The document discusses various principles and laws related to current electricity, focusing on Kirchhoff's laws for analyzing complex circuits and the Wheatstone bridge for measuring resistances. It explains the applications of potentiometers and galvanometers in circuitry, including how to convert a galvanometer into an ammeter or voltmeter, emphasizing the importance of these devices in electrical measurements. Additionally, it addresses potential sources of errors and the advantages of potentiometers over traditional voltmeters.

1. Current Electricity
Chapter : 9

2. Previously we have been studied Series and parallel combination of resistors.
In series combination
𝑅1 𝑅2 𝑅3
𝐸 𝐾
𝑅 = 𝑅1 + 𝑅2 + 𝑅3
In parallel combination
𝑅1 𝑅2 𝑅3 1
𝑅
=
1
𝑅1
+
1
𝑅2
+
1
𝑅3
𝐸
𝐾

3. A circuit containing several complex connections of electrical components
cannot be easily reduced into a single loop by using the rules of series and parallel
combination of resistors.
More complex circuits can be analyzed by using Kirchhoff’s laws. Gustav
Robert Kirchhoff (1824-1887) formulated two rules for analyzing a complicated
circuit.

4. Kirchhoff’s Laws of
Electrical Network

5. Before describing these laws we will define some terms used for electrical
circuits.
Junction
Any point in an electric circuit
where two or more conductors are joined
together is a junction.
Loop
Any closed conducting path in an
electric network is called a loop or mesh.
Branch
A branch is any part of the
network that lies between two junctions.
𝑅1 𝑅2 𝑅3
𝐸
𝐾
𝐴 𝐵 𝐶 𝐷
𝐹
𝐺
𝐻
𝐼
Here points A,B,C,D,E,F,G,H and
I are junctions
Here ABHIA, ABCGHIA,
ABCDFGHIA, BCGHB, BCDFGHB
and CDFGC are representing
loops
Here AI, BH, CG, DF are
representing some branches

6. Kirchhoff’s First Law: (Current law/ Junction law)
Statement
The algebraic sum of the
currents at a junction is zero in an
electrical network, i.e.,
𝑘=1
𝑛
𝐼𝑘 = 0
Sign Conventions
The currents arriving at the
junction are considered positive and
the currents leaving the junction are
considered negative.
Explanation
𝐼1
𝐼2
𝐼3
𝐼4
𝐼5
The currents 𝐼1, 𝐼2 are arriving at the
junction are considered positive and the
currents 𝐼3, 𝐼4 and 𝐼5 leaving the junction
are considered negative.
∴ 𝐼1 + 𝐼2 − 𝐼3 − 𝐼4 − 𝐼5 = 0
or 𝐼1 + 𝐼2 = 𝐼3 + 𝐼4 + 𝐼5
This law is simply a statement of
“conservation of charge”.

7. Kirchhoff’s Second Law: (Voltage law/ Loop law)
Statement
The algebraic sum of the potential
differences and the electromotive forces
in a closed loop is zero.
𝐸 + 𝐼𝑅 = 0
Sign Conventions
Choose the loop to apply
Kirchhoff’s law. Assume any direction.
3) Emf is positive if assumed direction
leaving positive terminal of battery.
4) Potential difference is negative if the
conventional current in the assumed
direction.
Explanation
For a loop ABFGA,
∴ 𝐸1 = 𝐼1𝑅1 + 𝐼3𝑅5 + 𝐼1𝑅3
∴ −𝐼3𝑅5 −𝐼2 𝑅4 + 𝐸2 − 𝐼2𝑅2 = 0

8. Wheatstone Bridge

9. Wheatstone’s bridge is generally
used to measure resistances in the range
from tens of ohm to hundreds of ohms.
The Wheatstone Bridge was
originally developed by Charles
Wheatstone (1802- 1875) to measure the
values of unknown resistances. It is also
used for calibrating measuring
instruments, voltmeters, ammeters, etc.
Wheatstone bridge is an
arrangement of four resistances which can
be used to measure one of them in terms
of rest. Here arms 𝐴𝐵 and 𝐵𝐶 are called
ratio arms and arms 𝐴𝐶 and 𝐵𝐷 are called
conjugate arms.

10. Balanced bridge :
The bridge is said to be balanced
when current in galvanometer is zero i.e.
no current flows through the
galvanometer or in other words
𝑉𝐵 = 𝑉𝐷.
In the balanced condition,
𝑃
𝑄
=
𝑆
𝑅
On mutually changing the position
of Resistance Box and galvanometer this
condition will not change.

