II PU PHYSICS 1

CURRENT ELECTRICITY
Electric current is the measurement of NET flow of charges per unit time across a unit area.

i.e, (current) I = Q/t

Steady Current: Electric currents are not generally steady. Hence we define current as the rate of
change of small charge )q with respect to small time interval )t
D
Hence, limD ®
D

Strength of electric current (I):

Electric current is the rate of flow of charges through a conductor.
I = dq/dt = q/t = ne/t
NOTE:

Electric current is measured in ‘ampere’ (coulomb sec-1)

 In metals and vacuum tubes, current is due to the flow of electrons.
 In electrolytes current is due to the flow of ions. In semiconductors the current is due to the
flow of electrons as well as holes. Discharge tubes employ electrons and ions for current.
 Positive charges always move from a region of higher
potential to a region of lower potential.
 BUT in conductors, electrons flow from a region of lower
potential to the region of higher potential.
 Yet, we take the direction of positive charges (opposite
to the flow of electrons) as the direction of current in
conductors.
 Electric current is a SCALAR quantity
 There are two types of electric currents:
a) Direct current: This is a unidirectional constant current.
b) Alternating current: A periodically changing current in magnitude and direction

ELECTRIC CURRENT IN CONDUCTORS:

 Metallic conductors are composed of atoms. Each atom is a system of equal and opposite
charges where, nucleus is of +ve charge and electrons of –ve charge.
 The electrons in a metal are practically free to move within the bulk material. These materials
develop electric currents in them when an electric field is applied.

Current in the absence of Electric field:

 In the absence of electric field, the electrons will be moving due to thermal motion during
which they collide with the fixed ions.
 An electron elastically collides with an ion. Its speed remains same after each collision.
 But the direction of its velocity changes completely random.
 The average velocity of collisions is given as

 Thus on the average, the number of electrons travelling in any direction will be equal to the
number of electrons travelling in the opposite direction. So, THERE WILL BE NO NET
ELECTRIC CURRENT.
II PU PHYSICS 2

Current in the Presence of Electric field:

 When an electric field is applied between the ends
off a conductor, the valence electrons of the metal
flow towards the +ve field, constituting a current in
the conductors in the background of fixed positive
ions.
 If we place two oppositely charges metal plates at the two ends of a cylindrical conductor, these
plates create an electric field due to which, the free electrons are accelerated and move towards
the positive plate.
 They will thus move to neutralize the charges. The electrons, as long as they are moving, will
constitute an electric current.
 Hence e in the situation considered, there will be a current for a very short while and no current
thereafter.

CURRENT IN STEADY ELECTRIC FIELD

 When the ends of a conductor are not neutralized by the moving electrons, there will be a
steady electric field in the
e body of the conductor.
 This will result in A CONTINUOUS CURRENT rather than a current for a short period of time.
 Mechanisms, which maintain a steady electric field, are called cells or batteries.

NOTE
 In the applied field, electrons experience a force due to which, they are accelerated in a
direction opposite to the direction of the applied field.
 In the course of their flow, electrons collide with each other, with the +ve ions and as well with
the other impurities present in the conductors.
 As a result, current through a conductor has a RESISTANCE due to the material and
geometry of the conductor.
 At each collision, they lose momentum and accelerated again. Due to these repeated collisions,
a backward drag acts over the electrons. As a result, eleelectrons
ctrons move slowly through the
conductors with an average uniform velocity of the order 10 -4 m/s
 As temperature increases, the rate of collision between the electrons also increases.
 A steady electric field is achieved through applying a constant Potential difference between the
ends of the conductor.

OHM’S LAW
“At
At constant temperature, the strength of electric current through a conductor is directly proportional to
potential difference between its ends
ends”
i.e, V µI
V=IR
Where, R is the proportionality constant called the resistance of the conductor.

Electrical resistance ( R ):
 Resistance
esistance of a conductor is defined as the ratio of p.d across its ends to the current flowing
through it.
 S.I.. unit of resistance is ohm ( )
Definition of one ohm:
 Resistance of the conductor is said to be one ohm if a p.d of one volt across its ends produce a
current of one ampere.
i.e, 1 = 1V/1A
II PU PHYSICS 3

 Resistance is symbolically represented as

THE DEPENDENCE OF R ON THE DIMENSIONS OF THE CONDUCTOR:

Dependence on length;
 Imagine placing two such identical slabs side by side so that
the length of the combination is 2l.
 The current flowing through the combination is the same as
that flowing through either of the slabs.
 If V is the potential difference across the ends of the first slab,
then V is also the potential difference across the ends of the second
slab
slab.
 The potential difference across the ends of the combination is
clearly the sum of the potential difference across the two individual slabs and hence equals 2V.2
 The current through the combination is I and the resistance of the combination RC is
ۥ
Which implies that doubling the length of conductor, its resistance can be doubled.

