1

2(a) Electric Current And Resistance During random motion these free electrons collide with the
ions/atoms of the conductor. The time elapse between two
When two conductors at different potentials are connected consecutive collisions of a free electron is known as
by a wire then charge flows from one conductor to another. relaxation time (𝝉).
Flowing electric charge constitute electric current
Let 𝐸 is electric field applied on the conductor, and 𝑒 is
Electric Current : Electric current is defined as the total charge on electron, then force on a free electron is
electric charge crossing any cross-section of a conductor per
unit time. i.e. 𝐹 = −𝑒 𝐸 ……...(2)

Total charge flowing

Electric current =
Time Taken
𝑞
or 𝐼= ……..(1)
𝑡

 Electric current is a scalar quantity.

 S.I unit of electric current is Ampere (A)
𝟏 𝐂𝐨𝐮𝐥𝐨𝐦𝐛
From eqn. (1) we get 𝟏 𝐀𝐦𝐩𝐞𝐫𝐞 =
𝟏 𝐒𝐞𝐜𝐨𝐧𝐝

i.e. current is said to be one ampere, if one coulomb of If m is mass and 𝑎 is acceleration of electron, then force
electric charge crosses a cross-section in one second. on electron is 𝐹 = 𝑚 𝑎…………(3)

Current carriers : The charged particles that constitute From eqns. (2) and (3) we get
electric current by flowing are known as current carriers.
𝑚 𝑎 = −𝑒 𝐸
1. In solid conductors the current carriers are electrons.
2. In liquids (electrolytes) the current carriers are positive 𝑎=
−𝑒 𝐸
……….(4)
and negative ions. 𝑚

3. In ionized gases the current carriers are positive ions Let 𝛕𝟏 ,𝛕𝟐 , 𝛕𝟑 … … … … . . 𝛕𝐧 be the relaxation time for n-
and electrons. electrons respectively, then velocities acquired by the
Direction of electric current : Direction of electric current electrons are given by
is taken along the direction of flow of positive charge or in 𝒗𝟏 = 𝒖𝟏 + 𝒂 𝝉𝟏
a direction opposite to the flow of negative charge.
𝒗𝟐 = 𝒖𝟐 + 𝒂 𝝉𝟐
Drift velocity : Drift velocity is defined as the average
velocity acquired by the free electrons in a conductor 𝒗𝟑 = 𝒖𝟑 + 𝒂 𝝉𝟑
under the influence of an external electric field.
………………..
At room temperature free electrons in a conductor move
randomly with certain velocities. These velocities are known 𝒗𝒏 = 𝒖𝒏 + 𝒂 𝝉𝒏
as thermal velocities. The average of these thermal
velocities is zero. Average velocity is given by
𝑣1 + 𝑣2 + 𝑣3 + ……….𝑣𝑛
𝑣𝑑 =
𝒏

𝒖𝟏 +𝒂 𝝉𝟏 + 𝒖𝟐 +𝒂 𝝉𝟐 +⋯.… 𝒖𝒏 +𝒂 𝝉𝒏
or 𝑣𝑑 =
𝒏

𝑢 1 +𝑢 2 +𝑢 3 …..𝑢 𝑛 +𝒂 𝝉𝟏 +𝝉𝟐 +𝝉𝟑 ….𝝉𝒏

or 𝒗𝒅 =
𝒏

𝑢 1 +𝑢 2 +𝑢 3 …..𝑢 𝑛 𝒂 𝝉𝟏 +𝝉𝟐 +𝝉𝟑 ….𝝉𝒏

or 𝒗𝒅 = +
𝒏 𝒏

By using eqn. (1) And

Hence if u1 , u2 , u3 , ……………un be the thermal
𝛕𝟏 +𝛕𝟐 +𝛕𝟑 ….𝛕𝐧
velocities of n-electrons, then average of these velocities is By putting =𝛕 (Average relax time )
𝐧
u 1 + u 2 + u 3 + ……….u n
u=
𝐧
= 𝟎.......(1) We get 𝐯𝐝 = 𝐚 𝛕
 2

Put value of 𝑎 from eqn. (4) then we get

−𝑒 𝐸
𝑣𝑑 = 𝜏
𝑚

This average velocity 𝑣𝑑 is known as drift velocity.

