we shall consider only mirrors
convex face is silvered, and the reflecting surface is concave, the
of small aperture. In diagrams, however, they will be made larger
mirror is called a concave mirror. When its concave face is silvered
for clarity.
and convex face is the reflecting face, the mirror is called a convex
mirror. Images formed by spherical mirrors
Let us consider various cases depending on the nature of the
object and the image
(i) Real object and real image

(a)

Before the discussion of reflection by curved mirrors, you shall

carefully comprehend the meaning of following terms
(i) Centre of curvature : Centre of curvature is the centre of
sphere of which, the mirror is a part.
(ii) Radius of curvature : Radius of curvature is the radius of
sphere of which, the mirror is a part.
(b)
(iii) Pole of mirror : Pole is the geometric centre of the mirror.
(iv) Principal axis : Principal axis is the line passing through the
pole and centre of curvature.
(v) Normal : Any line joining the mirror to its centre of curvature
is a normal. (ii) Real object and virtual image

(a)

(b)
 RAY OPTICS

object two of the following four rays are drawn passing through
the object. To construct the image of an extended object the image
of two end points is only drawn. The image of a point object lying
on principles axis is formed on the principal axis itself. The four
(c) rays are as under :

(d)

(iii) Virtual object and real image

Ray 1 : A ray through the centre of curvature which strikes the

(a)
mirror normally and is reflected back along the same path.
Ray 2 : A ray parallel to principal axis after reflection either actually
passes through the principal focus F or appears to diverge from it.
Ray 3 : A ray passing through the principal focus F or a ray which
appears to converge at F is reflected parallel to the principal axis.
Ray 4 : A ray striking at pole P is reflected symmetrically back in
the opposite side.
4.1 Sign conventions
(b)
(i) All distances are measured from the pole.
(ii) Distances measured in the direction of incident rays are
taken as positive while in the direction opposite of incident
rays are taken negative.
(iv) Virtual object and virtual image (iii) Distances above the principle axis are taken positive and
below the principle axis are taken negative.

Ray diagrams
We shall consider the small objects and mirrors of small aperture Same sign convention are also valid for lenses.
so that all rays are paraxial. To construct the image of a point

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RAY OPTICS

Position, size and nature of image formed by the spherical mirror

Use following sign while solving the problem

4.2 Relation between f and R

In figure, P is pole, C is centre of curvature and F is principal focus of a concave mirror of small aperture. Let a ray of light AB be
incident on the mirror in a direction parallel to the principal axis of the mirror. It gets reflected along. BF. Join CB. It is normal to
the mirror at B.
 RAY OPTICS

i.e., F is the centre of PC

1
? PF = PC, Using sign conventions,

PF = – f and PC = –R.
Therefore, –f = –R/2 or f = R/2
i.e., focal length of a concave mirror is equal to half the
? ‘ABC = i, angle of incidence radius of curvature of the mirror.
‘CBF = r, angle of reflection
4.3 Deriving the Mirror Formula
Now ‘BCF = ‘ABC = i (alternate angles)
Mirror formula can be derived for any of the cases of image
In 'CBF, as i = r (law of reflection)
formation shown before. When we derive a formula, we keep in
? CF = FB mind the sign conventions and substitute each value with sign.
But FB = FP (' aperture is small) This makes a formula suitable to be applied in any case. Here, we
? CF = FP shall derive the formula for two cases.

Real object and real image Real object and virtual image
(concave mirror) (convex mirror)

PO = – u (distance of object) PO = – u (distance of object)

PC = – R (radius of curvature) PI = + v (distance of image)
PI = – v (distance of image) PC = + R (radius of curvature)
In 'OAC, J= D + T ...(i) In 'OAC, T = D + J ...(i)
In 'OAI, E= D + 2T ...(ii) In 'OAI, 2T = D + E ...(ii)
From (i) and (ii) From (i) and (ii)
2 (J – D) = E – D 2 (D + J) = D + E
ŸE+ D = 2J ŸE– D= 2J

AP AP AP AP AP AP
E ,D ,J E ,D ,J
PI PO PC PI PO PC

AP AP 2AP AP AP 2AP
 
PI PO PC PI PO PC

1 1 1 1 1 1 1 2 1 1 1
 Ÿ   Ÿ 
v u R v u f v u R v u f
 RAY OPTICS

While deriving the above result, if we do not use sign convention, 1 1 1 –2 –2

results obtained will be different for different cases. From ,  we have –v dv –u du = 0
v u f
4.4 Magnification 2
dv §v·
or ¨ ¸
The linear magnification produced by a mirror is defined as du ©u¹
height of image 2
dv §v·
height of object or mL  ¨ ¸ m2
du ©u¹
I BBc If we differentiate the mirror formula
m
O AA c 1 1 1

v u f
with respect to time, we get
dv du
 v 2 .  u 2 0 (as f = constant)
dt dt

dv § v2 · du
or ¨¨ 2 ¸
¸ dt ...(iii)
dt ©u ¹
As every part of mirror forms a complete image, if a part of the
mirror is obstructed, full image will be formed but intensity will be
PB = – v (distance of image) reduced.
PA = – u (distance of object) 5. REFRACTION OF LIGHT
BcB BP
Now, 'A’AP ~ 'B’BP Ÿ
AcA AP

 PB  v v
Ÿ m
PA u u

By mirror formula, 1  1 1
v u f
When a ray of light is incident on the boundary between two
v v v f v transparent media, a part of it passes into the second medium
Ÿ 1  Ÿ m 1
u f f f with a change in direction.
1 1 1 u u f This phenomenon is called refraction.
Also,  Ÿ 1 Ÿm
v u f v f f u
5.1 Refractive Index
v f v f
?m Absolute refractive index of a medium is defined by the ratio of
u f f u
c
The magnification is negative when image is inverted and speed of light in vacuum to speed of light in the medium P ,
v
positive when image is erect.
where c is speed of light in vacuum and v is the speed of light in
If an object is placed with its length along the principal axis, the medium.
then so called longitudinal magnification becomes,
5.2 Law of Refraction (Snell’s Law)
I § v  v1 · dv
mL ¨¨ 2 ¸¸  (for small objects) A refracted ray lies in the plane of incidence and has an angle of
O © u 2  u1 ¹ du
refraction related to angle of incidence by P1sin i = P2 sin r. Where,
 RAY OPTICS

(i) i = angle of incidence in medium 1

(ii) P1 = refractive index of medium 1 (it is a dimensionless constant)
(iii) r = angle of refraction in medium 2
(iv) P2 = refractive index of medium 2
(v) If P1 = P2, then r = i. The light beam does not bend
(vi) If P1 > P2, then r > i. Refraction bends the light away from normal
(vii) If P1 < P2, then r < i. Refraction bends the light towards the normal
A medium having greater refractive index is called denser medium
while the other medium is called rarer medium.
We shall derive the expression for small angles (or you can say
that the object is being seen from top). By Snell’s law,
P2 × sin i = P1 × sin r or, P2 × i = P1 × r

AB AB AB AB P2 P1
i ,r Ÿ P2 u P1 u Ÿ
R A R A R A

The following possibilities may arise.

(i) When observer is in air and the object is in a medium of
refractive index P,

P 1 R
You have, ŸA
R A P

The three conditions required to find the unit vector along the
refracted ray = r (provided we are given the unit vector along the
incident ray = u, and the normal unit vector shown in the figure,
(ii) When observer is in a medium of refractive index P and
from medium–1 towards medium–2) are
the object is in air, you have
1. |r| = 1
I P
2. Snell’s law ŸA PR
R A
3. u, n and r are coplanar Ÿ STP = 0 = r . (u × n)

cos i = (u . n) ; cos r = (r . n)

5.3 Single Refraction from a Plane Surface

Real and Apparent Depth

When an object placed in a medium is seen from another medium,

its apparent position is different from the actual position. Consider
the following figure.

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 RAY OPTICS

5.4 Shift due to a Glass Slab (Double Refraction AC t

from Plane Surfaces) Proof : AB
cos r cos r
(i) Normal Shift : Here, again two cases are possible. (as AC = t)

t
Now, d = AB sin (i – r) = cos r [sin i cos r – cos i sin r]

or d = t [sin i – cos i tan r] ...(i)

sin i sin i
Further P or sin r
sin r P

sin i
? tan r
P  sin 2 i
2

An object is placed at O. Plane surface CD forms its image (virtual)

Substituting in eq. (i), we get,
at I1. This image acts as object for EF which finally forms the
image (virtual) at I. Distance OI is called the normal shift and its ª º
cos i
value is, d «1  » t sin i
« P 2  sin 2 i »¼
¬
§ 1·
OI ¨¨1  ¸¸ t
© P¹ Hence Proved.
This can be proved as under : Exercise : Show that for small angles of incidence,
Let OA = x then AI1 = Px (Refraction from CD)
§ P 1·
BI1 = Px + t d = ti ¨¨ ¸¸ .
© P ¹
BI1 t
BI x (Refraction from EF) Apparent distance from observer
P P
§ t· § h1 h 2 hn ·
? OI = (AB + OA) – BI t  x  ¨¨ x  ¸¸ = P obser ¨¨ P  P  ...... P ¸¸
© P¹ © 1 2 n ¹

§ 1· 5.5 Total Internal Reflection

¨¨1  ¸¸ t Hence Proved.
© P¹
Consider an object placed in a denser medium 2 (having refractive
(ii) Lateral Shift : We have already discussed that ray MA is
index P2) being seen from a rarer medium 1 (having refractive
parallel to ray BN. But the emergent ray is displaced
index P1)
laterally by a distance d, which depends on P, t and i and
its value is given by the relation,
§ cos i ·
d t ¨1  ¸ sin i
¨ ¸
© P  sin 2 i
2
¹

Different rays from the object are shown. As we move from A

towards C, angle of incidence goes on increasing. Therefore, the
angle of refraction goes on increasing. At B, angle of refraction
approaches 90°. This is called critical condition. After B, angle of
 RAY OPTICS

incidence increases, but angle of refraction cannot be greater

­° J  r, in fig.
than 90°. Therefore after point B, refraction of light does not take i D  E, E ®
place, only reflection of light takes place. This is called total internal °̄r  J , in fig. II
reflection.
P1 D  E P 2 E B J in fig. I and fig. II
5.6 Refraction through Curved Surfaces
Ÿ P 1D r P 2 J P 2  P1 E,
Spherical Refracting Surfaces
A spherical refracting surface is a part of a sphere. For example,
the plane face of cylindrical glass rod is curved to form a spherical
shape (as shown in the figure).