IN GENERAL, the resistance R is direct

directly proportional to the length of the conductor l.
µ

Dependence on thickness (area of cross section):

 Imagine dividing the slab into two by cutting it lengthwise

len so that the
slab can be considered as a combination of two identical slabs of length l, but
each having a cross sectional area of A/2.
 For a given voltage V across the slab, if I is the current through the
entire slab, then clearly the current flowing through each of the two half-
half
slabs is I/2.
/2.
 Since the potential difference across the ends of the half
half-slabs is V,, i.e., the same as across the
full slab, the resistance of each of the half-slabs R1 is
‚ ƒ € •

Thus, halving the area of the cross-section

section of a conductor doubles the resistance.

IN GENERAL, the resistance R is inversely proportional to the cross-sectional

sectional area A,

µ
RESISTIVITY ( D) OF A CONDUCTOR
CONDUCTOR;
Consider that, µ and µ combining these relations, we get

 Where D is called resistivity of the material of the conductor which does not depend on the
dimensions (shape and size) of the conductor.
 if 1 and 1
II PU PHYSICS 4

“ Thus, resistivity is numerically equal to the resistance of a conductor of unit length and unit
area of cross section”
 Resistivity is also known as SPECIFIC RESISTANCE.
 The SI unit of resistivity is ohm metre (Sm)
 The reciprocal of resistivity is called CONDUCTIVITY denoted by F.
„ … … . †‡ ˆ‡
 Resistivity of metals is low. It ranges from 10-8 to 10-6 ohm metre.
 Resistivity depends on temperature and nature of the material.

CURRENT DENSITY ( j ):
From Ohm’s law, we have
‚ ‰ ‚ ‰

Š
Where j = I/A ; Current per unit area is called current density.
The SI units of the current density are A/m2.
“Current density is defined as the current per unit area normal to the direction of current.”

Ohm’s law in terms of current density (j) and electric field (E):
Further, if E is the magnitude of uniform electric field in the conductor whose length is l, then the
potential difference V across its ends is El. Using these, the equation of Ohm’s law reads as
Š
‹ Š
‹ Š
Is another way of expressing Ohm’s law, V =RI
Š
‹ (where F = 1/D is conductivity)
„
OR, Š „‹

NOTE
 When electrons are moving in a conductor, they undergo continuous collision.
 At each collision, they lose momentum and accelerated again. Due to these repeated collisions,
a backward drag acts over the electrons.
As a result, electrons move slowly through the conductors with an average uniform velocity of
the order 10-4 m/s
 Relaxation time (): The average time between two successive collisions is called relaxation
time. (of the order 10-14s) Œ • åŽ •
Ž
 Drift velocity (vd ): The average uniform velocity with which, electrons move in a conductor is
called drift velocity.

EXPRESSION FOR ELECTRIC CURRENT:

Consider a conductor of length l, area of cross section A where electrons
are moving with a drift velocity vd . When current is set in the conductor,
let n be the number of electrons per unit volume.

The total number of electrons in a given volume V is given by

Ž Ž
II PU PHYSICS 5

Total charge carried by the electrons, • • Ž •

If ‘ Þ ‘ • then, • Ž ‘••
•
• Ž ‘••
By definition,
• •
\ Ž • ‘

DERIVATION OF OHM’S LAW:

Consider the electrons moving in a conductor in the influence of an electric field. The acceleration of
an electron due to the applied field is given by
’ •‹
“ “
Where, m = mass of electron, E = applied electric field and e = charge on electron
If vd is the drift velocity and ti is the relaxation time, then ‘ ” Œ
But, the average velocity of electrons before the application of electric field is zero.
i.e,
•

/Ž

‘ ” Œ

•‹Œ
‘ “
Consider the expression for current, Ž • ‘
•‹Œ Œ
Ž •‚ ‰ ‹
“ “

Ž • Œ
As ‹ we can write ™ š
“
“
›œ,
Ž • Œ
OR,
“
where , called as the resistance of the conductor.
Ž • Œ

NOTE:
“ “ “
 ™ š where
Ž • Œ Ž• Œ Ž• Œ
Ž• Œ
 As resistivity is the reciprocal of conductivity, , we have „
„ “
EXPRESSION FOR ELECRICAL CONDUCTIVITY (F):
Consider a conductor of length l, area of cross section A where electrons are moving with a drift
velocity vd . When current is set in the conductor, let n be the number of electrons per unit volume.