Relation of drift velocity with electric current : Consider Resistance (R) : Resistance is defined as the obstruction
a conductor of length 𝑙 and area of cross-section A. Let n offer to the flow of electric current by the conductor. The
be the number density of free electrons in the conductor. value of Resistance depends upon:-

1. Dimensions of the conductor.

2. Nature of material of the conductor.

3. Temperature of the conductor.

S.I. unit of the resistance is ohm (Ω)

𝑉
Total number of free electrons in the conductor is = 𝒏 𝑨 𝒍 From 𝑅= we get
𝐼

1 Volt
Let 𝒆 be the charge on electron, then total charge on the 1 Ohm =
1 Ampere
free electrons in the conductor is 𝒒 = 𝒏 𝑨 𝒍 𝒆 …….(1)
𝑞 i.e. resistance of a conductor is said to be 1 Ohm when
Electric current is 𝐼 = potential difference of 1 volt across its ends gives a current
𝑡
of 1 Ampere through it.
Put value of 𝑞 from eqn. eqn. (1) then we get
𝒏𝑨𝒍𝒆
Dimensional formula of resistance is [𝑴𝟏 𝑳𝟑 𝑻−𝟑 𝑨−𝟐 ]
𝐼=
𝑡
Cause of resistance : We know that, electric current flows
𝒍 through the conductor due to the movement of current
𝐼 = 𝑛𝑒𝐴𝒗𝒅 [ As = 𝒗𝒅 ]
𝒕 carriers. These current carriers collide with the atom or ions
of the conductor during their movement. These collisions
Ohm’s Law : Ohm’s law states that, electric current
give opposition to the flow of current carriers. This
flowing through a conductor is directly proportional to the
opposition to the flow of current carriers is known as
potential different across the ends of the conductor,
resistance of the conductor.
provided the physical conditions like temperature,
mechanical strain etc. remain constant OR vice versa Resistor : Resistor is a device which is used to offer a high
resistance. A resistor is commonly called as resistance.
i.e. Potential Difference ∝ Electric Current
Resistivity or Specific resistance :
or 𝑽 ∝𝑰
It can be studied that resistance (R) of a conductor is :
or 𝑉 = 𝑅𝐼
1. Directly proportional to the length 𝒍 of the conductor
Where R is a constant of proportionality and is known as i.e. 𝑹 ∝ 𝒍
Resistance of the conductor.
2. Inversely proportional to the area of cross-Section 𝑨 of
Ohmic Conductors : The conductors which obey ohm’s 𝟏
law are known as Ohmic conductors. the conductor. i.e. 𝑹 ∝
𝑨

Non-Ohmic conductors : The conductors which do not By combining we will get

obey ohm’s law are known as Non-Ohmic conductors. For
𝒍
ex. semiconductor diode, transistor, vacuum tube, liquid 𝑹∝
𝑨
electrolyte etc. are non-ohmic conductors.
𝒍
or 𝑹=𝝆
𝑨

Where 𝜌 is the constant of proportionality and is known as

the resistivity or specific resistance of the conductor.
 3

The value of 𝜌 depends upon :- Hence 𝝆 ∝

1
𝜏
1. Nature of material of the conductor.
…………………………………………………………………..
2. Temperature of the conductor.
Variation of Resistivity with Temperature: The variation
The value of 𝜌 does not depend upon the dimensions or of resistivity with temperature is different in conductors, semi
size of the conductor. conductors and insulators, which is explained below:

Definition of Resistivity or Specific Resistance (𝜌) : 1. In conductors : In a conductor there are large number
of free electrons. When temperature of the conductor is
In eqn. 𝑹=𝝆
𝒍
If increased, then collisions of the free electrons become
𝑨 faster. Due to which the relaxation time (  ) decreases.
1
𝒍= 𝟏 𝑎𝑛𝑑 𝑨 = 𝟏, then we get 𝑅 = 𝜌 Hence from 𝜌∝ the resistivity 𝜌 of the
𝜏
conductor increases.
i.e. resistivity (𝜌) is equal to the resistance (R) of conductor 2. In Semiconductors : In semiconductors there are very
of 1 unit length and 1 unit area of cross-section. less free electrons at room temperature. When
temperature of semiconductor is increased then more
 S.I unit of resistivity is ohm-meter (Ω-m)
electrons become free. Hence the number density (n)
 Dimensional formula of resistivity is [𝑀1 𝐿3 𝑇 −3 𝐴−2 ] of free electrons increases. Due to which the resistivity
𝜌 of the semiconductor decreases.
To deduce Ohm’s law : Electric current is given by

