Question

Write the method to determine the value of an unknown resistance by meter bridge and derive the necessary formula. Draw a circuit diagram.

Answer 1

Hint: When no current flows through the coil or, the potential difference across the galvanometer is zero the bridge is said to be in a balanced condition.

Complete answer:
The Meter bridge is the modification of Wheatstone’s network used to determine the value of unknown resistance.
The meter bridge consists of a thin, uniform, and homogenous conducting wire AC, rectangular wooden board between two thick L shaped metal strips C1 and C2 as shown in the diagram.

In the diagram AC- conducting wire
C1, C2, and C3- metal strips
E- cell
X- unknown resistance
R- resistance box
D- null point
K’- jockey
G- galvanometer
Rh- rheostat

To determine the unknown resistance, the circuit is connected as shown in the diagram. The unknown resistance X is connected in one gap (left gap) and a resistance box (known variable resistor) is connected in another gap (right gap).
A cell of emf E, key K, and rheostat Rh are connected to junction B of X and R also other terminal connected to the jockey.
The jockey is placed at points A and C and the deflection is checked in the galvanometer. This deflection must be on opposite sides otherwise rheostat and the value of resistance from the resistance box should be adjusted.
A suitable value of resistance R is taken in the resistance box and by touching the jockey at different points of the wire AC, a point D is obtained for which the galvanometer shows zero deflection, point D is called the null point.
Let $l_x$ and $l_y$ be the distances of the null point D (balancing lengths) measured in centimeters from end A and C of the wire AC respectively and σ be the resistance per unit length of wire AC.
Here X, R, and resistances of wire AD and wire CD from arms of Wheatstone’s network.
Therefore from balancing condition,
$$\Rightarrow {\displaystyle \frac{\text{X}}{\text{R}}}={\displaystyle \frac{\text{Resistance of wire AD}}{\text{Resistance of wire CD}}}$$
$$\Rightarrow {\displaystyle \frac{\text{X}}{\text{R}}}={\displaystyle \frac{\sigma l_X}{\sigma l_y}}={\displaystyle \frac{l_X}{l_y}}$$
$$\Rightarrow \text{X = R}\left({\displaystyle \frac{l_X}{l_y}}\right)$$
$$\Rightarrow \text{X = R}\left({\displaystyle \frac{l_X}{100-l_x}}\right)$$

$\therefore$ Hence the unknown resistance X can be determined by using the formula $$\text{X = R}\left({\displaystyle \frac{l_X}{100-l_x}}\right)$$.

Note:
Slide wire is known as a meter bridge. The meter bridge is an instrument that works on the principle of the Wheatstone bridge. A meter bridge is also a simple type of the potentiometer that is used to measure the resistance and it is being used in the school laboratories.