Class XII Physics – Study Guide
CHAPTER III:
Kirchhoff’s laws and their applications to circuits with resistors and sources of emf; Wheatstone bridge, meter
bridge and potentiometer; use of standard resistors and standard cells; internal resistance of sources of current; use
of resistors (shunts and multipliers) in ammeters and voltmeters.
Introduction
Ohm’s law is useful in dealing with simple electrical circuits only. Study of circuits containing more than one
source of emf becomes difficult with the help of Ohm’s law.
Consider the following two circuits. Neither can be solved by series-parallel combinations.
Many practical resistor networks cannot be reduced to simple, series-parallel combinations. The circuit below
shows a DC power supply with EMF E1 charging a battery with a smaller EMF E2 and feeding current to a light
bulb with resistance R. Here we cannot identify the resistances in series or in parallel. So, German physicist
Gustow Robert Kirchoff developed a technique in the form of Kirchhoff Rules.
Sign Convention in Applying Kirchhoff's Rules
The emf of a cell is positive when one moves in the direction of increasing potential (i.e., negative pole to
positive pole) through the cell and is negative when one moves from positive to negative.
The product of resistance and current, i.e., the IR term, in any arm of the circuit is taken negative if one
moves in a closed path, in the same direction of the assumed current; and positive if in the opposite
direction.
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Class XII Physics – Study Guide
Kirchhoff's Rules
i. First Rule: Kirchhoff's Junction Rule (Current Rule/Law)
The algebraic sum of the currents at a junction in a closed circuit is zero.
Therefore,
Hence
Or
i.e. Sum of currents entering a junction = Sum of currents leaving the
junction)
This rule is based on the fact that charge cannot be accumulated at any point
in a conductor in a steady situation. It is based on the Law of conservation of charge.
ii.
Second Rule: Kirchhoff's Loop Rule (Voltage Rule/Law)
The algebraic sum of the potential differences in any
loop including those associated with emfs and those of
resistive elements must be equal to zero.
)
Traversing in the clock wise direction,
In the upper loop (afeba)
In the lower loop (abdca)
=0
) =0
+(
This rule is based on energy conservation, i.e., the net change in the energy of a charge after completing
the closed path is zero. Otherwise, one can continuously gain energy by circulating charge in a particular
direction.
Wheatstone bridge Principle
Wheatstone Bridge Principle states the if four resistance P, Q,
R and S are arranged to form a bridge as shown in figure with
a cell E and one key K1 between the point A and C and
galvanometer G and tapping key K2 between the points B and
D, closing K1 first and K2 later on, if the galvanometer shows
no deflection then bridge is balanced.
Current distribution in the circuit is shown in the figure. Total
current by cell is I, it distributed to P and S as I1 and I-I1. Let
current through galvanometer is I3.
Current in Q is I1-I3 and in S current are I-I1+I3.
Resistance of galvanometer is G.
Now apply Kirchhoff’s 2nd law in ABDA, we get
I1P +I3G – (I-I1) R=0
…..…1
Now apply Kirchhoff’s 2nd law in BCDB, we get
(I1 –I3) Q – (I-I1+I3) S – I3 G = 0
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Class XII Physics – Study Guide
If value of R is such that the galvanometer shows no deflection that is Ig = 0. Putting this value in equation 1 and 2 we
get
This last equation relating the four resistors is called the balance condition for the galvanometer to give zero or null
deflection.
A practical device using this principle is called the meter bridge.
Slide Wire Bridge or Meter Bridge
It consists of a wire of length 1m and of uniform cross sectional area taut and clamped between two thick metallic
strips bent at right angles, as shown. The metallic
strip has two gaps across which resistors can be
connected. The end points where the wire is
clamped are connected to a cell through a key.
One end of a galvanometer is connected to the
metallic strip midway between the two gaps. The
other end of the galvanometer is connected to a
‘jockey’.
The jockey is essentially a metallic rod whose one
end has a knife-edge which can slide over the
wire to make electrical connection.
R is an unknown resistance whose value to be
determine. It is connected across one of the gaps. Across the other gap, we connect a standard known resistance S.