As aperture is small D | tan D, E | tan E , J | tan J

P1 tan D r P2 tan J P 2  P1 tan E

P1 P 2 P 2  P1
r ...(i)
P cO P c P cC
Applying sign convention i.e., u = – P’O
v = P’I and – P’I, in fig. I and fig. II respectively R = P’C
Substituting the above values in equation (i), we get
P o Pole of refracting surface P 2 P1 P 2  P1
 (For both fig. I and fig. II)
C o Centre of curvature v u R
PC o Radius of curvature 5.8 Linear Magnification for Spherical Refracting Surface
Principal axis : The line joining pole and centre of curvature.
A c Bc
m 
5.7 Relation between Object Distance and Image AB

Distance Refraction at Spherical Surfaces sin i P2

Now,
sin r P1
Consider the point object O placed in the medium with refractive
index equal to P1. As P1sin i = P2sin r and for small aperture i, r o 0

As i, r o 0, i | sin i | tan i, r | sin r | tan r

tan i P2 AB / PA P2
i.e. paraxial rays ŸP1 i = P2 r or
tan r P1 A cBc / PA c P1
 RAY OPTICS

towards the point, after refraction becomes parallel to

A cBc  PA c / P 2
Ÿ principal axis.
AB PA / P1

v / P2
Hence, m
u / P1

6. THIN LENS
A thin lens is defined as a portion of transparent refracting medium
bounded by two surfaces. One of the two surfaces must be curved.
Following figures show a number of lenses formed by different
refracting surfaces.
A lens is one of the most familiar optical devices for a human
being. A lens is an optical system with two refracting surfaces.
The simplest lens has two spherical surfaces close enough together (b) Second principal focus F2 : It is a point on principal axis,
that we can neglect the distance between them (the thickness of such that a ray moving parallel to principal axis, after
the lens). We call this a thin lens. refraction converges or diverges towards the point.

(vi) Focal Length : The distance between optical centre and

second principal focus is focal length. Assumptions and
sign conventions are same as these of mirrors with optical
centre C in place of pole P of the mirror.

6.2 Ray diagram

6.1 Terms Related with Lenses To construct the image of a small object perpendicular to the axis
of a lens, two of the following three rays are drawn from the top of
(i) Centre of curvature (C1 and C2) : The two bounding surfaces the object.
of a lens are each part of a complete sphere. The centre of the 1. A ray parallel to the principal axis after refraction passes
sphere is the centre of curvature. through the principal focus or appears to diverge from it.
(ii) Radius of curvature (R1 and R2) : The radii of the curved
surfaces forming the lens are called radii of curvature.
(iii) Principal axis : The line joining the two centres of
curvature is called principal axis.
(iv) Optical centre : A point on the principal axis of the lens
from which a ray of light passes undeviated.
(v) Principal foci : There are two principal foci of a lens.
(a) First principal focus F1 : It is a point on the principal axis,
such that a ray, diverging from the point or converging
 RAY OPTICS

2. A ray through the optical centre P passes undeviated 3. A ray passing through the first focus F1 become parallel
because the middle of the lens acts like a thin parallel- to the principal axis after refraction.
sided slab.

6.3 Image formation by Lens

Minimum distance between an object and it’s real image formed by a convex lens is 4f.
Maximum image distance for concave lens is it’s focal length.
RAY OPTICS

6.4 Lens maker’s formula and lens formula determine the values of R1 and R2 that are needed for a given
refractive index and a desired focal length f.
Consider an object O placed at a distance u from a convex lens as
Combining eqs. (iii) and (v), we get
shown in figure. Let its image I after two refractions from spherical
surfaces of radii R1 (positive) and R2 (negative) be formed at a 1 1 1
 ...(vi)
distance v from the lens. Let v1 be the distance of image formed v u f
by refraction from the refracting surface of radius R1. This image Which is known as the lens formula. Following conclusions can
acts as an object for the second surface. Using, be drawn from eqs. (iv), (v) and (vi).
1. For a converging lens, R1 is positive and R2 is negative.

§ 1 1 ·
Therefore, ¨¨  ¸¸ in eq. (v) comes out a positive
R
© 1 R 2 ¹

quantity and if the lens is placed in air, (P – 1) is also a

positive quantity. Hence, the focal length f of a converging
lens comes out to be positive. For a diverging lens however,
R1 is negative and R2 is positive and the focal length f
becomes negative.
P 2 P1 P 2  P1
 twice, we have
v u R

P 2 P1 P 2  P1
or  ...(i)
v1 u R1

P1 P 2 P1  P 2
and  ...(ii)
v v1 R2

Adding eqs. (i) and (ii) and then simplifying, we get

1 1 § P2 ·§ 1 1 ·
 ¨¨  1¸¸ ¨¨  ¸¸ ...(iii)
v u © P1 ¹ © R1 R 2 ¹

This expression relates the image distance v of the image formed

by a thin lens to the object distance u and to the thin lens properties
(index of refraction and radii of curvature). It is valid only for
paraxial rays and only when the lens thickness is much less then
R1 and R2. The focal length f of a thin lens is the image distance
2. Focal length of a mirror (fM = R/2) depends only upon the
that corresponds to an object at infinity. So, putting u = f and
radius of curvature R while that of a lens [eq. (iv)] depends
v = f in the above equation, we have
on P1, P2, R1 and R2. Thus, if a lens and a mirror are immersed
in some liquid, the focal length of lens would change while
1 § P2 ·§ 1 1 ·
¨ ¸¨ ¸
¨ P  1¸ ¨ R  R ¸ ...(iv) that of the mirror will remain unchanged.
f © 1 ¹© 1 2 ¹
3. Suppose P2 < P1 in eq. (iv), i.e., refractive index of the
If the refractive index of the material of the lens is P and it is placed medium (in which lens is placed) is more than the refractive
in air, P2 = P and P1 = 1 so that eq. (iv) becomes §P ·
index of the material of the lens, then ¨¨ 2  1¸¸ becomes a
P
©  ¹
1 § 1 1 ·
P  1 ¨¨  ¸¸ ...(v)
f negative quantity, i.e., the lens changes its behaviour. A
© R1 R 2 ¹
converging lens behaves as a diverging lens and vice-
This is called the lens maker’s formula because it can be used to versa. An air bubble in water seems as a convex lens but
 RAY OPTICS

behaves as a concave (diverging) lens. The shorter the focal length of a lens (or a mirror) the more it
converges or diverges light. As shown in the figure,
f1 < f2
and hence the power P1 > P2, as bending of light in case 1 is more
than that of case 2. For a lens,

1
P (in dioptre) = and for a mirror,,
f metre
6.5 Magnification
The lateral, transverse of linear magnification m produced by a lens 1
is defined by, P (in dioptre) =
f metre
height of image I
m Following table gives the sign of P and f for different type of lens
height of object O
and mirror.
A real image II’ of an object OO’ formed by a convex lens is shown
in figure. 8. COMBINATION OF LENS
height of image IIc v (i) For a system of lenses, the net power, net focal length and
height of object OOc u magnification given as follows :
P = P1 + P2 + P3 ............,

1 1 1 1
   ...........,
F f1 f 2 f 3

m = m1 × m2 × m3 × ............
(ii) When two lenses are placed co-axially at a distance d from
each other then equivalent focal length (F).

Substituting v and u with proper sign,

IIc I v I v
or m
OOc O u O u

v
Thus, m
u

7. POWER OF AN OPTICAL INSTRUMENT

1 1 1 d
By optical power of an instrument (whether it is a lens, mirror or a  
refractive surface) we mean the ability of the instrument to deviate F f1 f 2 f1f 2 and P = P1 + P2 – dP1P2
the path of rays passing through it. If the instrument converges
the rays parallel to the principal axis its power is said positive and 9. CUTTING OF LENS
if it diverges the rays it is said a negative power.
(i) A symmetric lens is cut along optical axis in two equal
parts. Intensity of image formed by each part will be same
as that of complete lens.
(ii) A symmetric lens is cut along principle axis in two equal
parts. Intensity of image formed by each part will be less
compared as that of complete lens. (aperture of each part
is 1 / 2 times that of complete lens)
 RAY OPTICS

? A + ‘MPN = 180° ...(i)

In triangle MNP, r1 + r2 + ‘MPN = 180° ...(ii)
From eqs. (i) and (ii), we have
r 1 + r2 = A ...(iii)

11.1 Deviation

Deviation G means angle between incident ray and emergent ray.

In reflection, G= 180 – 2i = 180 – 2r
in refraction, G= |i – r|

10. SILVERING OF LENS

On silvering the surface of the lens it behaves as a mirror. The
1 2 1
focal length of the silvered lens is  where
F f1 f m In prism a ray of light gets refracted twice one at M and
f1 = focal length of lens from which refraction takes place (twice) other at N. At M its deviation is i1 – r1 and at N it is i2 – r2.
These two deviations are added. So the net deviation is,
fm = focal length of mirror from which reflection takes place.
G= (i1 – r1) + (i2 – r2) = (i1 + i2) – (r1 + r2) = (i1 + i2) – A
11. PRISM
Thus, G= (i1 + i2) – A ...(iv)
A prism has two plane surfaces AB and AC inclined to each other
sin i1
as shown in figure. ‘A is called the angle of prism or refracting (i) If A and i1 are small : P , therefore, r1 will also be
angle. sin r1
small. Hence, since sine of a small angle is nearly equal to
the angle is radians, we have, i1 = Pr1
Also, A = r1 + r2 and so if A and r1 are small r2 and i2 will
sin i 2
also be small. From P , we can say, i2 = Pr2
sin r2

Substituting these values in eq. (iv), we have

G = (Pr1 + Pr2) – A = P (r1 + r2) – A = PA – A
or G= (P – 1) A ...(v)
(ii) Minimum deviation : It is found that the angle of deviation
The importance of the prism really depends on the fact that the G varies with the angle of incidence i1 of the ray incident
angle of deviation suffered by light at the first refracting surface, on the first refracting face of the prism. The variation is
say AB (in 2-dimensional figure) is not cancelled out by the shown in figure and for one angle of incidence it has a
deviation at the second surface AC (as it is in a parallel glass minimum value G min. At this value the ray passes
slab), but is added to it. This is why it can be used in a spectrometer, symmetrically through the prism (a fact that can be proved
an instrument for analysing light into its component colours.
theoretically as well as be shown experimentally), i.e., the
General Formulae angle of emergence of the ray from the second face equals
In quadrilateral AMPN, ‘ AMP + ‘ANP = 180° the angle of incidence of the ray on the first face.
 RAY OPTICS

Now, if minimum value of r2 is greater than Tc then obviously all

values of r2 will be greater than Tc and TIR will take place under all
conditions. Thus, the condition of no emergence is, (r2)min > Tc or
A – Tc > T

T
or A! ...(xii)
2

11.3 Dispersion and deviation of light by a prism

White light is a superposition of waves with wavelengths

extneding throughout the visible spectrum. The speed of light in
vacuum is the same for all wavelengths, but the speed in a material
i2 = i1 = i ...(vi) substance is different for different wavelengths. Therefore, the
It therefore, follows that index of refraction of a material depends on wavelength. In most
r1 = r2 = r ...(vii) materials the value of refractive index P decreases with increasing
From eqs. (iii) and (vii) wavelength.