Consider the expression for current and drift velocity respectively as

•‹Œ
Ž • ‘ and ‘ “
Where, E is the electric field applied, m is the mass of electrons and  is relaxation time.
•‹Œ Ž‹• Œ
Þ Ž •‚ ‰
“ “
Ž• Œ
• , ‹
“
By definition I is related to the magnitude |j| of the current density by |j| = I/A
II PU PHYSICS 6

Ž• Œ
|Š| |‹|
“
Vectorially, the direction of žŸ is parallel to ‹Ÿ. Hence,
Ž• Œ
žŸ ¡‹Ÿ¢
“
OR žŸ „ ¡‹Ÿ¢
Ž• Œ
Where „ is electrical conductivity.
“
Mobility ():
Mobility is defined as the magnitude of the drift velocity per unit electric field.
‘
£
‹
The SI unit of mobility is m2 V-1s-1
¥¦§ •Œ
As ¤ , we have £
¨ “
NOTE:
Ohmic devices:
Devices which obey Ohm’s law are called Ohmic devices.
Eg. Galvanometer, Voltmeter, Ammeter etc.
For ohmic devices, the graph plotted I vs V is a straight line

Non-Ohmic devices:
Devices which do not obey Ohm’s law are called non-Ohmic devices.
Eg. Thermistor, diode, vacuum tube, etc.
For non-Ohmic devices, graph plotted I vs V is non linear.

LIMITATIONS OF OHM’S LAW:

 Ohm’s law is applicable for metallic conductors when temperature remains constant.
 For certain materials like semiconductors, V is no longer proportional to I {fig (1) }
 Same current is not produced when we reverse the positive potential to negative in diodes.
 For materials like GaAs (Gallium Arsenide), there are two or more values of V for the same
current. {fig (2) }

(1) (2)

TEMPERATURE DEPENDENCE OF RESISTIVITY OF CONDUCTORS:

 Resistivity of conductors depends on temperature.
 If D is the resistivity of a metal at the temperature T and D0 is the resistivity at 00C, then
” © Dª
Where, " is the temperature coefficient of temperature.
Temperature Coefficient of Resistivity (")
 It is defined as the change in resistivity of a conductor to its value at zero degree Celsius per
‡
change in temperature. i.e, © ™ š
Dª
II PU PHYSICS 7

 The value of temperature coefficient of resistance is POSITIVE for

conductors.
 The graph of resistivity versus temperature is a straight line.
 Its value of © is about 10-3 for pure metals.
 SI Unit of © is per degree celcius OR per kelvin.
 © of an alloy is less than its constituent metals.
 At temperatures much lower than 0°C, the graph, however,
deviates considerably from a straight line
line.
NOTE:
 Temperature coefficient of resistance can also be determined by the change in resistance of the
metal with temperature (when the dimensions of metal remain same) as,
” © Dª
 If R1 is the resistance at t1 C and R2 is the resistance at t20C, then
0
³
©
• ³ •
 As temperature decreases, the temp.coeff.of resistivity decreases. At very low temperatures, the
resistivity of a metal becomes zero at a certain temperature, called CRITICAL TEMP (TC)
 At critical temp. the conductor becomes a SUPERCONDUCTOR
 (Eg: Mercury becomes superconductor at 4.2K)
TEMPERATURE DEPENDENCE OF RESISTIVITY OF SEMICONDUCTORS:
 Resistivity of a semiconductor decreases exponentially with temperature as
«
• ¬ where a & b are constants & T is absolute temp.
 © is negative for semiconductors.
conductors.
 The graphical variation of resistivity of semiconductors with temperature is
as shown in the fig.
NOTE:
´
 semiconductor is given by ©
cient of resistivity of the semi
The temp. coefficient ³
ª
 When R1 and R2 are the resistances at T1 K and T2 K respectively, then
- ¯° ± •
© …………..per kelvin
ª ‡ª
Difference between Temp. dependence of resistivity of conductors and semiconductors
Conductors Semiconductors
The dependence of D with temp. is linear The dependence of D with temp. is exponential
² is positive ² is negative
D increases with temp. D decreases with temp.
Number of charge carriers per unit volume Number of charge carriers per unit volume
(n) doesnot change much with temp. (n) changes with temp.
NOTE: in conductors, as D " 1/, only relaxation time decreases with rise in T to increase D.

