Question

The given circuit shows a uniform straight wire AB of 40 cm length fixed at both ends. In order to get zero reading in the galvanometer G, the free end of J is to be placed from B at:

• A. 32 cm
• B. 8 cm
• C. 16 cm
• D. 24 cm

Answer 1

The Correct Option is D

To find the position of the free end of J from B for a zero reading in the galvanometer G, we need to apply the principle of the Wheatstone bridge for balancing the circuit. 

The Wheatstone bridge principle states that for the bridge to be balanced, the ratio of resistances in one arm must be equal to that in the other arm. Let's denote the length of the wire as the resistance, which is uniform.

Given:

• Resistance in the left arm, \( R_1 = 8\, \Omega \)
• Resistance in the right arm, \( R_2 = 12\, \Omega \)
• Total length of wire \( AB = 40 \, \text{cm} \)

Let \( x \) be the distance from B to J where the galvanometer shows zero deflection.

Therefore, the length of AJ = \( 40 - x \) cm.

Balancing condition for the bridge:

\[ \frac{R_1}{R_2} = \frac{L_{AJ}}{L_{JB}} \]

Substituting the values, we get:

\[ \frac{8}{12} = \frac{40 - x}{x} \]

Solving for \( x \):

\[ \frac{2}{3} = \frac{40 - x}{x} \]

\[ 2x = 3(40 - x) \]

\[ 2x = 120 - 3x \]

\[ 5x = 120 \]

\[ x = \frac{120}{5} = 24 \, \text{cm} \]

Thus, the free end of J should be placed 24 cm from B to achieve a zero reading in the galvanometer.

Therefore, the correct answer is 24 cm.