A
r
2
Further at, G = Gm = (i + i) – A

A  Gm
or i ...(viii)
2
sin i
? P
sin r
If a beam of white light, which contains all colours, is sent through
§ A  Gm ·
sin ¨ ¸ the prism, it is separated into a spectrum of colours. The spreading
or P © 2 ¹ ...(ix) of light into its colour components is called dispersion.
A
sin
2 11.4 Dispersive Power

11.2 Condition of no emergence When a beam of white light is passed through a prism of
transparent material light of different wavelengths are deviated
In this section we want to find the condition such that a ray of
by different amounts. If Gr, Gy and Gv are the deviations for red,
light entering the face AB does not come out of the face AC for
yellow and violet components then average deviation is measured
any value of angle i1, i.e., TIR takes place on AC
by Gy as yellow light falls in between red and violet. Gv – Gr is
r 1 + r2 = A ? r2 = A – r1 called angular dispersion. The dispersive power of a material is
or (r2)min = A – (r1)max ...(x) defined as the ratio of angular dispersion to the average deviation
when a white beam of light is passed through it. It is denoted by
Now, r1 will be maximum when i1 is maximum and maximum
Z. As we know
value of i1 can be 90°.
G= (P – 1) A
sin i1 sin 90q
Hence, P
max

sin r1 max sin r1 max

1
? sin r1 sin T ? (r1)max = T
max
P

? From eq. (x), (r2)min = A – Tc ...(xi)

RAY OPTICS

This equation is valid when A and i are small. Suppose, a beam of

white light is passed through such a prism, the deviation of red,
yellow and violet light are
Gr = (Pr – 1) A, Gy = (Py – 1) A and Gv = (Pv – 1) A
The angular dispersion is Gv – Gr = (Pv – Pr) A and the average deviation
is Gy = (Py – 1) A. Thus, the dispersive power of the medium is,

P v  Pr Coma can be reduced by carefully working out the curvature

Z ...(i)
Py  function, or by blocking off the rays that create the ‘tail’ of the
comet shaped image.
12. MONOCHROMATIC ABERRATIONS “ Astigmatism : The shape of the image is different at different
IN MIRRORS AND LENSES distances. Suppose a point object is placed off the optical
axis of a converging lens. Then, as a lateral screen is moved
(INDEPENDENT OF WAVELENGTH) along the axis, at one point, the image is almost a line. At
other positions of the screen, the image changes into an
Spherical aberration : because of the fact that all rays are not
different shapes at different locations of the screen.
paraxial. The image of a point object formed by a spherical mirror
is a surface, whose 2-D view is called a ‘caustic curve’. When a Astigmatism can be reduced by using non-spherical
real image is seen on a screen and the screen is moved forward/ surfaces of revolution-such corrected lenses are called
‘anastigmatic’.
backward slightly, a disc image is formed which becomes smallest
at one position. The periphery of this smallest disc is called ‘the “ Curvature : Consider a point object placed off the optical
circle of least confusion’. Lenses too exhibit spherical aberration. axis of a lens. We have seen that image is spread out laterally
as well as longitudionally, with individual defects in each
We can reduce it by blocking non-paraxial rays but this reduces
direction. However, the best image is obtained on a curved
the brightness of the image. A ring shaped black paper is affixed
surface and not on a plane screen. This phenomenon is
on the lens so that only those rays pass through the ‘hole’ in the called ‘curvature’.
ring, which are paraxial. Parabolic mirrors do not exhibit any
“ Distortion : A square lateral object has images, which are
spherical aberration, hence all expensive reflecting telescopes use
either ‘barrel shaped’ or ‘curving in’ as shown. This is
parabolic mirrors.
because the lateral magnification itself depends on the actual
In lenses, spherical aberration can be reduced by using a distance of a portion of the object from the optical axis.
combination of convex and concave lenses, which cancel out These different magnifications of different portions produce
each other’s aberrations. this effect.

13. CHROMATIC ABERRATIONS IN LENSES

(DEPENDENT ON WAVELENGTH)
Coma : Consider a point object placed ‘off’ the optical axis. Most
These aberrations are absent in mirrors. In lenses, the focal length
of the rays focus at a single point, but others form images at
depends on the refractive index, which is different for different
different points so that the overall image is like that of a ‘comet’
colors. Hence, colored images are formed at different points if
( ) having a sharp ‘point’ image followed by a trail like that of a white light is emitted by the object. A proper combination of convex
comet. and concave lenses exactly cancel out each others chromatic
 RAY OPTICS

aberration (for light having two wavelengths only) so that the Magnifying power of a simple microscope is defined as the
final image is not split into colored images. Such a combination is ratio of the angles subtended by the image and the object
called an ‘achromatic doublet’. The distance along the optical on the eye, when both are at the least distance of distinct
axis between images of violet and red is called ‘axial or longitudional vision from the eye.
chromatic aberration’ = LCA (say):
E
For an incident parallel beam of white light, image distance = focal By definition, Magnifying power m ...(1)
D
length. From lens-makers formulae:
-df/f = dn/(n - 1) = Z = dispersive power of lens | (nV – nR)/(n – 1) For small angles expessed in radians, tan T | T
Ÿ LCA = 'f | Zf. For two thin lenses in contact, (1/F) = (1/f1) + D | tan D and E | tan E
?
(1/f2). Therefore, dF = 0 ŸZ1/f1 = –Z2/f2 o achromatic lens. An
achromatic ‘doublet’ or lens combination can be made by placing tan E
two thin lenses in contact, with one converging and the other ? m ...(2)
tan D
diverging, made of different materials.
For lateral objects, images of different colors have different sizes AB
as magnification itself depends on the focal length, which is In 'ABC, tan E
CB
different for different colors. The difference in the size of lateral
images of violet and red colors is called ‘lateral chromatic A1 B' AB
aberration’. In 'A1B’C, tan D
CB' CB'
Putting in (2), we get

AB CB' CB' v v
m u ...(3)
CB AB CB u u
where, CB’ = – v, distance of image from the lens, CB = –u,
distance of object from the lens

1 1 1
From lens formula, 
v u f
Multiply both sides by v

v v
1
u f
14. OPTICAL INSTRUMENTS
v
14.1 Simple Microscope or Magnifying Glass using (3), 1  m
f
A simple microscope is used for observing magnified images
of tiny objects. It consists of a converging lens of small v
or m 1
focal length. A virtual, erect and magnified image of the f
object is formed at the least distance of distinct vision from
the eye held close to the lens. That is why the simple § d·
But v = – d, ? m ¨1  ¸
microscope is also called a magnifying glass. © f¹

14.2 Compound Microscope

A compound microscope is an optical instrument used for

observing highly magnified images of tiny objects.
Construction : A compound microscope consists of two
converging lenses (or lens system); an objective lens O of
very small focal length and short aperture and an eye piece
E of moderate focal length and large aperture.
RAY OPTICS

where d is C2B’’ = least distance of distinct vision, fe is focal

length of eye lens. And

A' B' distance of image A' B' from C1

m0
AB distance of object AB from C1

C1 B' v0
C1 B  u0

Putting these values in (3), we get

v0 § d · v0 § d ·
m ¨1  ¸ ¨1  ¸
 u0 ¨ f ¸ | u0 | ¨© f e ¸ ...(4)
© e ¹ ¹

Magnifying power of a compound microscope is defined as As the object AB lies very close to F0, the focus of objective
the ratio of the angle subtended at the eye by the final lens, therefore,
image to the angle subtended at the eye by the object, when u0 = C1B | C1F0 = f0 = focal length of objective lens.
both the final image and the object are situated at the least As A’B’ is formed very close to eye lens whose focal length
distance of distinct vision from the eye. is also short, therefore,
In figure, C2B’’ = d. Imagine the object AB to be shifted to v0 = C1B’ | C1C2 = L = length of microscope tube.
B’’ so that it is at a distance d from the eye. If ‘A’’ C2 B’’
1 Putting in (4), we get
= E and ‘A1C2B’’ = D, then by definition,
L § d · L § d ·
E m ¨1  ¸ ¨1  ¸ ...(5)
Magnifying power, m ...(1)  f0 ¨ f ¸ | f 0 | ¨© f e ¸
D © e ¹ ¹

For small angles expressed in radians, tan T|T 14.3 Astronomical Telescope
? D| tan D and E| tan E An astronomical telescope is an optical instrument which is
used for observing distinct image of heavenly bodies like
tan E stars, planets etc.
From (1), m ...(2)
tan D
It consists of two lenses (or lens systems), the objective
A ' ' B' ' lens, which is of large focal length and large aperture and
In 'A’’B’’C2, tan E the eye lens, which has a small focal length and small
C 2 B' '
aperture. The two lenses are mounted co-axially at the free
Bcc AB ends of the two tubes.
In 'A1B’’C2, tan D 1
C 2 Bcc C 2 Bcc
Putting in (2), we get
ccBcc C 2 Bcc ccBcc ccBcc A' B'
m u u
C 2 Bcc AB AB A ' B' AB
m = me × m0
ccBcc
where m e , magnification produced by eye lens,
A' B'

A ' B' However, in astronomical telescope, final image being

and m 0 , magnification produced by objective lens.
AB inverted with respect to the object does not matter, as the
astronomical objects are usually spherical.
§ d ·
Now, m e ¨1  ¸ Magnifying Power of an astronomical telescope in normal
¨ f ¸
© e ¹ adjustment is defined as the ratio of the angle subtended at
 RAY OPTICS

the eye by the final image to the angle subtended at the eye, ? ‘A’C1B’ = D
by the object directly, when the final image and the object Further, let ‘A’’C2B’’ = E, where C2B’’ = d
both lie at infinite distance from the eye.
E
? By definition, Magnifying power, m ...(4)
E D
Magnifying power, m ...(1)
D As angles D and E are small, therefore, E| tan E and D| tan D
As angles D and E are small, therefore, D| tan D and E tan E
| tan E. From (4), m ...(5)
tan D
tan E A ' B'
From (1), m ...(2) In 'A’B’C2, tan E
tan D C 2 B'
A ' B'
In 'A’B’C2, tan E A ' B'
C 2 B' In 'A’B’C1, tan D
C1B'
A 'B'
In 'A’B’C1, tan D A' B' C1B'
C1B ' Putting in (5), we get m u
C 2 B' A' B'
A ' B' C1B' C1B'
Put in (2), m u C1 B' f0
C 2 B' A' B' C 2 B' m ...(6)
C 2 B'  ue
f0 where C1B’ = f0 = focal length of objective lens
or m ...(3)
 fe C2B’ = – ue, distance of A’B’, acting as the object for
eye lens.
where C1B’ = f0 = focal length of objective lens.
C2B’ = –fe = focal length of eye lens. 1 1 1
Now, for eye lens, 
v u f
Negative sign of m indicates that final image is inverted.
Taking ve = –d, u = –ue and f = + fe, we get
Memory Note
(i) In normal adjustment of telescope, distance between the 1 1 1

objective lens and eye lens = (f0 + fe).  d  ue fe
(ii) Angular magnification produced by the telescope = ED.
1 1 1 1 § fe ·
Clearly, visual angle E is much larger as compared to D.  ¨1  ¸
ue fe d fe © d¹
Figure shows the course of rays in an astronomical
telescope, when the final image is formed at the least distance f0 § f e ·
of distinct vision (d) from the eye) Putting in (6), we get m  ¨1  ¸
fe © d¹
Discussion :
(i) As magnifying power is negative, the final image in an
astronomial telescope is inverted i.e. upside down and left
turned right.
(ii) As intermediate image is between the two lenses, cross wire
(or measuring device) can be used.
(iii) In normal setting of telescope, final image is at inifiny.
Magnifying power is minimum.
Magnifying power of an astronomical telescope is defined When final image is at least distance of distinct vision,
as the ratio of the angle subtended at the eye by the final magnifying power is maximum. Thus
image at the least distance of distinct vision to the angle
subtended at the eye by the object at infinity, when seen ª f0 º f § f ·
directly. (M.P.)min. = – « »; (M.P.)max. = – 0 ¨1  e ¸
f
¬ e¼ fe © d¹
WAVE OPTICS

WAVE OPTICS
1. WAVEFRONT 2. HUYGENS’S PRINCIPLE
A source of light sends out disturbance in all directions. In a Huygen’s principle is a geometrical construction, which is used
homogeneous medium, the disturbance reaches all those to determine the new position of a wavefront at a later time from
particles of the medium in phase, which are located at the its given position at any instant. In order words, the principle
same distance from the source of light and hence at any instant, gives a method to know as to how light spreads out in the medium.
all such particles must be vibrating in phase with each other. Huygen’s principle is based on the following assumptions :
The locus of all the particles of medium, which at any instant 1. Each point on the given or primary wavefront acts as a source
are vibrating in the same phase, is called the wavefront. of secondary wavelets, sending out disturbance in all directions
in a similar manner as the original source of light does.
Depending upon the shape of the source of light, wavefront can
2. The new position of the wavefront at any instant (called
be the following types :
secondary wavefront) is the envelope of the secondary
1.1 Spherical wavefront wavelets at that instant.
The above two assumptions are known as Huygen’s
A spherical wavefront is produced by a point source of light. It is
principle or Huygens’construction.
because, the locus of all such points, which are equidistant from
the point source, is a sphere figure (a).