RESISTORS:
Commercially produced standard resistances for domestic use or in laboratories are called resistors.
They are of two major types:
Wire bound resistors and Carbon
arbon resistors
resistors.
Wire bound resistors:
 Wire bound resistors are made by winding the wires of an alloy, viz., manganin, constantan,
nichrome or similar ones.
II PU PHYSICS 8


The metals chosen for wire bound resistors are such that their resistivities are relatively
insensitive to temperature.
 These resistances are typically in the range of a fraction of an ohm to a few hundred ohms.
 To make standard resistances, we use the coils of Constantin (Cu+Ni+Fe+Mn), Manganin
(Cu+Ni+ Mn) or Nichrome (Ni+Ch+ Fe) wireswires.
 Because their resistivity is high and their " is low. As a result, their resistance will not vary
much with the temperature.
Carbon resistors:
 Carbon resistors are compact, inexpensive and thus find extensive use in electronic circuits.
 Carbon resistors are
re small in size and hence their values are given using a colour code.

COLOR CODING OF CARBON RESISTORS

A type of resistances is
COLOR CODE COLOR CODE
available as carbon
Black 0 aaaaa Green 5 resistors, compact in size
Brown 1 Blue 6 and shape with a standard
value of resistance.
Red 2 Violet 7
Carbon resistors have four coloured bands. Ist band
Orange 3 Grey 8 refers to Ist digit of the resistor, 2nd band to the 2nd digit
Yellow 4 White 9 and the 3rd band refers rs to the power of ten to be raised.
Fourth band is called tolerance band, which gives the %
COLOR TOLERANCE
error in determining the standard value of resistor from the first
Gold ½5% three bands.
Silver ½10% The coding of resistors can be given as shown in the table.
NOTE: To remember the color code,
No color ½20%
“BBROY of Great Britain has a Very Good Wife”
Eg. For the color code –yellow
yellow orange red gold, we have, R = 43 X 102 ± 5%

ELECTRICAL ENERGY, POWER:

Consider an electrical circuit consisting of a cell of p.d V, connected across a conductor of ends A and
B carrying a current I.
p.d across the conductor, ³ ¹
Let Q ammount of charges flow across AB in a time duration of )t,, such that, Q = I.)t
I.
Work done by the cell in carrying the charge Q from end A to B is the difference in potential energy of
the charges.
i.e, D¶ •· ¸ ³ •· ¹ ¸ • D•
From work-energy
energy theorem, we have, (difference in PE) = KE of charges
Due to the resistance of the conductor,
ductor, any increase in kinetic energy of charges in the conductor
would result in the frequency of collisions
collisions. The energy gained by the charges thus is shared with the
atoms. The atoms vibrate more vigorously, i.e., the conductor heats up. Thus, in an actual
act conductor,
an amount of energy dissipated as heat in the conductor
conductor.
Energy dissipated per unit time is the POWER dissipated across AB.
D¶ D•
¾
D• D•
Thus ¿ À Á
Since º » , we can write, ¼ œ
OR ¾
II PU PHYSICS 9

Also, ¾ ™ š OR ¾
The power necessary for the current in a circuit is supplied by the source. i.e cell/battery.
POWER LOSS IN TRANSMISSION:
 The power generated in power stations is supplied to the domestic and industrial purposes
through transmission cables hundreds of miles away.
 As the resistance varies with length of conductor, the longer length of cables offer a higher
power loss.
 Consider a device which works on supplying a power of P watts such that,
¾
Let Rc be the resistance of the transmission cables through which the power is delivered to the
device.
¾
The dissipated power in cables is ¾ ™ š

 OR, ¾ µ
 Thus, to drive a device of power P, the power wasted in the connecting wires is inversely
proportional to V 2.
Why transmission lines carry current at higher voltages?
 The transmission cables from power stations are hundreds of miles long and their resistance
Rc is considerable.
 To reduce Pc, these wires carry current at enormous values of V and this is the reason for the
high voltage danger signs on transmission lines.