11. Balancing condition for Wheatstone bridge/network
Figure shows Wheatstone bridge.
Using Kirchhoff’s loop law for a loop 𝐴𝐵𝐷𝐴 we
get,
−𝐼1𝑃 − 𝐼𝑔𝐺 + 𝐼2𝑆 = 0 …(1)
Using Current law at junction A, we get
𝐼 = 𝐼1 + 𝐼2
Using Kirchhoff’s loop law for a loop 𝐵𝐶𝐷𝐵 we
get,
−(𝐼1 − 𝐼𝑔)𝑄 + (𝐼2 + 𝐼𝑔)𝑅 + 𝐼𝑔𝐺 = 0 …(2)

12. Balancing condition for Wheatstone bridge/network
Balancing condition for Wheatstone bridge is
Thus equation (1) and (2) becomes
−𝐼1𝑃 + 𝐼2𝑆 = 0
∴ 𝐼1𝑃 = 𝐼2𝑆 …(3)
𝐼𝑔 = 0
And
−(𝐼1 − 𝐼𝑔)𝑄 + (𝐼2 + 𝐼𝑔)𝑅 + 0 = 0
∴ 𝐼1𝑄 = 𝐼2𝑅 …(4)
Thus, dividing equation (3) by (4), we get
𝑃
𝑄
=
𝑆
𝑅
This is the balancing condition
for Wheatstone bridge.

13. Application of Wheatstone
bridge

14. Meter bridge
Construction
𝑋 𝑅
𝐿𝑒𝑓𝑡 𝐺𝑎𝑝 𝑅𝑖𝑔ℎ𝑡 𝐺𝑎𝑝
𝐶
𝐽
𝐺
𝐸 𝐾
A meter bridge consists of a wire of length 1 𝑚 and of uniform cross-sectional area stretched out and clamped between two thick
metallic strips bent at right angles with two gaps across which resistors are to be connected.
Usually, an unknown resistance X is connected in the left gap and a resistance box is connected in the other gap, as
shown in figure.
One end of a galvanometer (𝐺) is connected to the metallic strip midway between the two gaps. The other end of the
galvanometer is connected to a jockey which moves along the wire to make electrical connection.
The end points (𝐴 and 𝐵) where the wire is clamped are connected to a cell (𝐸) through a key (𝐾).

15. Meter bridge
Working
A suitable resistance R is selected from resistance box. By tapping jockey on wire
AB, point D is obtained where the galvanometer shows zero deflection.
Let 𝐴𝐷 = 𝑙𝑋 and 𝐵𝐷 = 𝑙𝑅.
𝑋 𝑅
𝐿𝑒𝑓𝑡 𝐺𝑎𝑝 𝑅𝑖𝑔ℎ𝑡 𝐺𝑎𝑝
𝐶
𝐽
𝐺
𝐸 𝐾
𝐴 𝐵
𝐷
𝑙𝑋 𝑙𝑅

16. Meter bridge
Working
where 𝑅𝐴𝐷 and 𝑅𝐵𝐷 are resistance of the parts AD and DB of the wire resistance
of the wire.
Let, 𝜌− specific resistance, 𝐴− cross-sectional area of the wire.
𝐷
𝑋 𝑅
𝐿𝑒𝑓𝑡 𝐺𝑎𝑝 𝑅𝑖𝑔ℎ𝑡 𝐺𝑎𝑝
𝐶
𝐽
𝐺
𝐸 𝐾
𝐴 𝐵
𝑙𝑋 𝑙𝑅
Then,
𝑅𝐴𝐷=𝜌𝑙𝑋/𝐴 and 𝑅𝐵𝐷=𝜌𝑙𝑅/𝐴
Then balancing condition is,
X
R
=
𝑅𝐴𝐷
𝑅𝐷𝐵

17. Meter bridge
Working
𝑋
𝑅
=
𝜌𝑙𝑋/𝐴
𝜌𝑙𝑅/𝐴
=
𝑙𝑋
𝑙𝑅
𝑋 𝑅
𝐿𝑒𝑓𝑡 𝐺𝑎𝑝 𝑅𝑖𝑔ℎ𝑡 𝐺𝑎𝑝
𝐶
𝐽
𝐺
𝐸 𝐾
𝐴 𝐵
𝑙𝑋 𝑙𝑅
Thus, balancing condition is,
∴ 𝑋 = 𝑅
𝑙𝑋
𝑙𝑅
Since, 𝑙𝑋 + 𝑙𝑅 = 100 𝑐𝑚
∴ 𝑋 = 𝑅
100−𝑙𝑅
𝑙𝑅
Thus, by knowing 𝑅, 𝑙𝑋 and 𝑙𝑅, we
can determine unknown resistance 𝑋.