1.2 Cylindrical wavefront

When the source of light is linear in shape (such as a slit), a
cylindrical wavefront is produced. It is because, all the points,
which are equidistant from the linear source, lie on the surface Key points
of a cylinder figure (b). Huygen’s principle is simply a geometrical construction
1.3 Plane wavefront to find the position of wavefront at a later time.
A small part of a spherical or a cylindrical wavefront originating 3. PRINCIPLE OF SUPER POSITION
from a distant source will appear plane and hence it is called a
plane wavefront figure (c). When two or more than two waves superimpose over each
other at a common particle of the medium then the resultant
1.4 Ray of light displacement (y) of the particle is equal to the vector sum of
An arrow drawn normal to the wavefront and pointing in the the displacements (y1 and y2) produced by individual waves.
i.e. y y1  y 2
G G G
direction of propagation of disturbance represents a ray of
light. A ray of light is the path along which light travels. In
3.1 Graphical view
figure thick arrows represent the rays of light.
Since the ray of light is normal to the wavefront, it is sometimes called
as the wave normal.

Key points
(i)
The phase difference between any two points on a
wavefront is zero.
 WAVE OPTICS

Resultant amplitude : After superimposition of the given

waves resultant amplitude (or the amplitude of resultant wave)
is given by A a 12  a 22  2a 1a 2 cos I
For the interfering waves y1 = a 1 sinZ t and y2 = a 2 cosZ t,
(ii)
Phase difference between them is 90o. So resultant amplitude

A a12  a 22

3.2 Phase/Phase difference/Path difference/Time difference Resultant intensity : As we know intensity v (Amplitude) 2
(i) Phase : The argument of sine or cosine in the expression Ÿ I1 ka12 , I 2 ka 22 and I = kA 2 (k is a proportionality
for displacement of a wave is defined as the phase. For constant). Hence from the formula of resultant amplitude, we
displacement y = a sin Z t ; term Z t = phase or get the following fo rmula of resultant i ntensity
instantaneous phase
I I1  I 2  2 I1 I 2 cos I
(ii) Phase difference (I) : The difference between the phases
of two waves at a point is called phase difference i.e. if The term 2 I1 I 2 cos I is called interference term. For
1
= a1 sin Zt and y2 = a2 sin (Zt + I) so phase difference = I
incoherent interference this term is zero so resultant intensity
(iii) Path difference (') : The difference in path length’s of I = I1 + I2.
two waves meeting at a point is called path difference
3.4 Coherent sources
O
between the waves at that point. Also ' uI The sources of light which emits continuous light waves of the
2S
same wavelength, same frequency and in same phase or having a
(iv) Time difference (T.D.) : Time difference between the
constant phase difference are called coherent sources.
T
waves meeting at a point is T.D. uI 4. INTERFERENCE OF LIGHT
2S
When two waves of exactly same frequency (coming from two
3.3 Resultant amplitude and intensity coherent sources) travels in a medium, in the same direction
If suppose we have two waves y1 = a1 sin Zt & y2 = a2 sin (Zt + I); simultaneously then due to their superposition, at some points
where a1, a 2 = Individual amplitudes, I = Phase difference intensity of light is maximum while at some other points intensity
between the waves at an instant when they are meeting a is minimum. This phenomenon is called Interference of light.
point. I1, I2 = Intensities of individual waves 4.1 Types of Interference

Constructive interference Destructive interference

(i) When the waves meets a point with same phase, (i) When the wave meets a point with opposite phase, destructive
constructive interference is obtained at that point interference is obtained at that point (i.e. minimum light)
(i.e. maximum light)
(ii) Phase difference between the waves at the point of (ii) I = 180° or (2n – 1) S; n = 1,2, ....
observation I = 0° or 2 nS or (2n + 1) S; n = 0, 1,2, .....
O
(iii) Path difference between the waves at the point of (iii) ' 2n  1 (i.e. odd multiple of O/2)
2
observation ' = nO (i.e. even multiple of O/2)
(iv) Resultant amplitude at the point of observation (iv) Resultant amplitude at the point of observation will be
will be maximum minimum
a1 = a 2 Ÿ Amin = 0 Amin = a1 – a2
If a1 = a 2 = a0 Ÿ Amax = 2a0 If a1 = a2 Ÿ Amin = 0
(v) Resultant intensity at the point of observation (v) Resultant intensity at the point of observation will be minimum
will be maximum
2 2
I max I1  I 2  2 I1 I 2 I max I1  I 2 I min I1  I 2  2 I1 I 2 I min I1  I 2

If I1 = I2 = I0 Ÿ Imax = 2 I0 If I1 = I2 = I0 Ÿ Imin = 0
WAVE OPTICS

4.2 Resultant intensity due to two identical waves = Wavelength of monochromatic light emitted from
source
For two coherent sources the resultant intensity is given by

I I1  I 2  2 I1I 2 cos I
For identical source I1 = I2 = I 0

I
Ÿ I I0  I0  2 I0 I0 cos I 4 I0 cos 2
2

T
[1 + cosT 2 cos 2 ]
2

(1) Central fringe is always bright, because at central

Ÿ In interference redistribution of energy takes place in position I= 0° or '= 0
the form of maxima and minima. (2) The fringe pattern obtained due to a slit is more bright
than that due to a point.
I max  I min
Ÿ Average intensity : I av I1  I 2 a 12  a 22 (3) If the slit widths are unequal, the minima will not be
2
co mplete da rk. Fo r very large wi dth uniform
Ÿ Ratio of maximum and minimum intensities : illumination occurs.
2
§ I1  I 2 · § I1 / I 2  1 ·
2 2 2 (4) If one slit is illuminated with red light and the other slit
I max ¨ ¸ ¨ ¸ § a1  a 2 · § a1 / a 2  1 ·
¨ ¸ ¨ ¸ is illuminated with blue light, no interference pattern is
I min ¨ I  I ¸ ¨ I / I 1 ¸ ¨ a a ¸ ¨ a / a 1 ¸
© 1 2 ¹ © 1 2 ¹ © 1 2 ¹ © 1 2 ¹ observed on the screen.
(5) If the two coherent sources consist of object and it’s
§ I max ·
¨ 1 ¸ reflected image, the central fringe is dark instead of
I1 a1 ¨ I min ¸ bright one.
also ¨ ¸
I2 a2 ¨ I max  1 ¸ 5.1 Path difference
¨ I ¸
© min ¹
Path difference between the interfering waves meeting at a
Ÿ If two waves having equal intensity (I1 = I2 = I0) meets yd
at two locations P and Q with path difference '1 and '2 point P on the screen is given by x d sin T where x is
D
respectively then the ratio of resultant intensity at point
the position of point P from central maxima.
I1 § S' ·
cos 2 cos 2 ¨ 1 ¸
P and Q will be
IP 2 © O ¹
IQ I § S' ·
cos 2 2 cos 2 ¨ 2 ¸
2 © O ¹

5. YOUNG’S DOUBLE SLIT EXPERIMENT (YDSE)

Monochromatic light (single wavelength) falls on two narrow
slits S 1 and S 2 which are very close together acts as two
coherent sources, when waves coming from two coherent
sources (S 1, S2) superimposes on each other, an interference For maxima at P : x nO ;
pattern is obtained on the screen. In YDSE alternate bright
where n = 0, r 1, r 2, …….
and dark bands obtained on the screen. These bands are called
Fringes. 2n  1 O
and For minima at P : x ;
d = Distance between slits 2
D = Distance between slits and screen where n = r 1, r 2, …….
 WAVE OPTICS

Note :- If the slits are vertical, the path difference (x) is If film is put in the path of upper wave, fringe pattern shifts
d sinT , so as T increases, ' also increases. But if slits upward and if film is placed in the path of lower wave, pattern
are horizontal path difference is d cos T , so as T shift downward.
increases, x decreases.

D E
Fringe shift = P 1 t P 1 t
d O
Ÿ Additional path difference = (P – 1)t

P 1 t
Ÿ If shift is equivalent to n fringes then n
O
5.2 More about fringe nO
or t
(i) All fringes are of equal width. Width of each fringe is P 1
OD O Ÿ
E and angular fringe width T Shift is independent of the order of fringe (i.e. shift of
d d zero order maxima = shift of nth order maxima.
(ii) If the whole YDSE set up is taken in another medium then Ÿ Shift is independent of wavelength.
changes so E changes
6. ILLUSTRATIONS OF INTERFERENCE
Oa Ea 3
e.g. in water O w Ÿ Ew Ea
Pw Pw 4 Interference effects are commonly observed in thin films when
their thickness is comparable to wavelength of incident light (If it
1 is too thin as compared to wavelength of light it appears dark and
(iii) Fringe width E v i.e. with increase in separation between if it is too thick, this will result in uniform illumination of film). Thin
d
layer of oil on water surface and soap bubbles shows various
the sources, E decreases.
colours in white light due to interference of waves reflected from
(iv) Position of n th bright fringe from central maxima the two surfaces of the film.
nO D
xn nE ; n = 0, 1, 2, ....
d
(v) Position of n th dark fringe from central maxima
2 n  1 OD 2n  1 E
xn ; n = 1, 2, 3....
2d 2

(vi) In YDSE, if n1 fringes are visible in a field of view with

light of wavelength O1, while n2 with light of wavelength
2
in the same field, then n1O1 = n2O2 .