Effective resistance: Effective resistance or equivalent resistance is the single resistance, which
effectively replaces the effective resistance of the entire combination.

RESISTANCES IN SERIES COMBINATION:

Consider three resistances R1 , R2 and R3 connected in
series across a p.d of V volts. Let I be the current flowing
through the combination.
 In series combination, current is same and
 The total p.d is the sum of pd across each
resistance.
i.e. ” ” Â ” ” Â ” ” Â
If RS is the effective resistance of the entire combination. From Ohm’s law we have,
à œÄ
\ Å ” ” Â
Æ ” ” Â

“In series combination, equivalent resistance is the sum of individual resistances”.

NOTE:
 If many resistances are in series, then effective resistance is given by
Å ” ” Â”Ç Ž
If œÈ œÉ œÊ Ç œË then, Å Ž
II PU PHYSICS 10

 p.d across each resistor is given by , and

Ì Ì
RESISTANCES CONNECTED IN PARALLEL:
Consider three resistances R1 , R2 and R3 connected in parallel
across a p.d of V volts. Let I be the current flowing through
the combination.
 In parallel combination, p.d across each resistance is
same.
 Total current through the combination is the sum of
current through each resistance
i.e, ” ” Â ” ”
Â
Let RP be the effective resistance of the entire combination.
From Ohm’s law we have
¾

\ ” ”
¾ Â

™ ” ” š
¾ Â

™ ” ” š
¾ Â
“In parallel combination, reciprocal of equivalent resistance is the sum of reciprocals of individual resistances”

NOTE:
È È È È È
 If many resistances in parallel, then ™ ” ” ”Ç š ;
ÍÎ ÍÏ ÍÐ ÍÑ ÍÒ
 If œÈ œÉ œÊ Ç œË then, ¾ Ž
 If two resistances are in parallel then, ¾ Ì

 If same resistors, R are connected in parallel, then ¾

 If n equal resistances each of resistance R are connected to form triangle (or) Square (or)
ˇÈ
Polygon then effective resistance between any two adjacent corners is œÓ œ
Ë

BRANCH CURRENT:
It is the current flowing through any of the resistors when many resistors are connected in parallel.
Consider two resistances R1 , R2 connected in parallel across a p.d
of V volts. Let I be the current flowing through the combination.
Let I1and I2 be the currents flowing through R1 and R2 respectively.

We have, ™ š Similarly, ™ š
Ì Ì

NOTE:
 This rule is applicable only for parallel combination.
 When many resistances are connected in parallel to a cell, then current through different
¾ ¾ ¾
branches is given by , , Â and so on.
Â
II PU PHYSICS 11

ELECTRO MOTIVE FORCE (EMF) OF A CELL ( ):

“e.m.f of a cell is the work done by the cell on a unit positive charge to make it complete one cycle of the
circuit”
S.I unit of e.m.f is volt (V)

emf of a cell is said to be one volt if one joule of work is done by the cell on a unit positive charge to
make it complete one cycle of the circuit.

Internal resistance of a cell (r):

Internal resistance is the effective resistance offered by material of the cell to the flow of charges
between its electrodes.

NOTE:
 Internal resistance of a cell increases with external resistance & with the continuous usage of
the cell.
 emf is the potential difference between the positive and negative electrodes in an open circuit.
 In practical calculations, internal resistances of cells in the circuit may be neglected when the
current I is such that >> I r.
 The actual values of the internal resistances of cells vary from cell to cell. The internal
resistance of dry cells, however, is much higher than the common electrolytic cells.

Terminal p.d :
It is the p.d between the terminals of an external resistance in the circuit.

Ohm’s law applied to a circuit:

Consider an electrical network in which, a resistance R is connected across a
cell of e.m.f  and of internal resistance r. Let I be the current flowing in the
circuit. Let V be the terminal p.d across R.

Then from Ohm’s law, à œ

Similarly, let V’ be the p.d across internal r then, Ã ¢ Ô

By definition, Õ Ã ” â Ô Ô OR Õ³ Ö
œ” Ô œ”Ô

Õ
OR
ÌÖ

Õ Õ
Or, Ö
ÌÖ ™ Ì š

Õ
Ö
™ Ì š

CASE (I):
 When Ô 0, then from above equation, Õ
Thus, terminal p.d is equal to e.m.f when the internal resistance of cell is zero.

CASE (II):
 When œ ¥ or when the circuit is open, then Õ
Thus, e.m.f of the circuit is the p.d across the resistance when circuit is open.
II PU PHYSICS 12

COMBINATION OF CELLS IN SERIES:

Consider first two cells in series where one terminal of the two cells is joined together leaving the
other terminal in either cell free. 1 , 2 are the emf’s of the two cells and r1, r2 their internal
resistances, respectively.
Let V (A), V (B), V (C) be the potentials at points A, B and C shown in Fig.
Then V (A) – V (B) is the potential difference between the positive and negative terminals of the first
cell and hence,
¹ ³ ¹ Õ ³ Ö
¹× ¹ ³ × Õ ³ Ö

× ³ ¹ ” ¹ ³ ×
Õ ”Õ ³ Ö ”Ö

× Õ•Ø ³ Ö•Ø
Where, Õ•Ø Õ ”Õ and Ö•Ø Ö ” Ö
NOTE:
 If instead we connect the two negative electrodes of the cell, Eq. would change to
Õ•Ø Õ ³ Õ and Ö•Ø Ö ” Ö

COMBINATION OF CELLS IN PARALLEL:

Consider first two cells in parallel where same terminals of the two cells are joined together across
each other. Let 1 , 2 are the emf’s of the two cells and r1, r2 their internal resistances,respectively.
Let I1 and I2 be the currents given out by the cells as shown in the fig such that,
”
Let V(B1) and V(B2) be the potentials at the points B1 and B2
respectively such that, p.d between them across the both the cells is
same.
p.d between B1 and B2 across the First cell is given by
Õ ‡
¹ ³ ¹ Õ ³ Ö OR
Ö
p.d between B1 and B2 across the Second cell is given by
Õ ‡
¹ ³ ¹ Õ ³ Ö OR
Ö
Consider,
Õ ³ Õ ³
” ”
Ö Ö

Õ Õ
Ù ” Ú³ Ù ” Ú
Ö Ö Ö Ö

Õ Ö ”Õ Ö Ö ”Ö
‚ ‰³ ‚ ‰
Ö Ö Ö Ö

Ö ”Ö Õ Ö ”Õ Ö
‚ ‰ ‚ ‰³
Ö Ö Ö Ö

Ö Ö Õ Ö ”Õ Ö Ö Ö
‚ ‰‚ ‰³ ‚ ‰
Ö ”Ö Ö Ö Ö ”Ö

Õ Ö ”Õ Ö Ö Ö
‚ ‰³ ‚ ‰
Ö ”Ö Ö ”Ö
II PU PHYSICS 13

OR,

Where, and

NOTE:
 When n cells are connected in parallel,
 and if then,
 If the negative terminal of the second is connected to positive terminal of the first, Equations
would still be valid with

KIRCHHOFF’S RULES
Electrical network: An electrical network is the combination of various electrical components in a
circuit.
Node: it is the intersection of two or more conductors in an electrical network.
Mesh or loop: It is a closed path for the current in a network such that it does not contain another
closed within it.

NOTE:
 According to the sign convention given by Kirchhoff, the currents entering a node are taken to
be +ve and the currents leaving a node are taken to be –ve.

Kirchhoff’s Current rule (JUNCTION RULE):

“The algebraic sum of currents at a node in an electrical network is zero.”
i.e, S 0
OR
“The sum of currents entering a junction is always equal to the
sum of currents leaving the junction.”
Explanation: If I1, I2, I3 are the currents entering a node, and I4
and I5 are the currents leaving it, then from I law, we have
0

NOTE:
 According to the sign convention given by Kirchhoff, while analyzing a mesh in a particular
direction, if the direction of current in a branch is parallel to the direction of motion, such
currents are taken to be +ve. Otherwise they are taken to be –ve.
 Kirchhoff’s current law signifies the law of conservation of charges.

Kirchhoff’s Voltage Law (LOOP RULE):

“The algebraic sum of changes in potential around any closed loop involving resistors and cells in the
loop is zero”
OR
“ In any mesh of an electrical network, the algebraic sum of the potential diff. across each branch is
equal to the algebraic sum of e.m.fs in that mesh.”
i.e, S S
II PU PHYSICS 14

Using sign convention we have,

NOTE: Kirchhoff’s voltage law signifies the law of

conservation of energy.

WHEATSTONE’S BRIDGE:
Wheat stones Bridge/network consists of four resistors P, Q, R and connected in cyclic order in the
form of a quadrilateral as shown in the fig. A galvanometer of resistance G is connected between the
ends B and D.