The jockey is connected to some point D on the wire, a distance l cm from the end A. The jockey can be moved along
the wire. The portion AD of the wire has a resistance , where is the resistance of the wire per unit centimeter.
The portion DC of the wire similarly has a resistance (
). The four arms AB, BC, DA and CD [with
resistances R, S, and (
)] form a Wheatstone bridge with AC as the battery arm and BD the galvanometer
arm.
If the jockey is moved along the wire, then there will be one position where the galvanometer will show no current. Let
the distance of the jockey from the end A at the balance point be l = l1. The four resistances of the bridge at the balance
point then are R, S, l1 and (100–l1). The balance condition gives
=
(
)
=
Thus, once we have found out l1, the unknown resistance R is known in terms of the standard known resistance S by
=
Slide Wire Bridge is practical form of Wheat stone bridge.
Jockey is slide over AC wire that is why it is called Slide Wire Bridge
It cannot be used to measure very high and very low resistance.
Balance position of Meter Bridge not change with interchange of cell and galvanometer.
Wheatstone is most sensitive when resistance of all four arms is same
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Class XII Physics – Study Guide
POTENTIOMETER
The electromotive force (emf) of a cell is its terminal voltage when no current is flowing through it. The terminal
voltage of a cell is the potential difference between its electrodes. A voltmeter cannot be used to measure the emf
of a cell because a voltmeter draws some current from the cell. To measure a cell's emf a potentiometer is used
since in a potentiometer measurement no current is flowing. It employs a null method of measuring potential
difference, so that when a balance is reached and the reading is being taken, no current is drawn from the source
to be measured.
PRINCIPLE
The fall in potential (potential drop) across a part of a wire is directly proportional to the length (l) of that part
provided the wire has uniform cross- sectional area and constant current is passed through it.
E l
E=Kl
K is the Potential Gradient.
Potential gradient is defined as the potential drop per unit length of a wire.
Let
resistance per unit length i.e.
=
So
=
……………….[ ]
From ohm’s law we have
V=IR
Replacing [i] we get
=
= ,
Therefore
=
………………[ii]
i.e.
Thus equation (ii) gives the potential difference applied across the length of the wire. K is the constant,
therefore,
, is the principle of working of potentiometer.
CONSTRUCTION
Where
J = Jockey
K = Key
R = Resistance of potentiometer wire,
= Resistance per unit length.
Rh = Variable resistance which controls
the current through the wire AB.
Note
i. The specific resistance ( ) of potentiometer wire must be high but its temperature coefficient of resistance
must be low.
ii. The value of known potential difference must be greater than the value of unknown potential difference to be
measured.
iii. The potential gradient must remain constant. For this the current in the primary circuit must remain constant.
iv. The diameter of potentiometer wire must be uniform everywhere.
Potential gradient directly depends upon
The resistance per unit length (R/L) of potentiometer wire.
The radius of potentiometer wire (i.e. Area of cross-section)
The specific resistance of the material of potentiometer wire (i.e. )
The current flowing through potentiometer wire (I)
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Applications of Potentiometer
1. To measure e.m.f. of a cell
A cell of emf. “E” which is to be measured is connected
with potentiometer.
The positive terminal of “E” is connected to point A and
negative terminal is connected to jockey through
galvanometer.
After making connections, jockey touches at different
points along the wire AC.
At one point, galvanometer shows null deflection.
Measure the length (l) of null point “J” from point A
(AJ = l). Since galvanometer shows null deflection current flowing through galvanometer is zero.
Therefore a cell of emf. “E” is in open circuit.
The emf of cell is balanced by the balancing length AJ= l.
EMF of the unknown cell is therefore measured as follows
Let potential gradient of the potentiometer wire be K
At null point balancing length AJ=l
Therefore
So emf of the unknown cell is E=Kl
2. To compare e.m.f of two cells
Let the known emf be “E1” and unknown emf “E2”
which are connected to the circuit via two-way switch.
Let the potential gradient of the wire be “K”.
When the switch touches at “A”
“E1” is in closed circuit while “E2” is in open circuit.
At null-point let the balancing length be l1.
The emf of “E1” is therefore
1=
…………….( )
When the switch touches at “B”
“E2” is in closed circuit while “E1” is in open circuit.
At null-point let the balancing length be l2.
The emf of “E2” is therefore
2=
…………….( )
Dividing (i) and (ii) we get
=
=
Since l1, l2 can be obtained from the reading and that E1 is also known, we can easily compare and find the
value of E2.
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Class XII Physics – Study Guide
3. To compare two resistances
In the following setup, resistances R1 and R2 are to be
compared.
Let the potential gradient of the wire be “K”.
When the switch is closed at “b” and “d”
Current flows through “R1”.
Let the balancing length be “l1”.
Potential difference across “R1” is
=
=
………….( )
When the switch is closed at “a” and “c”
Current flows through “R2”.
Let the balancing length be “l2”.
Potential difference across “R2” is
=
=
………….( )
R1
R2
Dividing (i) by (ii), we get
=
=
=
=
=
4. To determine internal resistance of a cell
In this setup the internal resistance “r” is to be determined.
Let the potential gradient of the wire be “K”.
When the key is open
We get the emf (E) of the cell.
Let the balancing length be l1.
=
………( )
When the key is closed
We get the terminal voltage (V).
Let the balancing length be l2.
=
………..( )
We also have
E = V + Ir
=
=
=
(
)
………..(
= )
1 ……………………….( )
Substituting (i) and (ii) in (iii), we get
=
1
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=
…………………….
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Class XII Physics – Study Guide
AMMETER and VOLTMETER
Ammeters and Voltmeters are based on the galvanometer- a device that is used
to detect small currents. Galvanometers that are calibrated to measure currents
of different amounts are called ammeters. When they are modified to measure
voltages, they are called voltmeters.
GALVANOMETER
The galvanometer is named after Italian physicist Luigi Galvani. It is an
instrument used to detect, measure, and determine the direction of small
electric currents by means of mechanical effects produced by a currentcarrying coil in a magnetic field.
Galvanometer is modified to be either ammeter or voltmeter.
AMMETER
Ammeter is an electrical measuring device, which is used to measure
electric current through the circuit. It is the modified form of
galvanometer.
An ammeter is always connected in series to a circuit.
Conversion of galvanometer into ammeter
Since Galvanometer is a very sensitive instrument therefore it can’t measure heavy currents. In order to convert a
Galvanometer into an Ammeter, a very low resistance known as "shunt" resistance is connected in parallel to
Galvanometer. Value of shunt is so adjusted that most of the current passes through the shunt. In this way a
Galvanometer is converted into Ammeter and can measure heavy currents without fully deflected.
Value of shunt resistance
Let resistance of galvanometer = Rg and it gives full-scale deflection when
current Ig is passed through it. Then,
Vg = IgRg -------(i)
Let a shunt of resistance (Rs) is connected in parallel to galvanometer. If
total current through the circuit is I.
Then current through shunt
Is = (I-Ig)
potential difference across the shunt:
Vs= IsRs
or
Vs = (I – Ig)Rs -------(ii)
But
Vs =Vg
(I - Ig)Rs = IgRg
Therefore
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Class XII Physics – Study Guide
VOLTMETER
Voltmeter is an electrical measuring device, which is used to measure
potential difference between two points in a circuit.
Voltmeter is always connected in parallel to a circuit.
Conversion of galvanometer into voltmeter
Since Galvanometer is a very sensitive instrument, therefore it cannot measure high potential difference. In order
to convert a Galvanometer into voltmeter, a very high resistance known as "series resistance" is connected in
series with the galvanometer.
Value of series resistance
Let resistance of galvanometer = Rg and resistance Rx (high) is connected in
series to it. Then combined resistance = (Rg + Rx).
If potential between the points to be measured = V and if galvanometer
gives full-scale deflection, when current "Ig" passes through it. Then,
V = Ig (Rg + Rx)
V = IgRg + IgRx
V – IgRg = IgRx
Rx = (V – IgRg)/Ig
Thus Rx can be found.
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