18. Source of errors
1. The cross section of the wire may not be uniform.
2. The ends of the wire are soldered to the metallic strip where contact resistance is
developed, which is not taken into account.
3. The measurements of 𝑙𝑋 and 𝑙𝑅 may not be accurate.
1. The value of R is so adjusted that the null point is obtained to middle one third of
the wire (between 34 cm and 66 cm) so that percentage error in the measurement
of 𝑙𝑋 and 𝑙𝑅 are minimum and nearly the same.
2. The experiment is repeated by interchanging the positions of unknown resistance X
and known resistance box R.
3. The jockey should be tapped on the wire and not slided. We use jockey to detect
whether there is a current through the central branch. This is possible only by
tapping the jokey.
To minimize the errors

19. Potentiometer

20. Potentiometer
Potentiometer is a device used
1. to compare the emfs of two cells,
2. find the emf of a cell,
3. to find the internal resistance of a cell and
4. to measure potential difference.
A potentiometer consists of uniform
wire of length 4 to 10 𝑚 arranged between A
and B as 4 to 10 𝑚 wires each of length 1m on
a wooden board. The wire has specific
resistance and low temperature coefficient of
resistance.

21. Principle of Potentiometer
The potential difference between
two points on the wire is directly
proportional to the length of the wire
between them provided the wire is of
uniform cross section, the current through
the wire is the same and temperature of
the wire remains constant.
Explanation
Consider a potentiometer
consisting a long wire AB of length L and
resistance R having uniform cross
sectional area A.
A cell of emf 𝐸 having internal
resistance 𝑟 is connected across AB as
shown in the figure. When the circuit is
switched on, current 𝐼 passes through the
wire.

22. Explanation
Current through AB is
𝐼 = 𝐸/(𝑅 + 𝑟).
Potential Difference across AB is
𝑉𝐴𝐵 = 𝐼𝑅
∴ 𝑉𝐴𝐵 =
𝐸𝑅
(𝑅+𝑟)
Therefore, the potential difference
per unit length of the wire is,
∴
𝑉𝐴𝐵
𝐿
=
𝐸𝑅
𝑅+𝑟 𝐿
As long as E remains constant
𝑉𝐴𝐵
𝐿
i.e. potential gradient (k) along wire 𝐴𝐵
also remains constant.
Consider a point C on the wire at
distance 𝑙 from the point A, as shown in
the figure. The potential difference
between A and C is 𝑉𝐴𝐶.
Therefore, 𝑉𝐴𝐶 =𝑘𝑙 i.e. 𝑉𝐴𝐶 ∝𝑙.

23. Uses of Potentiometer

24. A) To Compare emf of Cells
There are two methods to compare emf of cells.
I] Direct Method/Individual Cell Method And
II] Sum and Difference Method

25. I) Individual Cell Method
𝐾
𝐸
𝑅ℎ
𝐴
𝐵
𝐸1
𝐸2
𝐾1
𝐾2
𝑃
𝑄
Key 𝐾 is closed and then, key 𝐾1 is closed and key
𝐾2 is kept open.
The null point for cell 𝐸1 is obtained, by touching
jockey on wire AB, say at 𝑃. Let 𝐴𝑃 = 𝑙1
∴ 𝐸1 = 𝑘𝑙1
Let 𝑘 be the potential
gradient
∴ 𝐸2 = 𝑘𝑙2
∴ 𝐸1/𝐸2 = 𝑘𝑙1/𝑘𝑙2
∴ 𝐸1/𝐸2 = 𝑙1/𝑙2
Thus, we can compare the emfs of
the two cells. If any one of the emfs is
known, the other can be determined.

26. I) Sum and Difference Method
𝐾
𝐸
𝑅ℎ
𝐴
𝐵
𝐸1
𝑃
𝑄
𝐸2
1 2
3
4
In this case, cell 𝐸1 and 𝐸2 will assist
each other. (i.e. 𝐸1 + 𝐸2)
∴ 𝐸1 + 𝐸2 = 𝑘𝑙1
∴ 𝐸1 + 𝐸2/𝐸1 − 𝐸2 = 𝑘𝑙1/𝑘𝑙2
By componendo and
dividendo method, we get,
∴
𝐸1
𝐸2
=
𝑙1 + 𝑙2
𝑙1 − 𝑙2
Thus, emf of two cells can be compared.