5.3 Shifting of fringe pattern in YDSE

If a transparent thin film of mica or glass is put in the path of

one of the waves, then the whole fringe pattern gets shifted.
WAVE OPTICS

6.1 Thin films If v = actual frequency, v’ =Apparent frequency, v = speed of

In thin films interference takes place between the waves reflected source w.r.t stationary observer, c = speed of light
from it’s two surfaces and waves refracted through it. Source of light moves Source of light moves
towards the stationary away from the stationary
observer (v << c) observer (v << c)
(i) Apparent frequency (i) Apparent frequency
§ v· § v·
Qc v ¨1  ¸ and Qc v ¨1  ¸ and
© c¹ © c¹
Apparent wavelength Apparent wavelength
§ v· § v·
O c O ¨1  ¸ Oc O¨ 1  ¸
© c¹ © c¹
Interference in reflected Interference in refracted
light light (ii) Doppler’s shift : Apparent (ii) Doppler’s shift : Apparent
wavelength < actual wavelength > actual
Condition of constructive Condition of constructive
wavelength, So spectrum of wavelength, So spectrum
interference (maximum interference (maximum
the radiation from the source of the radiation from the
intensity) intensity)
of light shifts towards the source of light shifts
O O
' 2P t cos r 2n r 1 ' 2P t cos r 2n red end of spectrum. This towards the violet end of
2 2
For normal incidence r = 0 For normal incidence is called Red shift Doppler’s spectrum. This is called

O v
so 2Pt 2n r 1 2Pt = nO shit 'O O. Violet shift Doppler’s shift
2 c

Condition of destructive Condition of destructive v

'O O.
interference interference c
(minimum intensity) (minimum intensity)
O O 8. DIFFRACTION OF LIGHT
' 2P t cos r 2n ' 2P t cos r 2n r 1
2 2 It is the phenomenon of bending of light around the corners
For normal incidence 2Pt = nO For normal incidence of an obstacle/aperture of the size of the wavelength of light.
O
2Pt = (2n ± 1)
2

The Thickness of the film for interference in visible light

is of the order of 10,000 Å.

7. DOPPLER’S EFFECT IN LIGHT

The phenomenon of apparent change in frequency (or
wavelength) of the light due to relative motion between the source
of light and the observer is called Doppler’s effect.
According to special theory of relativity

v' 1r v / c
v 1  v 2 / c2
 WAVE OPTICS

8.1 Types of diffraction (ii) Minima occurs at a point on either side of the central
maxima, such that the path difference between the
The diffraction phenomenon is divided into two types
waves from the two ends of the aperture is given by
' = nO; where n = 1, 2, 3 ..... i.e. d sin T = nO;
Fresnel diffraction Fraunhofer diffraction

(i) If either source or screen (i) In this case both source nO

Ÿ sin T
or both are at finite distance and screen are effectively d
from the diffracting device at infinite distance from (iii) The secondary maxima occurs, where the path
(obstacle or aperture), the the diffracting device. difference between the waves from the two ends of the
diffraction is called Fresnel
type. O
aperture is given by ' 2n  1 ; where n = 1, 2, 3 .....
(ii) Common examples : (ii) Common examples : 2
Diffraction at a straight edge Diffraction at single slit,
narrow wire or small opaque double slit and diffraction O 2n  1 O
i.e. d sin T 2n  1 Ÿ sin T
disc etc. grating. 2 2d

8.3 Comparison between interference and diffraction

Interference Diffraction
Results due to the superposition Results due to the super-
of waves from two coherent position of wavelets from
source. different parts of same
wave front. (single coherent
8.2 Diffraction of light at a single slit source)
In case of diffraction at a single slit, we get a central bright All fringes are of same width All secondary fringes are of
band with alternate bright (maxima) and dark (minima) bands
OD
of decreasing intensity as shown E same width but the central
d
maximum is of double the
width
OD
E0 2E 2
d

All fringes are of same intensity Intensity decreases as the

order of maximum increases.
Intensity of all minimum may be Intensity of minima is not
zero. Positions of nth maxima zero. Positions of nth
and minima secondary maxima and

nOD , OD
Xn Xn Bright 2n  1 ,
Bright
d d
OD nOD
Xn Dark 2n  1 Xn Dark
d d

Path difference for nth maxima for nth secondary maxima

O
2O D ' = nO ' 2n  1
(i) Width of central maxima E 0 and angular width 2
d
Path difference for nth minima Path difference for nth
2O
d ' = (2n – 1)O minima ' = nO
WAVE OPTICS

8.4 Diffraction and optical instruments 9.2 Polarised light

The objective lens of optical instrument like telescope or The light having oscillations only in one plane is called Polarised or
microscope etc. acts like a circular aperture. Due to diffraction plane polarised light.
of light at a circular aperture, a converging lens cannot form a (i) The plane in which oscillation occurs in the polarised light
point image of an object rather it produces a brighter disc is called plane of oscillation.
known as Airy disc surrounded by alternate dark and bright
(ii) The plane perpendicular to the plane of oscillation is called
concentric rings.
plane of polarisation.
(iii) Light can be polarised by transmitting through certain
crystals such as tourmaline or polaroids.

9. 3 Polarization by Scattering
When a beam of white light is passed through a medium
containing particles whose size is of the order of
1.22 O wavelength of light, then the beam gets scattered.
The angular half width of Airy disc = T (where D =
D When the scattred light is seen in a direction
aperture of lens) perpendicular to the direction of incidence, it is found
to be plane polarized (as detected by the analyser).
The lateral width of the image fT (where f = focal length of
The phenomenon is called polarization by scattering.
the lens)

Diffraction of light limits the ability of optical

instruments to form clear images of objects when they
are close to each other.

9. POLARISATION OF LIGHT
Light propagates as transverse EM waves. The magnitude of
electric field is much larger as compared to magnitude of
magnetic field. We generally prefer to describe light as electric
field oscillations.
9.4 Polarization of Light by Reflection
9.1 Unpolarised light
The light having electric field oscillations in all directions in When unpolarized light is reflected from a surface, the
the plane perpendicular to the direction of propagation is reflected light may be completely polarised, partially
called Unpolarised light. The oscillation may be resolved into polarized or unpolarized. This would depend on the
horizontal and vertical component. angle of incidence.
The angle of incidence at which the reflected light is
completely plane polarized is called polarizing angle or
Brewster’s angle. (i p)
 WAVE OPTICS

9.5 Polaroids
It is a device used to produce the plane polarised light. It is based
on the principle of selective absorption and is more effective than
the tourmaline crystal. or
It is a thin film of ultramicroscopic crystals of quinine idosulphate
with their optic axis parallel to each other.

(i) Polaroids allow the light oscillations parallel to the I = I0 cos 2 T and and A 2 A 20 cos 2 T ŸA = A0 cos T
transmission axis pass through them. If T = 0°, I = I0, A = A0,
(ii) The crystal or polaroid on which unpolarised light is
If T = 45°, I =I0/2, A A0 / 2
incident is called polariser. Crystal or polaroid on which
polarised light is incident is called analyser. If T = 90°, I = 0, A = 0
(ii) If Ii = Intensity of unpolarised light.
Ii
So I 0 i.e. if an unpolarised light is converted into
2
plane polarised light (say by passing it through a
polaroid or a Nicol-prism), its intensity becomes half
Ii
and I cos 2 T
2

I max  I min
Percentage of polarisation u100
I max  I min

9.7 Brewster’s law

Brewster discovered that when a beam of unpolarised light is
reflected from a transparent medium (refractive index =P), the
When unpolarised light is incident on the reflected light is completely plane polarised at a certain angle
polariser, the intensity of the transmitted polarised light of incidence (called the angle of polarisation Tp).
is half the intensity of unpolarised light. Also P = tan Tp Brewster’s law
(i) For i < Tp or i > Tp
(iii) Main uses of polaroids are in wind shields of automobiles,
sun glassess etc. They reduce head light glare of cares
and improve colour contrast in old paintings. They are
also used in three dimensional motion pictures and in optical
stress analysis.
9.6 Malus law
This law states that the intensity of the polarised light
transmitted through the analyser varies as the square of the
cosine of the angle between the plane of transmission of the
analyser and the plane of the polariser.

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WAVE OPTICS

Both reflected and refracted rays becomes partially

1
polarised Resolving Power (R.P.) =
Limit of Revolution
(ii) For glass T P | 57q , for water T P | 53q
11.1 Resolving power of Microscope
10. VALIDITY OF RAY­OPTICS
2 P sin T
R. P. of microscope
When a parallel beam of light travels upto distances as large O
as few metres it broadens by diffraction of light travels.

10.1 Fresnel Distance

Fresnel distance is the minimum distance a beam of

light can travel before its deviation from straight line
path becomes significant/noticeable.

a2
ZF
O

Since wavelength of light is very small deviation is

very small and light can be assumed as travelling in a
straight line. 11.2 Resolving power of Telescope
Hence we can ignore broadening of beam by diffraction
upto distances as large as a few meters, i.e., we can 1 D
R.P. of telescope
assume that light travels along straight lines. Hence dT 1.22 O
ray optics can be taken as a limiting case of wave optics.
where D is aperture of telescope.
Hence Ray optics can be taken as a limiting case of
waveoptics.

11. RESOLVING POWER

When two point objects are close to each other their

images diffraction patterns are also close and overlap
each other.

The minimum distance between two objects which can

be seen seperately by the object instrument is called
limit of resolution of the instrument.
 SEMI CONDUCTOR

SEMI CONDUCTOR
supplied, the electrons can easily jump from valence band to
1. ENERGY BANDS IN SOLIDS
conduction band. For example when the temperature is increased
In case of a single isolated atom, there are single energy levels in the forbidden band is decreased so that some electrons are
case of solids, the atoms is arranged in a systematic space lattice liberated into the conduction band.
and hence the atom is greatly influenced by neighbouring atoms.
The closeness of atoms results in the intermixing of electrons of 12..3 Conductors
neighbouring atoms of course, for the valence electrons in the
In case of conductors, there is no forbidden band and the valence
outermost shells which are not strongly bounded by nucleus.
Due to intermixing the number of permissible energy levels band and conduction band overlap each other. Here plenty of free
increases or there are significant changes in the energy levels. electrons are available for electric conduction. A slight potential
Hence in case of a solid, instead of single energy levels associated difference across the conductor cause the free electrons to constitute
with the single atom, there will be bands of energy levels. electric current. The most important point in conductors is that due
to the absence of forbidden band, there is no structure to establish
1.1 Valence Band, Conduction Band & Forbidden Energy Gap holes. The total current in conductors is simply a flow of electrons.
The band formed by a series of energy levels containing the
2. SEMICONDUCTORS
valence electrons is known as valence band. The valency band
may be defined as a band which is occupied by the valence Thus a substance which has resistively in between conductors
electrons or a band having highest occupied band energy. and insulators is known as semiconductor.
The conduction band may also be defined as the lowest unfilled Semiconductors have the following properties.
energy band. The separation between conduction band and (i) They have resistively less than insulators and more than
valence band is known as forbidden energy gap. There is no conductors.
allowed energy state in this gap and hence no electron can stay
(ii) The resistance of semiconductor decreases with the
in the forbidden energy gap.
increase in temperature and vice versa.
1.2 Insulators, Semiconductors and Conductors (iii) When suitable metallic impurity like arsenic, gallium etc.
is added to a semiconductors, its current conducting
On the basis of forbidden band, the insulators, semiconductors properties change appreciably.
and conductors are described as follows:
2.1 Effect of temperatue of Semiconductors
1.2.1 Insulators
At very low temperature (say 0 K) the semiconductor crystal
In case of insulators, the forbidden energy band is very wide.
behaves as a perfect insulator since the covalent bonds are very
Due to this fact electrons cannot jump from valence band to
strong and no free electrons are available. At room temperature some
conduction band. In insulators the valence electrons are bond
of the covalent bonds are broken due to the thermal energy supplied
very tightly to their parent atoms. Increase in temperature enables
to the crystal. Due to the breaking of the bonds, some electrons
some electrons to go to the conduction band.
become free which were engaged in the formation of these bonds.
The absence of the electron in the covalent bond is represented
by a small circle. This empty place or vacancy left behind in the
crystal structure is called a hole. Since an electron unit negative
charge, the hole carries a unit positive charge.