CONDITION FOR BALANCE OF THE WHEATSTONE’S NETWORK:

If the resistances in a Wheatstone’s bridge are adjusted such that the current through the
galvanometer is zero, the network is said to be balanced.

Using Kirchhoff’s I law at node B,

And at the node D, €

Applying Kirchhoff’s II law to the mesh ABDA,

• ‚ ƒ „ ………..(1)

Applying Kirchhoff’s II law to the mesh BCDB,

… †‡ … †ˆ ‚ „ ………………..(2)

When the network is balanced, current through galvanometer is zero ( ‰ 0

(1) Reduces to, • ƒ „ Þ • ƒ ………………(3)
(2) Reduces to, ‡ ˆ „ Þ ‡ ˆ ..……………(4)

Dividing eqn (3) and (4),

• ƒ
‡ ˆ

NOTE:
 By the above condition for balance, we can determine the unknown resistance of a resistor if
the other three resistors are known.
 Balanced condition is unaffected
o When the cell and galvanometer are interchanged.
o When the galvanometer is replaced by another high or low resistance.
o When the emf source is changed.
II PU PHYSICS 15

METER BRIDGE:
 Meter bridge works on the principle of Wheatstone’s
network.
 It consists of a wire of uniform resistance of length
between the terminals A and B as shown in the figure.
 Meter bridge has a left gap connected to A through a
metal strip in which an unknown resistance X is
connected.
 The mid strip is connected to a galvanometer at the terminal C. Galvanometer is then
connected to the wire via a sliding pencil jockey.
 The right gap of the meter bridge is connected by a standard resistance R to the terminal B.
 By adjusting the position of sliding contact on the wire the position D is determined where
galvanometer shows zero deflection.
 Now the meter bridge is said to be balanced. Length AD=l represents a certain resistance and
so is the remaining length DB= (1- l )

Using Wheatstone’s condition for balance,

Š ‹ ƒ‹
Œ •ƒ, Š
ƒ ‹ ‹
POTENTIOMETER
This is an instrument similar to meter bridge where the length of the wire is more than a meter.

POTENTIOMETER ( TO COMPARE THE EMF OF TWO CELLS)

Consider a potentiometer consisting of a wire of length
AB.
The potential difference between any two points at a
distance l from A is directly proportional to l
i.e, V=Nl
where, N is the potential drop per unit length of the
wire.
Consider two cells of emfs E1 and E2 connected across
AB as shown in the figure through a two way key (K1
and K2) as shown in the figure.
When K1 is closed:
When the first key is closed, the galvanometer is
connected only to the cell E1. The jockey is moved along the wire such that the deflection in it shows
zero at a point C1 having a length say, l1 from A.
Applying KIRCHHOFF’s Loop rule to the mesh AE1K1GC1, we get Nl1 =E1……(1)
When K2 is closed:
When the second key is closed, the galvanometer is connected only to the cell E2. The jockey is moved
along the wire such that the deflection in it shows zero at a point C2 having a length say, l2 from A.
Applying KIRCHHOFF’s Loop rule to the mesh AE2K2GC2, we get Nl2 =E2…….(2)
Comparing (1) and (2), we get
Ž ‹
Ž ‹

NOTE:
II PU PHYSICS 16

 This simple mechanism thus allows one to compare the emf’s of any two sources.
 In practice one of the cells is chosen as a standard cell whose emf is known to a high degree of
accuracy. The emf of the other cell is then easily calculated from above Eq.

POTENTIOMETER (TO FIND THE INTERNAL RESISTANCE OF A CELL)

We can also use a potentiometer to measure internal
resistance of a cell. For this, the cell (emf ) whose internal
resistance (r)) is to be determined is connected across a
resistance box through a key K2, as shown in the figure.
When key K2 is open: balance is obtained at length l1 (C1).
Then, E = l1
When key K2 is closed: the cell sends a current (I ( )
through the resistance box (R). If V is the terminal potential
difference of the cell and balance is obtained at length l2
(C2), such that V = l2

Ž ‹
Hence, we have
‹
But, Ž ƒ and ƒ
ƒ ‹
This implies,
ƒ ‹
‹
OR ƒ• •
‹

NOTE:
 Using above Equation, we can find the internal resistance of a given cell.
 The potentiometer has the advantage that it draws no current from the voltage source being
measured. As such it is unaffected by the internal resistance of the source.