27. A) To find internal resistance of Cell
𝐾
𝐸
𝑅ℎ
𝐴
𝐵
𝐸1
𝑟
𝑅
𝐾1
𝑆
𝑃 𝑄
𝑅
Let 𝑘 be the potential gradient of potentiometer wire.
Initially, the key 𝐾 is closed and 𝐾1
is open.
The null point is
then obtained. Let 𝑙1 be
length of the potentiometer
wire between the null
point (say P) and the point
𝐴. This length corresponds
to emf 𝐸1 .
𝑃
∴ 𝐸1 = 𝑘𝑙1

28. A) To find internal resistance of Cell
Now, the key 𝐾 and 𝐾1 are closed
The null point is then
obtained. Let 𝑙2 be length of
the potentiometer wire
between the null point (say
Q) and the point 𝐴. This
length corresponds to
terminal potential difference
of cell 𝐸1 (say 𝑉).
𝐾
𝐸
𝑅ℎ
𝐴
𝐵
𝐸1
𝑟
𝑅
𝐾1
𝑆
𝑃 𝑄
𝑅
𝑃
∴ 𝑉 = 𝑘𝑙2
∴
𝐸1
𝑉
=
𝑙1
𝑙2

29. A) To find internal resistance of Cell
By Ohm’s law, we get, 𝑉 = 𝐼𝑅
And 𝐸1 = 𝐼(𝑅 + 𝑟)
𝐾
𝐸
𝑅ℎ
𝐴
𝐵
𝐸1
𝑟
𝑅
𝐾1
𝑆
𝑃 𝑄
𝑅
𝑃
∴
𝐼(𝑅 + 𝑟)
𝐼𝑅
=
𝑙1
𝑙2
∴
𝑅 + 𝑟
𝑅
=
𝑙1
𝑙2
∴ 1 +
𝑟
𝑅
=
𝑙1
𝑙2
∴ 𝑟 = 𝑅
𝑙1
𝑙2
− 1
This equation gives the internal resistance of the cell.

30. As shown in the figure, potential 𝑉 is set up between
points 𝐴 and 𝐵 of a potentiometer wire. One end of a device is
connected to positive point A and the other end is connected to a
slider that can move along wire 𝐴𝐵. The voltage 𝑉 divides in
proportion of lengths 𝑙1 and 𝑙2 as shown in the figure.
C] Applications of Potentiometer
Voltage Divider

31. Audio Control
Sliding potentiometers, are commonly used in
modern low-power audio systems as audio control
devices. Both sliding (faders) and rotary
potentiometers (knobs) are regularly used for
frequency attenuation, loudness control and for
controlling different characteristics of audio signals.
Potentiometer as a sensor
If the slider of a potentiometer is connected to
the moving part of a machine, it can work as a motion
sensor. A small displacement of the moving part
causes changes in potential which is further amplified
using an amplifier circuit. The potential difference is
calibrated in terms of the displacement of the moving
part.

32. Advantages of a Potentiometer over a Voltmeter
Merits
i) Potentiometer is more sensitive than a voltmeter.
ii) A potentiometer can be used to measure a potential difference as well as an
emf of a cell. A voltmeter always measures terminal potential difference, and as it
draws some current, it cannot be used to measure an emf of a cell.
iii) Measurement of potential difference or emf is very accurate in the case of a
potentiometer. A very small potential difference of the order 10–6 volt can be measured
with it. Least count of a potentiometer is much better compared to that of a voltmeter.
Demerits
Potentiometer is not portable and direct measurement of potential difference or
emf is not possible.

33. Galvanometer

34. Galvanometer is an electro mechanical instrument which is used for the
detection of electric currents through electric circuits.
The primary purpose of galvanometer is the detection of electric current not
the measurement of current.
MOVING COIL GALVANOMENTER
It can only measure small amount of currents.

35. MOVING COIL GALVANOMENTER
A moving coil galvanometer is an instrument used for detection and
measurement of small electric currents.

36. Moving coil galvanometers are used in all kinds of different equipment
from aeroplane cockpit to sound level (VU) meters in recording studios
Cockpit of an
aeroplane
Moving coil galvanometers are used in the cockpit of an areoplane

37. Principle of Galvanometer
It works on the principle that when a current loop is placed in an external
magnetic field, it experiences torque, and the value of torque can be changed by
changing the current in the loop

38. Galvanometer
Moving coil galvanometer consists of
permanent horse-shoe magnets, coil, soft
iron core, pivoted spring, non-metallic
frame, scale and pointer.
When an electric current passes
through the coil, it deflects.
The deflection is proportional to the
current passing through the coil. The
deflection of the coil can be read with the
help of a pointer attached to it.
Position of the pointer on the scale
provided indicates the current passing
through the galvanometer or the potential
difference across it.
Thus, a galvanometer can be used
as an ammeter or voltmeter with suitable
modification.
The galvanometer coil has a
moderate resistance (about 100 ohms)
and the galvanometer itself has a small
current carrying capacity (about 1 mA).