2.2 Mechanism of conduction of Electrons and Holes

When the electrons are liberated on breaking the covalent bonds,

they move randomly through the crystal lattice.
1.2.2 Semiconductors
When an electric field is applied, these free electrons have a
In semiconductors, the forbidden band is very small. Germanium steady drift opposite to the direction of applied field. This
and silicon are the examples of semiconductors. A semiconductor constitute the electric current. When a covalent bond is broken,
material is one whose electrical properties lies between insulators a hole is created. For one electron set free, one hole is created.
and good conductors. When a small amount of energy is This thermal energy creates electron-hole pairs-there being as

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SEMI CONDUCTOR

many holes as free electrons. These holes move through the The doping material is either pentavalent atoms (bismuth,
crystal lattice in a random fashion like liberated electrons. When antimony, arsenic, phosphorus which have five valence
an external electric field is applied, the holes drift in the direction electrons) or trivalent atoms (gallium, indium, aluminium, boron
of applied field. Thus they constitute electric current. which have three valence electrons). The pentavalent doping
atom is known as donor atom because it donates one electron to
There is a strong tendency of semiconductor crystal to form a
the conduction band of pure semiconductor.
covalent bonds. Therefore, a hole attracts an electron from the
neighbouring atom. Now a valence electron from nearby covalent The doping materials are called impurities because they alter the
bond comes to fill in the hole at A. This results in a creation of structure of pure semiconductor crystals.
hole at B. The hole has thus effectively shift from A to B. This 2.4.2 N–Type Extrinsic Semiconductor
hole move from B to C from C to D and so on.
When a small amount of pentavalent impurity is added to a pure
This movement of the hole in the absence of an applied field is semiconductor crystal during the crystal growth, the resulting
random. But when an electric field is applied, the hole drifts crystal is called as N-type extrinsic semiconductor.
along the applied field.
In case of N-type semiconductor, the following points should be
2.3 Carrier Generation and Recombination remembered
(i) In N-type semiconductor, the electrons are the majority
The electrons and holes are generated in pairs. The free electrons
carriers while positive holes are minority carriers.
and holes move randomly within the crystal lattice. In such a
random motion, there is always a possibility that a free electron (ii) Although N-type semiconductor has excess of electrons
may have an encounter with a hole. When a free electron meets but it is electrically neutral. This is due to the fact that
electrons are created by the addition of neutral pentavalent
a hole, they recombine to re-establish the covalent bond. In the
impurity atoms to the semiconductor i.e., there is no
process of recombination, both the free electron and hole are
addition of either negative changes or positive charges.
destroyed and results in the release of energy in the form of heat.
The energy so released, may in turn be re-absorbed by another 2.4.3 P–Type Extrinsic Semiconductor
electron to break its covalent bond. In this way a new electrol-
hole pair is created. When a small amount of trivalent impurity is added to a pure
crystal during the crystal growth, the resulting crystal is called a
T hus the process of breaking of covalent bonds and P-type extrinsic semiconductor.
recombination of electrons and holes take place simultaneously.
In case of P-type semiconductor, the following points should be
When the temperature is increased, the rate of generation of
remembered
electrons and holes increases. This is turn increases, the densities
of electrons and hole increases. As a result, the conductivity of (i) In P-type semiconductor materials, the majority carriers
semiconductor increases or resistivity decreases. This is the are positive holes while minority carriers are the electrons.
reason that semiconductors have negative temperature coefficient (ii) The P–type semiconductor remains electrically neutral
of resistance. as the number of mobile holes under all conditions remains
equal to the number of acceptors.
2.4 Pure or Intrinsic Semiconductor and
Impurity or Extrinsic Semiconductors 2.5 P–N Junction Diode
A semiconductor in an extremely pure from is known as intrinsic When a P-type material is intimately joined to N-type, a P-N
semiconductor or a semiconductor in which electrons and holes junction is formed. In fact, merely-joining the two pieces a P-N
are solely created by thermal excitation is called a pure or intrinsic junction cannot be formed because the surface films and other
semiconductor. In intrinsic semiconductor the number of free irregularities produce major discontinuity in the crystal structure.
electrons is always equal to the number of holes. Therefore a P-N junction is formed from a piece of semiconductor
(say germanium) by diffusing P-type material to one half side
2.4.1 Extrinsic Semiconductors
and N-type material to other half side.When P-type crystal is
The electrical conductivity of intrinsic semiconductor can be placed in contact with N-type crystal so as to form one piece, the
increased by adding some impurity in the process of assembly so obtained is called P-N junction diode.
crystallization. The added impurity is very small of the order of
2.5.1 Forward Bias
one atom per million atoms of the pure semiconductor. Such
semiconductor is called impurity or extrinsic semiconductor. The When external d.c. source is connected to the diode with p–section
process of adding impurity to a semiconductor is known as doping. connected to +ve pole and n–section connected to –e pole, the
 SEMI CONDUCTOR

junction diode is said to be reverse biased. The upper end of RL will be at +ve potential w.r.t. the lower end.
The magnitude of output across RL during first half at any instant
2.5.2 Reverse Bias will be proportional to magnitude of current through RL, which in
turn is proportional to magnitude of forward bias and which
When an external d.c. battery is connected to junction diode with
ultimately depends upon the value of a.c. input at that time.
P–section connected to –ve pole and n–section connected to
+ve pole, the junction diode is said to be reverse biased.

P–N JUNCTION is such a device (any way) which offers

low resistance when forward biased and behaves like an
insulator when reverse biased.

Thus output across RL will vary in accordance with a.c. input.

Symbol :
During second half, junction diode get reverse biased and hence
no–output will be obtained. Thus a discontinuous supply is
obtained.

2.7 Full Wave Rectifier

A rectifier which rectifies both halves of a.c. input is called full

wave rectifier.
2.6 Junction Diode as Rectifier
2.7.1 Principle
An electronic device which converts a.c. power into d.c. power
is called a rectifier. Junction Diode offers low resistive path when forward biased
and high resistive path when reverse biased.
2.6.1 Principle

Junction diode offers low resistive path when forward biased 2.7.2 Arrangement
and high resistance when reverse biased. The a.c. supply is fed across the primary coil (P) of step down
2.6.2 Arrangement transformer. The two ends of S–coil (secondary) of transformer
are connected to P-section of junction diodes D1 and D2. A load
The a.c. supply is fed across the primary coil (P) of step down
resistance RL is connected across the n–sections of two diodes
transformer. The secondary coil ‘S’ of transformer is connected
and central tapping of secondary coil. The d.c. output is obtained
to the junction diode and load resistance RL. The output d.c.
across secondary.
voltage is obtained across RL.

2.6.3 Theory 2.7.3 Theory

Suppose that during first half of a.c. input cycle the junction Suppose that during first half of input cycle upper end of s-coil is
diode get forward biased. The conventional current will flow in at +ve potential. The junction diode D1 gets forward biased,
the direction of arrow heats. while D2 gets reverse biased. The conventional current due to
D1 will flow along path of full arrows.
When second half of input cycle comes, the conditions will be
exactly reversed. Now the junction diode D2 will conduct and
the convensional current will flow along path of dotted arrows.
Since current during both the half cycles flows from right to left
through load resistance RL, the output during both the half cycles
will be of same nature.
The right end of RL is at +ve potential w.r.t. left end. Thus in full
wave rectifier, the output is continuous.
SEMI CONDUCTOR

The majority carriers (e–) in emitter are repelled towards base due
to forward biase. The base contains holes as majority carriers
but their number density is small as it is doper very lightly (5%)
as compared to emitter and collector. Due to the probability of e–
and hole combination in base is small. Most of e– (95%) cross
into collector region where they are swept away by +ve terminal
of battery VCB.

Corresponding to each electron that is swept by collector, an

electron enters the emitter from -ve pole of collector – base battery.

If Ie, Ib, Ic be emitter, base and collector current respectively then

using Kirchoff first law

Ie Ib  Ic

2.8.2 Action p–n–p Transistor

The p–type emitter is forward biased by connecting it to +ve

pole of emitter – base battery and p–type collector is reverse
2.8 Transistor biased by connected it to –ve pole of collection - base battery. In
It is three section semiconductor, in which three sections are this case, majority carriers in emitter i.e. holes are repelled towards
combined so that the two at extreme ends have the same type of base due to forward biase. As base is lightly doped, it has low
majority carriers, while the section that separates them has the number density of e–. When hole enters base region, then only
majority carriers in opposite nature. The three sections of 5% of e– and hole combination take place. Most of the holes
transistor are called emitter (E), Base (B), collector (C). reach the collector and are swept away by –ve pole of VCB battery.

Symbol :

2.8.1 Action of n-p-n Transistor 2.9 Common base Amplifier

In this base of the transistor is common to both emitter and

collector.
(a) Amplifier ckt. using n-p-n transistor : The emitter is
forward biased using emitter bias battery (Vcc) & due to
this, the resistance of input circuit bias battery (Vcc), due
to this, resistance of output circuit is large.

Fig. shows that, the n-type emitter is forward biased by connecting

it to -ve pole of VEB (emmitter-base battery) and n-type collector
is reverse biased by connected it to +ve pole of VCB (collector-
base battery).
 SEMI CONDUCTOR

Low input voltage is applied across emitter – base ckt. and and Ic be the emitter current, base current and collector
amplified circuit is obtained across collector - base circuit. If Ie, current respectively. Then according to Kirchhoff’s first
Ib, Ic be the emitter, base and collector current than law
Ie = Ib + Ic
Ie Ib  I c …(i)

When current Ic flows in collector circuit, a potential drop IcRc

occurs across the resistance connected in collector - base circuit
and base collector voltage will be

Vcb Vcc  Ic R c …(ii)

(b) Amplifier circuit using p–n–p Transistor

3. When the positive half cycle of input a.c. signal voltage

comes, it supports the forward biasing of the emitter-base
circuit. Due to this, the emitter current increases and
consequently the collector current increases. As a result
of which, the collector voltage Vc decreases.
4. Since the collector is connected to the positive terminal of
VCE battery, therefore decreases in collector voltage means
the collector will become less positive, which means
1. When the positive half cycle of input a.c. signal voltage negative w.r. to initial value. This indicates that during
comes, it supports the forward biasing of the emitter–base positive half cycle of input a.c. signal voltage, the output
circuit. Due to this, the emitter current increases and signal voltage at the collector varies through a negative
consequently the collector current increases. half cycle.
2. As Ic increases, the collector voltage Vc decreases. 5. When negative half cycle of input a.c. signal voltage
3. Since the collector is connected to the negative terminal comes, it opposes the forward biasing of emitter-base
of VCC battery of voltage VCB, therefore, the decrease in circuit, due to this the emitter current decreases and hence
collector voltage means the collector will become less collector current decreases; consequently the collector
negative. This indicates that during positive half cycle of voltage Vc increases i.e., the collector becomes more
input a.c. signal voltage, the output signal voltage at the positive. This indicate that during the negative half cycle
collector also varies through the positive half cycle. of input a.c. signal voltage, the output signal voltage varies
through positive half cycle.
4. During negative half cycle of input a.c. signal voltage,
the output signal voltage at the collector also varies 2.11 Common base Amplifier
through the negative half cycle. Thus in common base
a.c. current gain : It is defined as the ratio of change in collector
transitor amplifier circuit the input signal voltage and the
current as constant collector voltage. It is denoted by Dac
output collector voltage are in the same phase.