𝑰 = 𝒏 𝒆 𝑨𝒗𝒅
𝑒𝐸
Put 𝑣𝑑 = 𝜏 then we get
𝑚

𝑒𝐸
𝑰 = 𝒏 𝒆𝑨 𝜏
𝑚

𝑉
Put 𝐸= then we get
𝑙
3. In insulators : In insulators there are no free electrons
𝑒𝐸
𝑰 = 𝒏 𝒆𝑨 𝜏 at room temperature. When temperature of insulator is
𝑚
increased then very few electrons become free to show
𝑉 𝑚𝑙 small conductivity. Hence resistivity of the insulator
or =
𝑙 𝑛𝑒 2 𝜏𝐴 decreases on increasing the temperature.
𝑉 𝑚𝑙
or =R [ Where 𝑅 = is constant ] Temperature coefficient of resistivity (𝛼𝜌 ) :Temperature
𝐼 𝑛𝑒 2 𝜏𝐴
coefficient of the resistivity is defined as the change in
or 𝑉 = 𝐼𝑅 resistivity of the conductor per unit its original
resistance per degree change in its temperature.
Which is Ohm’s law
Let 𝜌 is the resistivity of a conductor at temperature 𝑇
…………………………………………………………………..
𝑚𝑙
Let 𝜌′ is the resistivity of a conductor at temperature 𝑇′
Important Results : As 𝑅= ….(1)
𝑛𝑒 2 𝜏𝐴
Then temperature coefficient of resistivity of the conductor is
For a conductor 𝑚, 𝑛, 𝑙, 𝐴 𝑎𝑛𝑑 𝑒 and are constants. given by
1 𝝆′ −𝝆
Hence 𝑅 ∝ 𝛼𝜌 =
𝜏 𝝆(𝑻′ −𝑻)
𝒍
Also 𝑹 = 𝝆 …..(2)  For conductors the resistivity increases with increase in
𝑨
temperature. Hence 𝛼𝜌 is positive for the conductors.
By comparing eqns.(1) and (2) we get
𝑚
 For semiconductors and insulators the resistivity
𝝆=
𝑛𝑒 2 𝜏
decreases with increase in temperature. Hence 𝛼𝜌 is
negative for the semiconductors and insulators.
For a conductor 𝑚, 𝑛, 𝑎𝑛𝑑 𝑒 and are constants.
 4

Alloys like mangnin, Constantan, Eureka has very small Let 𝑽𝟏 , 𝑽𝟐 𝒂𝒏𝒅 𝑽𝟑 is the potential difference
value 𝛼𝜌 . Due to which these alloys are used for making across resistance 𝑹𝟏 , 𝑹𝟐 𝒂𝒏𝒅 𝑹𝟑 respectively.
standard resistances. Then total potential difference across the series combination
is
Variation of Resistance with Temperature: ( Same as the
of resistivity with temperature) 𝑉 = 𝑽𝟏 + 𝑽𝟐 +𝑽𝟑 ……(1)

Temperature coefficient of Resistance : ( Same as the Let 𝑰 is current in the circuit, then using Ohm’s we get
temperature coefficient of the resistivity )
𝑉 = 𝐼𝑅 𝑽𝟏 = 𝑰 𝑹 𝟏
Definitions :
And 𝑽𝟐 = 𝑰 𝑹 𝟐 𝑽𝟑 = 𝑰 𝑹 𝟑
1. Current density(J): Current density at a point is
Put these values in eqn. (1) then we get
defined as the magnitude of the electric current flowing
per unit area of cross-section of a conductor, around 𝑰𝑹 = 𝑰 𝑹𝟏 + 𝑰 𝑹𝟐 + 𝑰 𝑹𝟑
that point, when this area is normal to the current. i.e.
𝑰
𝑹 = 𝑹𝟏 + 𝑹𝟐 + 𝑹𝟑 ……….(2)
𝑱=
𝑨
Where 𝑹 is equivalent resistance of the series combination
 S.I. unit of current density is 𝐴𝑚2 . It is a vector of three resistances 𝑹𝟏 , 𝑹𝟐 𝒂𝒏𝒅 𝑹𝟑 .
quantity. Its direction is along the direction of the electric
current. 𝑭𝒐𝒓 𝒏 − 𝒓𝒆𝒔𝒊𝒔𝒕𝒂𝒏𝒄𝒆𝒔 𝒊𝒏 𝒔𝒆𝒓𝒊𝒆𝒔