39. Conversion of MCG into an
Ammeter

40. A galvanometer can be converted to an ammeter by connecting a low resistance,
called SHUNT (S) in parallel to the galvanometer (G).
Conversion of a galvanometer to ammeter
What is a shunt?
A low resistance connecting
parallel to the galvanometer to
protect it from large current is
known as shunt.
 Ammeter is used for measuring the current
in an electric circuit.
What is an ammeter?

41. A galvanometer can be converted to an ammeter by connecting a low
resistance, called SHUNT (S) in parallel to the galvanometer (G).
Conversion of a galvanometer to ammeter
The shunt resistance is calculated
as follows. In the arrangement shown in
the figure, 𝐼𝑔 is the current through the
galvanometer.
Therefore, the current through 𝑆 is
𝐼 − 𝐼𝑔 = 𝐼𝑠.
Since 𝑆 and 𝐺 are parallel,
𝐺𝐼𝑔 = 𝑆𝐼𝑠.
𝐺𝐼𝑔 = 𝑆 𝐼 − 𝐼𝑔 .
𝐺 = 𝑆
𝐼−𝐼𝑔
𝐼𝑠
.

42. (i) If the current 𝐼 is 𝑛 times
current 𝐼𝑔 , then 𝐼 = 𝑛𝐼𝑔. Using this in the
above expression we get
Conversion of a galvanometer to ammeter
𝐺 = 𝑆
𝐼−𝐼𝑔
𝐼𝑔
.
𝐺 = 𝑆
𝑛𝐼𝑔−𝐼𝑔
𝐼𝑔
. = 𝑆(𝑛 − 1)
𝑆 =
𝐺
𝑛−1
This is the required shunt to
increase the range n times.

43. (ii) Also if 𝐼𝑠 is the current
through the shunt resistance, then the
remaining current 𝐼 − 𝐼𝑠 will flow
through galvanometer. Hence
Conversion of a galvanometer to ammeter
𝐺 = 𝑆
𝐼−𝐼𝑔
𝐼𝑔
.
𝐺(𝐼 − 𝐼𝑠) = 𝑆𝐼𝑠.
∴ 𝐺𝐼 − 𝐺𝐼𝑠 = 𝑆𝐼𝑠.
∴ 𝐺𝐼 = 𝑆𝐼𝑠 + 𝐺𝐼𝑠 = (𝑆 + 𝐺)𝐼𝑠.
∴
𝐼𝑠
𝐼
=
𝐺
𝑆+𝐺
This equation gives the fraction of
the total current through the shunt
resistance.

44. Use of Shunt
a. It is used to divert a large part of total current by providing an alternate
path and thus it protects the instrument from damage.
b. It increases the range of an ammeter.
c. It decreases the resistance between the points to which it is connected.

45. Conversion of MCG into an
Voltmeter

46. A galvanometer can be converted to an voltmeter by connecting a high
resistance, in series to the galvanometer (G).
Conversion of a galvanometer to ammeter
The value of the resistance 𝑋 can
be calculated as follows. If 𝑉 is the voltage
to be measured, then
𝑉 = 𝑉
𝑥 + 𝑉
𝑔.
𝑉 = 𝐼𝑔𝑋 + 𝐼𝑔𝐺.
𝐼𝑔𝑋 = 𝑉 − 𝐼𝑔𝐺.
𝑋 =
𝑉−𝐼𝑔𝐺
𝐼𝑔
.
where 𝐼𝑔 is the current flowing
through the galvanometer.

47. Conversion of a galvanometer to ammeter
𝑋 =
𝑉−𝐼𝑔𝐺
𝐼𝑔
.
If 𝑛𝑣 =
𝑉
𝑉𝑔
=
𝑉
𝐼𝑔𝐺
i.e. the factor by which the voltage
range is increased
It can be shown that
𝑋 = 𝐺(𝑛𝑣 − 1).

48. Use of High Resistance
A voltmeter is connected across the points where potential difference is to be
measured. If a galvanometer is used to measure voltage, it draws some current (due
to its low resistance), therefore, actual potential difference to be measured
decreases. To avoid this, a voltmeter should have very high resistance. Ideally, it
should have infinite resistance.

49. Comparison of Ammeter
and Voltmeter