2.10 Common Emitter Amplifier § 'I c ·

D ac ¨ ¸
¨ 'I ¸ [VCB = const.]
Amplifier circuit using n–p-n transitor © e ¹

1. The input (emitter base) circuit is forward biased with Voltage gain : It is defined as the ratio of change in output voltage
battery VBB of voltage VEB, and the output (collector- to the change in input voltage. It is denoted by A.
emitter) circuit is reversed biased with battery VCC of
voltage VCE. Due to this, the resistance of input circuit is 'I c R out 'I c R out
Av = u
low and that of output circuit is high. Rc is a load resistance 'I e R in 'I e R in
connected in collector circuit.
Or Av = DAC × resistance gain,
2. When no a.c. signal voltage is applied to the input circuit
where Rout/Rin is called resistance gain.
but emitter base circuit is closed let us consider, that Ie, Ib
SEMI CONDUCTOR

Power gain : It is defined as the ratio of change in output power

to the change in input power. Therefore,

change in output power 'I c R out

a.c. power gain =
change in input power 'I e R in

'I 2 c R out
u
'I 2 e R in

Or a.c. power gain = D2ac × resistance gain.

4. DIGITAL SIGNALS
2.12 Common Emitter Amplifier
Signals having either of the two levels, 0 or 1, are called digital
a.c. current gain : It is defined as the ratio of the change in signals.
collector to the change in base current. It is denoted by Eac.

§ 'Ic ·
Therefore, Eac = ¨ 'I ¸ [Vce = const.]
© b ¹v

Its value is quite large as compared to 1 and lies between 15 to 50.

Voltage gain : It is the ratio of the change in output voltage to the
change in input voltage. It is denoted by A.

'I c u R out 'I c R out

Av = u
'I b u R in 'I b R in

Or Av = Eac × resistance gain.

a.c. power gain : It is the ratio of the change in output power to
the change in input power. 5. LOGIC GATES
A digital circuit which either stops a signal or allows it to pass
2
change in output power 'I c R out through it is called a gate. A logic gate is an electronic circuit
a.c. power gain = = 2
change in input power 'I b R in which makes logical decisions. Logic gate has one or more inputs
but one output. Logic gates are the basic building blocks for
Or a.c. power pain = E2ac × resistance gain. most of the digital systems. Variables used at the input and output
are 1’s and 0’s. These are three basic logic gates:
2.13 Relation between D and E
(i) OR gate (ii) AND gate
For both the types of amplifier, we have
(iii) NOT gate.
ie = ib + ic
5.1 OR Gate
Dividing both sides of the above equation by Ic, we get
OR gate is an electronic device that combines A and B to give Y
ie ib
1 as output. In this figure two inputs are A and B and output is Y.
ic ic In Boolean algebra OR is represented by +.
? 1/D = (1/E) + 1 or 1/E = (1/D) 1 = (1D)/D
or E = D/ (1D)

3. ANALOG SIGNALS
Truth Table: A truth table may be defined as the table which
Signals which varies continuously with time is called analog gives the output state for all possible input combinations.
signal. A typical analog signal is shown in figure. Circuit used
Logic operations of OR gate are given in its truth table for all
for generating analog signal is called analog electronic circuit.
possible input combinations.
 SEMI CONDUCTOR

Input Output
A B Y
0 0 0
0 1 1
1 0 1 Truth Table :

1 1 1 A B Y’ Y
0 0 0 1
5.2 AND Gate
0 1 1 0
In an AND gate there are two or more inputs and one output. In
1 0 1 0
Boolean algebra AND is represented by a dot (.).
1 1 1 0

Boolean expression for NOT gate is and it is read as Y = A  B

and it is read as Y equals A OR B negated. A NOR function is the
reverse of OR function.
Truth Table

In put Output
A B Y
0 0 0
0 1 0 Truth Table :
1 0 0
In put Output
1 1 1
A B Y
5.3 NOT Gate 0 0 1
NOT gate is an electronic circuit which has one input and one 0 1 0
output. This circuit is so called because output is NOT the same 1 0 0
as input. 1 1 0

5.5 NAND Gate

A NAND gate has two or more inputs and one output. Actually
a NAND gate is a NOT–AND gate. If a NOT gate is connected at
the output of a AND gate, we get NAND gate as shown in figure
Boolean expression for NOT gate is Y = A .
and its truth table is given in table.
Truth Table:

In put Output
A Y
0 1
1 0 A B Y’ Y

0 0 0 1
5.4 NOR Gate
0 1 0 1
A NOR gate has two or more inputs and one output. Actually 1 0 0 1
NOR gate is a NOT-OR gate. If a NOT gate is connected at the 1 1 1 0
output of an OR gate, we get NOR gate as shown in figure and its
truth table in table.

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SEMI CONDUCTOR

Truth Table :
Boolean expression for NAND gate, is Y = A . B and is read as Y
In put Output
equals A and B negated.
Logical symbol of NAND gate is shown in figure and its truth A B Y
table in table. 0 0 1
0 1 1
1 0 1
1 1 0

Like NOR gate, NAND gate can also be used to realize all basic
gates : OR, AND and NOT. Hence it is also known as universal
Gate.
 Chapter 15
Communication System
Introduction
We live in the world of information. Information needs to be communicated from one entity to another entity.
This act of sending and receiving message from one place to another place, successfully, is called
communication.
The word successful in the above definition, implies many things like
o Common understanding by the sender and the receiver in interpreting the information
o Quality in communication, which implies no addition, deletion or modification of the actual information
The growing needs of human beings in the field of communication imposed demands on
o Complexity of information
o Speed of transmission

Evolution in communication
The table below shows us how physical messengers who travelled from one place to another changed to the
current day situation where information comes to your doorstep anytime with easy access.
Time period Event Remarks
· Announcement to common people · Messengers travelled from one place to
When Kings · Peace and war message from one another
ruled country to another · Drum beaters announced Kings decisions
Invention of Telegraph by F.B.Morse Messengers physically going from one place
1835 and Sir Charles Wheatstone to another reduced
Invention of Telephone by Alexander
1876 Graham Bell and Antonio Meucci Even now this communication is very useful
Wireless Telegraphy by Jagadis Leap in communication history from using
1895 Chandra Bose and G Marconi wires to wireless
1936 Television broadcast by John Logi Baird Being used even today
1955 Radio FAX by Alexander Bain Being used even today
First internet where file transfer from one
computer to another computer was
1968 ARPANET by JCR LIcklider possible
1975 Fiber Optics by Bell Laborataries More economical means of communication
Information access made so easy in modern
1989-91 World Wide Web by Tim Berners-Lee world

Communication System
The general form of communication system is depicted below:
As we see here, the basic elements of communication includes transmitter, Channel and the receiver. The
transmitter and the receiver may be located geographically at different places. The Channel connects the
transmitter and the receiver.
Information Source – The source produces signal of the information which needs to be communicated.
Signal – Information in electrical form suitable for transmission is called signal.
Transmitter – Converts the source signal into suitable form for transmission through the channel.
Channel – The channel connecting the transmitter and the receiver is a physical medium. The channel can be
in the form of wires, cables or wireless.
Noise – When the transmitted signal propagates along the channel, it may get distorted due to channel
imperfection.
Thus, noise refers to unwanted signals that tend to disturb the process of communication from the transmitter
to the receiver.
Receiver – Due to noise and other factors, the corrupted version of signal arrives at the receiver. The receiver
has to reconstruct the signal into recognizable form of the original message for delivering it to the user. The
signal at the receiver forms the output.

Modes of communication
Point to point communication – There is a single link between the transmitter and the receiver.
Communication takes place between single transmitter and receiver
Example – Telephone

Broadcast mode – There are large number of receivers though information is sent by a single transmitter.
Example – Television and Radio

Communication – Terminology
1. Transducer – Any device which converts energy in one form to another form is called transducer.
Electrical transducer: A device that converts some physical variable like pressure, displacement, force,
temperature, into corresponding variations in electrical signal. Hence, the output of this would be an
electrical signal.
2. Signal Types – Information in electrical form suitable for transmission called signal, is of two types
Analog signal –
o Continuous variations of voltage and current. Hence, single valued functions of time.
o Sine wave is a fundamental analog signal
o Example – Sound and picture signals in television

Digital signal –
o Digital step value based
o Binary system where 0 represents low level and 1 represents high level is used
o Universal digital coding methods like BCD – Binary Coded Decimal and ASCII – American Standard
Code of Information Interchange is used in common
 3. Amplitude –
The maximum extent of vibration or oscillation from the position of equilibrium

4. Frequency –
The frequency is the number of waves which pass a fixed place in a given amount of time.

5. Phase –
The two waves depicted below have a phase difference indicated by the phase shift which is the fraction of
the wave cycle which has elapsed relative to the origin.