2. Electrical conductance (𝑪):- The reciprocal of the 𝑹 = 𝑹𝟏 + 𝑹𝟐 + 𝑹𝟑 ……… 𝑹𝒏

resistance of a conductor is called its conductance .
(2)Resistances in parallel : Parallel arrangement of three
1 resistances 𝑹𝟏 , 𝑹𝟐 𝒂𝒏𝒅 𝑹𝟑 is shown in the following
i.e. 𝐶=
𝑅 circuit.
S.I. unit of conductivity is mho or Simen

3. Electrical conductivity (𝝈):- The reciprocal of the

resistivity of a conductor is called its conductivity.
1
i.e. 𝜎=
𝜌

S.I. unit of conductivity is mho𝑚−1 or 𝑆𝑖𝑚𝑒𝑛. 𝑚𝑒𝑡𝑒𝑟 −1

4. Mobility (𝝁) :- Mobility is defined as the drift velocity of Let 𝑰𝟏 , 𝑰𝟐 𝒂𝒏𝒅 𝑰𝟑 is the current through the
the of the current carriers per unit electric field applied. resistance 𝑹𝟏 , 𝑹𝟐 𝒂𝒏𝒅 𝑹𝟑 respectively, Then total
𝑣𝑑
current in the circuit is
i.e. 𝜇 =
𝐸
𝐼 = 𝑰𝟏 + 𝑰𝟐 + 𝑰𝟑 …….(1)
S.I unit of mobility is 𝑚𝑠 −1 𝑁 −1 𝐶
If V is potential difference across the parallel
Combinations of Resistances : Resistances can be combination, then using Ohm’s we get
combined in two ways (1) In Series combination (2) In 𝑽 𝑽
Parallel combination 𝑰=
𝑹
𝑰𝟏 =
𝑹𝟏

(1)Resistances in series : Series arrangement of three 𝑽 𝑉

𝑰𝟐 = 𝑰𝟑 =
resistances 𝑹𝟏 , 𝑹𝟐 𝒂𝒏𝒅 𝑹𝟑 is shown in the following 𝑹𝟐 𝑹𝟑
circuit.
Put these values in eqn. (1) then we get
𝑽 𝑽 𝑽 𝑉
= + +
𝑹 𝑹𝟏 𝑹𝟐 𝑹𝟑

𝟏 𝟏 𝟏 1
or = + + ………(2)
𝑹 𝑹𝟏 𝑹𝟐 𝑹𝟑

Where 𝑹 is equivalent resistance of the parallel

combination of three resistances 𝑹𝟏 , 𝑹𝟐 𝒂𝒏𝒅 𝑹𝟑 .
 5

For n -resistances in parallel E.M.F. of cell (E) :- E.M.F. of a cell is defined as the
maximum potential difference between the electrodes of the
𝟏 𝟏 𝟏 1 𝟏
= + + ……… cell when no current is drawn from the cell or the cell is in
𝑹 𝑹𝟏 𝑹𝟐 𝑹𝟑 𝑹𝒏
open circuit. The e.m.f. of a cell depends upon:-
Important : In series the current through each resister is
1. Nature of the electrodes of cell.
same and in parallel potential difference across each
2. Nature of the electrolyte used.
resistance is same.
3. Concentration of the electrolyte.
4. And temperature of the electrolyte.
Color code for carbon resistor : In a carbon resistors the
value of its resistance and its accuracy are indicated on it in Terminal potential difference (V): Terminal potential
color codes. difference of a cell is defined as the potential difference
between the electrodes of the cell when current is drawn
from the cell or when cell is in closed circuit.

