Metre bridge- Law of combination of resistors

# Our Objective:

To verify the laws of resistances in series and parallel.

# The Theory:

## Metre Bridge

The metre bridge, consists of a one metre long wire of uniform cross sectional area, fixed on a wooden block. A scale is attached to the block. Two gaps are formed on it by using thick metal strips in order to make the Wheat stone’s bridge. The terminal B between the gaps is used to connect galvanometer and jockey.

The metre bridge is operates under Wheatstone’s principle. Here, four resistors P, Q, R, and S are connected to form the network ABCD.

In the balancing condition, there is no deflection on the galvanometer. Then,

$\frac{P}{Q} =\frac{R}{S}$
A resistance wire is introduced in gap G1 and the resistance box is in gap G2. One end of the galvanometer is connected to terminal D and its other end is connected to a jockey. As the jockey slides over the wire AC, it shows zero deflection at the balancing point (null point).

If the length AB is l, then the length BC is ( 100-l ).

Then, according to Wheatstone’s principle;

$\frac{X}{R} =\frac{l}{(100-l)}$
Now, the unknown resistance can be calculated as,

## $X =\frac{Rl}{(100-l)}$ Resistors in Series

When two or more resistors are connected such a way that one end of one resistor is connected to the starting end of the other, then the circuit is called a Series Circuit.

When the two resistors X1 and X2 are connected in series in a circuit, the current I passing through each resistor is same.

Using Ohm’s Law, the potential difference V1 across X1 is:

$V_{1}=IX_{1}$

$V_{2}=IX_{2}$
Let Xs be the effective resistance of the two resistors in series, and V be the potential difference across the ends.

$V=V_{1}+V_{2}$

$V=IX_{s}$

$IX_{s}=IX_{1}+IX_{2}$

$IX_{s}=I(X_{1}+X_{2})$

$Hence, \;X_{s}=X_{1}+X_{2}$

Thus, when a number of resistors are connected in series, the effective resistance is equal to the sum of the individual resistances. This is called the law of combination of resistances in series.

$i.e, \;X_{s}=X_{1}+X_{2}+X_{3}+X_{4}............+X_{n}$
Adding resistors in series always increases the effective resistance.

## Resistors in Parallel

If the starting ends of two resistors are joined to a point and the terminal ends of the two are combined and given connection to a source of electricity,those circuits are called Parallel Circuit.

When the two resistors X1 and X2 are connected in parallel in a circuit, the potential difference across X1 and X2 are the same.

Then the current passing through the circuit is,

$I_{p}=I_{1}+I_{2}$

$i.e,\;\frac{V}{X_{p}} = \frac{V}X_{1}+\frac{V}{X_{2}}$

$i.e,\;\frac{1}{X_{p}} = \frac{1}X_{1}+\frac{1}{X_{2}}$
If there are ‘n’ number of resistors with different resistances connected in parallel, then we have,

$i.e,\;\frac{1}{X_{p}} = \frac{1}X_{1}+\frac{1}{X_{2}}+\frac{1}{X_{3}}+\frac{1}{X_{4}}.................+\frac{1}{X_{n}}$
That is, for a set of parallel resistors, the reciprocal of their equivalent resistance equals the sum of the reciprocals of their individual resistances. Thus, resistance decreases in parallel combination.

# Learning Outcomes:

• The student learns the following concepts:
• Resistance in a circuit.
• When two resistors are connected in series, its equivalent resistance increases.
• Law of combination of resistors connected in series.
• When two resistors are connected in parallel, its equivalent resistance decreases.
• Law of combination of resistors connected in parallel.

Cite this Simulator: