Calculate Unknown Resistance in Meter Bridge with End Corrections

To determine the unknown resistance $X$ using a meter bridge, we use the principle of a balanced Wheatstone bridge.

At the null point, the ratio of the resistances in the two gaps is equal to the ratio of the resistances of the corresponding wire segments:
$\frac{X}{R} = \frac{R_A}{R_B}$

Since the resistance of a uniform wire is directly proportional to its effective length, we include the end corrections $\alpha$ and $\beta$ for ends $A$ and $B$, respectively:
$\frac{X}{R} = \frac{l_1 + \alpha}{l_2 + \beta}$

Given data:

• Standard resistance, $R = 10\text{ }\Omega$
• Distance of the null point from end $A$, $l_1 = 52\text{ cm}$
• Remaining length of the wire towards end $B$, $l_2 = 100 - l_1 = 100 - 52 = 48\text{ cm}$
• End-correction at end $A$, $\alpha = 1\text{ cm}$
• End-correction at end $B$, $\beta = 2\text{ cm}$

Substitute these values into the bridge equation:
$\frac{X}{10} = \frac{52 + 1}{48 + 2}$

$\frac{X}{10} = \frac{53}{50}$

Solving for $X$:
$X = 10 \times \frac{53}{50} = \frac{53}{5} = 10.6\text{ }\Omega$

Thus, the determined value of $X$ is $10.6\text{ ohm}$.

Correct Option: B