When current is drawn from the cell then potential

Following table gives the color codes in carbon resistors :- difference between its electrodes decreases. Hence V is
always less than E.
Color Letter for Number Multiplier
Memory Internal resistance of a cell (r): It is the resistance offered
to the electric current by the electrolyte and electrodes of
Black B 0 100 the cell. The internal resistance of a cell depends upon:-

Brown B 1 101 1. Distance between the electrodes of cell.

2. Area of the electrodes immersed in the electrolyte.
Red R 2 102 3. Nature of the electrodes
4. Nature of the electrolyte used.
Orange O 3 103
Expression for internal resistance:- Consider a cell
yellow Y 4 104
Let 𝐸 = e. m .f cell
Green G 5 105
r = Internal resistance cell.
Blue B 6 10 6
R = External resistance
Violet V 7 10 7
𝐼 = Current in the circuit
Grey G 8 10 8

White W 9 109

For accuracy

Color Tolerance

Gold 5%

Silver 10%
Using 𝑽 = 𝑰𝑹 we get
No color 20%
𝑬 = 𝑰(𝑹 + 𝒓)
𝑬
or 𝑰= …..…(1)
(𝑹+𝒓)
Sentence B B ROY Great Britain Very Good Wife
Wearing Gold Silver Necklace helps to remember the Potential difference across the resistance R is
tables
𝑽 = 𝑰𝑹
Electrolytic cell: It is a device which makes the flow of
electric current in a closed circuit. It consist of a vessel of Put value of I from eqn. (1) then we
insulating material which contain an electrolyte and two
𝑬
metal electrodes immersed in the electrolyte. 𝑽= 𝑹
(𝑹+𝒓)
 6

or 𝑽 𝒓 + 𝑹 = 𝑬𝑹 Special case : For n-cells in series

or 𝑽 𝒓 + 𝑽𝑹 = 𝑬𝑹 𝐸𝑒𝑓 = 𝐸1 + 𝐸2 + 𝐸3 + ⋯ … . . … 𝐸𝑛

or 𝑽 𝒓 = 𝑹 (𝑬 − 𝑽) 𝑟𝑒𝑓 = 𝑟1 + 𝑟2 + 𝑟3 + ⋯ … . . … 𝑟𝑛
(𝑬−𝑽)
or 𝒓= 𝑹 If each cell has equal e.m.f. E and equal internal
𝑽
resistance r then
𝑬
or 𝒓= −𝟏 𝑹 𝐸𝑒𝑓 = 𝐸 + 𝐸 + 𝐸 + ⋯ . 𝐸 = 𝑛𝐸
𝑽

This is the expression for the internal resistance of a cell 𝑟𝑒𝑓 = 𝑟 + 𝑟 + 𝑟 + ⋯ . 𝑟 = 𝑛𝑟

Grouping of cells :Cells can be grouped in three ways (i) 𝒏𝑬
in series (ii) in parallel (iii) in mixed grouping Then eqn. (3) becomes 𝑰 =
𝒏𝒓 + 𝑹

(1)Two Cells in series : The series grouping of two cells of ………………………………………………………………

e.m.f. 𝑬𝟏 𝒂𝒏𝒅 𝑬𝟐 and internal resistances
𝒓𝟏 𝒂𝒏𝒅 𝒓𝟐 respectively is shown in fig. below. Where (2)Two Cells in parallel : The parallel grouping of two
R is external resistance. cells of e.m.f. 𝑬𝟏 𝒂𝒏𝒅 𝑬𝟐 and internal resistances
𝒓𝟏 𝒂𝒏𝒅 𝒓𝟐 respectively is shown below. Where R is
external resistance.

Let I is current through the cells, Then potential difference

between points A and B is
.Let 𝑰𝟏 𝒂𝒏𝒅 𝑰𝟐 are the currents through the two cell
𝑉𝐴𝐵 = 𝐸1 − 𝐼 𝑟1 …..(1) respectively, then total current in the circuit is given by

Potential difference between points B and C is 𝐼 = 𝑰𝟏 + 𝑰𝟐 ……….(1)