Signal propagation – Terminology

1. Attenuation – The loss of strength of the signal while propagating through a medium is known as
attenuation.
2. Amplification – The process of increasing the amplitude of the signal by using an electronic circuit is
called amplification. This also increases the strength of the signal. Hence, it compensates the
attenuation of the signal.
3. Range – It is the largest distance between the source and the destination upto which the signal is
received with sufficient strength.
4. Bandwidth – It refers to the frequency range for which the equipment operates.
5. Modulation – If the information signal is of low frequency, it cannot be transmitted to long distances.
Hence, at transmission point, it is super imposed on high frequency wave. This high frequency wave
acts as a carrier of information. This is modulation
Sinusoidal wave modulation-
There are 3 types of modulation, namely 1. Amplitude modulation 2.Frequency modulation and 3.
Phase modulation
Amplitude modulation –
The amplitude of the carrier wave is varied in accordance with the information signal
 Frequency modulation –
The frequency of the carrier wave is varied in accordance with the information signal

Phase modulation –
The phase of the carrier wave is varied in accordance with the information signal

Pulse wave modulation-

There are 3 types of pulse wave modulation – namely (a) Pulse amplitude modulation (b) Pulse width
modulation (c) Pulse position modulation

6. Demodulation – The process of retrieval of information from the carrier wave at the receiver is termed
as demodulation. This is a reverse of modulation.
7. Repeater – A repeater is a combination of receiver and a transmitter.
A repeater picks up the signal from the transmitter, amplifies and retransmits it to the receiver. Thus
repeaters are used to extend the range of communication system
Example – Communication satellite is a repeater station in space.
Propagation of electromagnetic waves
While communication using radio waves, the transmitter antenna radiates electromagnetic waves. These
waves travel through the space and reach the receiving antenna at the other end. We have considered below
some of the wave propagation methods in brief.
Ground or Surface wave propagation:

o In this mode of wave propagation, ground has a strong influence on propagation of signal waves from the
transmitting antenna to the receiving antenna. The signal wave glides over the surface of the earth
o While propagating on the surface of the earth, the ground wave induces current in the ground. It also
bends around the corner of the objects on the earth
o Due to this, the energy of the ground wave is gradually absorbed by the earth and the power of the
ground wave decreases
o The power of the ground wave decreases with the increase in the distance from the transmitting station.
This phenomenon of loss of power of the ground wave is called attenuation
o The attenuation of ground waves increases very rapidly with the increase in its frequency
o Thus, ground wave communication is not suited for high frequency signal wave and for very long range
communication
o To radiate signals with high efficiency, the antennas should have a size comparable to the wavelength of
the signal
Sky waves:

o The ionosphere plays a major role in sky wave propagation. We know that the earth’s atmosphere is
divided into various regions like – Troposphere, Stratosphere, Mesosphere and Ionosphere.
o The ionosphere is also called as thermosphere as temperature increases rapidly here and it is the
outermost part of the earth’s atmosphere.
o Above troposphere, we have various layers like D (part of stratosphere), E (part of stratosphere), F1 (part
of mesosphere), F2 (part of ionosphere)
o The ionosphere is called so because of the presence of large number of ions or charged particles.
Ionisation occurs due to the absorption of the ultraviolet and other high energy radiation coming from the
sun, by the air molecues
o The phenomenon of bending of electromagnetic waves in this layer so that they are diverted towards the
earth is helpful in skywavepropogation. This is similar to total internal reflection in optics
o The radiowaves of frequency range from 1710 kHz to 40 MHz are propagated in sky wave propagation
Space waves:

o The space waves travel in straight line from the transmitting antenna to the receiving antenna.
o Hence, space waves are used for line of sight communication such as television broadcast, microwave link
and satellite communication
o The line of sight communication is limited by (a) the line of sight distance (b) the curvature of the earth
o At some point by the curvature of the earth, the line of sight propagation gets blocked.
o The line of sight distance is the distance between transmitting antenna and receiving antenna at which
they can see each other. It is also called range of communication d M
o The range of space wave communication can be increased by increasing the heights of the transmitting
antenna and receiving antenna.
o The maximum line of sight distance (range of communication) dM between two transmitting antenna of
height hT and the receiving antenna of height hR above the earth is given by
𝑑𝑀 = √(2𝑅ℎ𝑇 ) + √2𝑅ℎ𝑅

Modulation and its necessity

Any message signal, in general, is not a single frequency sinusoidal. But it spreads over a range of frequencies
called the signal bandwidth.
Suppose we wish to transmit an electronic signal, in the audio frequency range, say 20 Hz to 20kHz range, over
a long distance we need to consider factors like
a. Size of antenna:
o Antenna is needed for both transmission and reception
o Antenna should have a size comparable to the wavelength of the signal, atleastλ/4 where λ is the
wavelength of the signal
o In the above audio frequency range, if we consider frequency 15,000= ‫ ט‬Hz. Then λ = c / / 108* 3= ‫ט‬
15,000 = 20,000 m
o Hence, antenna length = λ/4 = 20,000 / 4 = 5000 m.
o It is practically impossible to design an antenna of height 5000m
o So the transmission frequency should be raised in such a way that the length of the antenna is within
100m which is feasible for practical purpose
o This shows that there is a need for converting low frequency signal to high frequency before
transmission

b. Effective power radiated by the antenna:

o Effective power rated by the antenna = P = E/t
o Also, E = h‫ = ט‬hc/λ
o Hence, P = E/t = hc/λ * c/λ
o Studies reveal that if l is the linear length of the antenna, then P is proporational to (l/λ) 2
o Hence, for good transmission, high power and hence small wavelength and high frequency waves are
required
o High frequency waves becomes inevitable in this case also
c. Avoiding mixing of signals from different transmitters:
o When many transmitters are transmitting baseband information signals simultaneously, they all gets
mixed up
 o There is no way to distinguish between them
o Possible solution is communication at high frequencies and allotting a band of frequencies for each
transmitter so that there is no mixing
o This is what is being dene for different radio and TV broadcast stations
Hence, we understand the necessity of modulation.
Band width
Bandwidth is also defined as the amount of data that can be transmitted in a fixed amount of time
Signals – Bandwidth:
o The message signal can be voice, music, picture or computer data
o Each of the above have different frequency ranges
o The speech signals frequency range from 300Hz to 3100Hz. Hence, bandwidth = 3100 -300 = 2800 Hz
o Any music requires bandwidth of 20kHz because of high frequencies produced by musical instruments
o Video signals for transmission of picture requires 4.2 MHz of bandwidth
o The Television signal which contains both voice and picture is usually allocated a bandwidth of 6MHz
bandwidth for transmission
Transmission Medium – Bandwidth:
o Different types of transmission media offers different bandwidth
o Coaxial cables, widely used wire medium offers bandwidth of approximately 750 MHz
o Communication through free space using radio waves offers wide range from hundreds of kHz to few GHz
o Optical fibres are used in the frequency range of 1THz to 1000 THz (THz – Tera Hertz; 1THz = 1012Hz)
o As mentioned earlier, to avoid mixing of signals, allotting a band of frequencies to a specific transmitter is
in practise
o The International Telecommunication Union administers this frequency allocation
o Services like FM Broadcast, Television, Cellular Mobile Radio and Satellite communication operate under
fixed frequency bands
Let us now consider amplitude modulation in detail.
Amplitude modulation
As we know, in amplitude modulation, the amplitude of the carrier wave is varied in accordance with the
amplitude of the information signal or modulating signal.
For sinusoidal modulating wave,
m(t) = Amsinωmt --------------------------------(1)
where, Am – Amplitude of modulating signal
ωm- 2πωm – Angular frequency of modulating signal
For carrier wave,
Cm(t) = Acsinωct --------------------------------(2)
Where, Ac – Amplitude of carrier wave
ωm- 2πωc– Angular frequency of carrier wave
For carrier wave, the amplitude is changed by adding the amplitude of the modulating signal which is
Ac + Amsinωmt
Cm(t) = (Ac+ Amsinωmt) sin ωct --------------------------------(3)
Multiply and Divide equation (3) RHS by Ac
Cm(t) = Ac (Ac/Ac+Am/Ac sin ωmt) sin ωct ---------------------------(4)
Replace Am / Ac = µ
µ is called Amplitude modulation index and is always less than or equal to 1 to avoid distortion.
Cm(t) = Acsin ωct + µAc sin ωct sin ωmt ---------------------------(5)
We know that sin A sin B = ½ [ cos (A-B) – cos (A + B)]

Hence, sin ωctsin ωmt = [cos (ωc-ωm)t – cos(ωc+ωm)t]

Cm(t) = Acsin ωct + µAc/2[cos (ωc-ωm)t – cos(ωc+ωm)t]
 Cm(t) = Acsin ωct + µAc/2cos (ωc-ωm)t – µAc/2cos(ωc+ωm)t--------(6)

Equation (6) shows that the amplitude modulated signal consists of

1. Carrier wave of frequency ωc
2. Sinusoidal wave of frequency (ωc-ωm)
3. Sinusoidal wave of frequency (ωc+ωm)
The two additional waves are called side bands. The frequency of these bands are called side band frequencies
Frequency of lower side band = (ωc-ωm)
Frequency of upper side band = (ωc+ωm)

The band width of the AM wave is Frequency of lower side band minus Frequency of upper side band
(ωc+ωm) - (ωc-ωm) = 2ωm (Twice the frequency of modulating signal)
Graphical representation

Production of amplitude modulated wave

We know that modulating signal is represented by

m(t) = Amsinωmt --------------------------------(1)
where, Am – Amplitude of modulating signal
ωm- 2π‫ט‬m – Angular frequency of modulating signal
Similarly, carrier wave is represented by
Cm(t) = Acsinωct --------------------------------(2)
Where, Ac – Amplitude of carrier wave
ωm- 2πωc– Angular frequency of carrier wave

Modulating signal is added to the carrier wave, Hence, the representation is

x(t) =Amsinωmt + Acsinωct
The above signal is passed to a square law device (non-linear device)
y(t) =B x(t) + C [x(t)]2
B,C – Arbitrary constants
Substitute for x(t) in y(t) and use formula (A + B)2 = A2 + B2 + 2AB
y(t) =B[Amsinωmt + Acsinωct] + C[Amsinωmt + Acsinωct]2
=B[Amsinωmt + Acsinωct] + C[Am2sin2ωmt + Ac 2sin 2ωct + 2 Am Acsin ωmtsin ωct]
We know sin A sin B = ½ [cos (A-B) – cos (A+B)]
Hence, sinωctsin ωmt = ½ [ cos (ωc – ωm)t – cos (ωc+ ωm)t ]
Also, sin2A = ( 1- cos 2A) /2
Hence,
sin 2ωct = (1 – cos 2ωct) / 2
sin 2ωmt = (1 – cos 2ωmt) / 2

Therefore y(t) can be re-written as

y(t) =B[Amsinωmt + Acsinωct] + cAm2/ 2(1– cos2ωmt) + cAc2/ 2( 1 – cos2ωct) + 2 Am Ac (c/2)[ cos (ωc – ωm)t –cos
(ωc+ ωm)t ] ]
y(t) =BAmsinωmt + BAcsinωct + c/ 2[Am 2 + Ac2] – cAm 2/ 2 cos2ωmt – c Ac2/ 2 cos2ωct + cAm Accos (ωc – ωm)t – c
Am Accos (ωc+ ωm)t
In the above equation, there is a d.c. term ½ c [Am 2 + Ac2] and sinusoidal waves of frequency ωc, ωm, 2ωm,(ωc –
ωm) and (ωc+ ωm)
The signal is passed through band pass filter centered at ωc
This rejects the low and high frequencies. In the above case, the filter rejects d.c, ωc,ωm, 2ωm,(ωc – ωm). The
frequencies ωc, (ωc – ωm) and (ωc+ ωm) are passed. This is amplitude modulated wave.
This wave cannot be passed as such. It needs to be amplified and then fed to an antenna of appropriate size
for radiation.

Detection of amplitude modulated wave

o Detection is the process of recovering the modulating signal from the modulated carrier wave.
o The transmitted message gets attenuated in propagating through the channel

o The receiving antenna receives the signal which is then amplified.

o The carrier frequency is changed to a lower frequency by Intermediate frequency (IF) stage

Detection Process:
o It is then passed through the detector.
Hence, INPUT – Modulated carrier wave of frequencies ωc, (ωc+ωm) and (ωc-ωm)
OUTPUT – Original signal m(t) of frequency ωm
We know that Rectifier consists of a simple circuit, which gives the input and output as indicated below:

The envelope detector gives the envelope of the given signal

Block Diagram for Detection of Amplitude modulated wave