𝑉𝐵𝐶 = 𝐸2 − 𝐼 𝑟2 …..(2) Pot. difference across the cell 𝐸1 is

Potential difference between points A and C is 𝑉 = 𝐸1 − 𝐼1 𝑟1

𝑉𝐴𝐶 = 𝑉𝐴𝐵 + 𝑉𝐵𝐶 𝐸1 −𝑉

or 𝐼1 = ……….(2)
𝑟1
Using eqns. (1) and (2) we get
Pot. difference across the cell 𝐸2 is
𝑉𝐴𝐶 = ( 𝐸1 − 𝐼 𝑟1 ) +(𝐸2 − 𝐼 𝑟2 )
𝑉 = 𝐸2 − 𝐼2 𝑟2
𝑉𝐴𝐶 = 𝐸1 + 𝐸2 – ( 𝐼𝑟1 + 𝐼𝑟2 ) 𝐸2 −𝑉
or 𝐼2 = ……….(3)
𝑟2
𝑉𝐴𝐶 = 𝐸1 + 𝐸2 – 𝐼( 𝑟1 + 𝑟2 )
Put values of 𝐼1 𝑎𝑛𝑑 𝐼2 from eqns. (2) and (3) in eqn. (1)
𝑉𝐴𝐶 = 𝐸𝑒𝑓 – 𝐼( 𝑟𝑒𝑓 )
𝐸1 −𝑉 𝐸2 −𝑉
Then we get 𝐼= +
𝑟1 𝑟2
Where 𝐸𝑒𝑓 = 𝐸1 + 𝐸2 is effective e.m.f.
𝐸1 𝑉 𝐸2 𝑉
𝑟𝑒𝑓 = ( 𝑟1 + 𝑟2 ) is effective internal resistance or 𝐼= − + −
𝑟1 𝑟1 𝑟2 𝑟2
of the series combination of two cells.
𝐸1 𝐸2 𝑉 𝑉
or 𝐼= + − +
𝑬𝒆𝒇 𝑟1 𝑟2 𝑟1 𝑟2
Hence current in the circuit will be 𝑰 = ..….(3)
𝒓𝒆𝒇 + 𝑹
𝑟2 𝐸1 +𝑟1 𝐸2 𝑟2 +𝑟1
or 𝐼= −𝑉
𝑟1 𝑟2 𝑟1 𝑟2
…………………………………………………………………..
 7

or 𝐼𝑟1 𝑟2 = (𝑟2 𝐸1 + 𝑟1 𝐸2 ) − 𝑉(𝑟2 + 𝑟1 ) 𝐸𝑒𝑓

=
𝑛𝐸
or 𝐸𝑒𝑓 = E ….….(13)
𝑟/𝑛 𝑟
or 𝑉(𝑟2 + 𝑟1 ) = (𝑟2 𝐸1 + 𝑟1 𝐸2 ) − 𝐼𝑟1 𝑟2
Using eqns. (12) and (13) in eqn.(6) we get
𝑟2 𝐸1 +𝑟1 𝐸2 𝑟1 𝑟2
or 𝑉= −𝐼 𝑬𝒆𝒇
𝑟1 + 𝑟2 𝑟1 + 𝑟2 𝑰=
𝒓/𝒏 + 𝑹
or 𝑉 = 𝐸𝑒𝑓 – 𝐼( 𝑟𝑒𝑓 ) 𝒏𝑬𝒆𝒇
or 𝑰=
𝒓 + 𝒏𝑹
Where
……………………………………………………………………
𝑟2 𝐸1 +𝑟1 𝐸2
𝐸𝑒𝑓 = is effective e.m .f…..…….(4)
𝑟1 + 𝑟2 (3) Mixed grouping of cells : Mixed grouping of cells is
𝑟1 𝑟2 shown in diagram. There are 𝒎 rows of the cells in parallel
𝑟𝑒𝑓 = is effective internal resistance..…(5)
𝑟1 + 𝑟2 and each row contains 𝒏 cells
𝑬𝒆𝒇
Hence current in the circuit is 𝑰 = ……(6) Let 𝐸 = e.m.f of each cell
𝒓𝒆𝒇 + 𝑹
𝑟 = internal resistance of each cell
………………………………………………………………….
𝑹 = external resistance
Special case : For n-cells in series parallel

Dividing eqn. (4) by (5) we get

𝐸𝑒𝑓 𝑟2 𝐸1 + 𝑟1 𝐸2
=
𝑟𝑒𝑓 𝑟1 𝑟2
𝐸𝑒𝑓 𝑟2 𝐸1 𝑟1 𝐸2
or = +
𝑟 𝑒𝑓 𝑟1 𝑟2 𝑟1 𝑟2

𝐸𝑒𝑓 𝐸1 𝐸2
or = + ……….(7)
𝑟 𝑒𝑓 𝑟1 𝑟2

From eqn. (5) the reciprocal of 𝑟𝑒𝑓 is It is clear from the circuit diagram that

1 𝑟1 + 𝑟2 𝐸𝑒𝑓 = 𝒏𝑬 …..(1)
=
𝑟𝑒𝑓 𝑟1 𝑟2 𝑛𝑟
and 𝑟𝑒𝑓 = ….(2)
𝑚
1 1 1
or = + ….…(8)
𝑟 𝑒𝑓 𝑟1 𝑟2 Total current in the circuit is

For n-cells in parallel eqns. (7) and (8) will become 𝑰=

𝑬𝒆𝒇
𝒓𝒆𝒇 + 𝑹
𝐸𝑒𝑓 𝐸1 𝐸2 𝐸𝑛
= + + ………. …..(9) 𝒏𝑬
𝑟 𝑒𝑓 𝑟1 𝑟2 𝑟𝑛
Using eqns. (1) and (2) we get 𝑰 = 𝑛𝑟
+𝑹
𝑚
1 1 1 1
= + +……….. ……(10) 𝒎𝒏𝑬
𝑟 𝑒𝑓 𝑟1 𝑟2 𝑟𝑛
or 𝑰= …….(1)
𝒏𝒓 + 𝒎𝑹
If each cell has same e .m .f (E) and same internal
resistance (r) then eqns. (9) and (10) will become Current 𝐼 in the circuit is maximum if

𝐸𝑒𝑓 𝐸 𝐸 𝐸 (𝑛𝑟 + 𝑚 𝑅) = 𝑚𝑖𝑛𝑖𝑚𝑢𝑚

= + +……….
𝑟 𝑒𝑓 𝑟 𝑟 𝑟
or 𝑛𝑟 2 + 𝑚𝑅 2 -2 𝑚𝑛𝑟𝑅 + 2 𝑚𝑛𝑟𝑅 = 𝑚𝑖𝑛
𝐸𝑒𝑓 𝑛𝐸
or = ………(11) 2
𝑟 𝑒𝑓 𝑟
or 𝑛𝑟 − 𝑚𝑅 + 2 𝑚𝑛𝑟𝑅 = 𝑚𝑖𝑛
1 1 1 1 𝑛
And = + +……….. =
𝑟 𝑒𝑓 𝑟 𝑟 𝑟 𝑟 It is possible if 𝑛𝑟 − 𝑚𝑅 = 0
𝑟
or 𝑟𝑒𝑓 = ……….(12) or 𝑛𝑟 = 𝑚𝑅
𝑛

Put this value of 𝑟𝑒𝑓 in eqn. (11) we get

 8
𝑛𝑟
or 𝑅= 8. If radius of a copper wire is doubled keeping length
𝑚
same. What is the effect on its (i) resistance (ii)
i.e. external resistance 𝑅 is equal to total internal resistivity?
resistance of all the cells. 9. Find conductivity of a wire of resistivity 20 ohm-m.
10. Join three resistances of 2Ω ohm each to obtain the
Hence for the maximum current in the mixed grouping of total resistance of 3Ω.
the cells ,the external resistance should be equal to the 11. A wire of resistance R is bent from middle by 1800 and
total internal resistance of all the cells. both halves are twisted with each other. What will be its
new resistance?
Assignment 12. A carbon resistor is marked with the rings of red blue
green and golden colors. What is the value of its
1. Define electric current. Gives its S.I unit. Is it vector or resistance?
scalar? 13. A carbon resistor of 45kΩ is marked with the rings of
2. What are current carriers? Name the current carriers in different colors. Write the sequence of the colors.
solids, liquids and gases?
3. What is meant by drift velocity of electrons and Numerical
relaxation time? Establish relation between them.
4. Derive a relation between electric current and drift 1. A solution of sodium chloride discharges 5× 1016 𝑁𝑎+
velocity. and 4× 1016 𝐶𝑙− ions in 2 seconds. Find current
5. State Ohm’s law. What are (i) Ohmic conductors.(ii) passing through the solution. [Ans. 7.2× 10−3 amp]
non-ohmic conductors. Draw V-I graphs for Ohmic and 2. How many electrons pass through a lamp in one minute
non-ohmic conductor. if current is 600mA. (e=1.6 × 10−19 𝐶)
6. What is resistance? On what factors does its depend? [ 2.25 × 1020 ]
Define S.I unit of resistance. 3. Find resistance of a hollow cylindrical pipe of length 2.0
7. Define resistivity. On what factors does it depend? m with inner and outer radii of 10cm and 20 cm
8. Explain the effect of temperature on the resistivity of (i) respectively. The resistivity of the material of pipe is 2
conductors (ii) semiconductors (iii) insulators. × 10−8 Ω𝑚. [1.74 × 10−6 ohms]
9. Define temperature coefficient of resistivity. What can 4. A wire of resistance R is stretched to make its length
you say about the temperature coefficient of (i) double. What is its new resistance and resistivity.
conductors (ii) semiconductors (iii) insulators? [𝐴𝑛𝑠. 4𝑅]
10. Why mangnin or constantan is used for making 5. A wire carries current of 2A, when potential difference of
standard resistances? 4.0 volts is applied across it. What is its conductance if
11. What is the diference between e.m.f. and terminal length of the wire is 3m and area of cross- section is
potential difference of a cell. On what factors e.m.f. of a 5m𝑚2 . Calculate its conductivity.
cell depends? −1
[𝐴𝑛𝑠. 2.4 × 105 Ω 𝑚−1 ]
12. What is internal resistance of a cell? On what factors 6. Find electric field in a copper wire of area of cross-
does it depend? Derive the expression for the internal section is 2m𝑚2 , if it carries a current of 2A and its
resistance of a cell in terms of e.m.f. and terminal resistivity is 1.5 × 10−8 Ω𝑚.
potential difference. [𝐴𝑛𝑠. 1.5 × 10−2 𝑁𝐶 −1 ]
13. Obtain the expression for the effective current in the 7. The copper wire has resistance 2Ω at 200 C. Find
series combination of cells. resistance at 500 C if temperature coefficient of copper
14. Obtain the expression for the effective current in the 0
parallel combination cells. is 4 × 10−2 𝐶 −1 [ 𝐴𝑛𝑠. 24 Ω]
8. The resistance of a conductor is 6Ω at 500 C and 7 at
Conceptual Questions C. Calculate the mean temperature coefficient of
the resistance of the material. Find resistance of the
1. Why copper wires are used as connecting wires?
2. What are the properties of a standard resistance? conductor at C.
3. What is the order of number of free electrons in a 0
[Ans. 𝛼 = 1 300 𝐶 −1 , R=5.14Ω]
metal? 9. At what temperature the resistance of copper become
4. What do you mean by linear resistor? double of its value at C. if temperature coefficient of
5. What is the effect of temperature on the relaxation time 0
in a conductor? copper is 4 × 10−3 𝐶 −1 [Ans. 2500 𝐶]
6. What is the effect of temperature on the drift velocity of 10. A parallel combination of three resistance takes a
free electrons in a conductor? current of 7.5 A at 30 volts. If two resistors are 10 Ω
7. What does the slope of a V-I graph for a linear and 12 Ω, find third one. [Ans.15Ω]
resistor give? 11. A wire of resistance R is bent to form a circle of radius r.
Find resistance between the points A and B.
 9

[Ans. R/4Ω]
12. A wire of resistance 12 is bent to form an equilateral
triangle. Find effective resistance between two corners [Ans.4.8Ω]
of the triangle. [Ans.2.67Ω] (v)
13. The total resistance of two resistors connected in series
is 9Ω and when connected in parallel the total
resistance becomes 2Ω. Find value of each resistance.
[Ans. 6Ω, 3Ω]
14. A battery of 6 volt can supply a current of 3A through a
resistance of 2Ω. What current does it supply through :
another resistance of 3 Ω. Also calculate internal [Ans.10Ω]
resistance of the cell. [ 𝐴𝑛𝑠. 𝑟 = 0 𝐼 = 2𝐴]
15. Find equivalent resistance between points A and B in
following circuits:
(i)

[Ans. 3Ω]
(ii)

[Ans.12.72Ω]
(iii)

[Ans.5Ω]
(iv)