Question

In the following circuit diagram, when the \(3\Omega\) resistor is removed, the equivalent resistance of the network: 
 

• A. Increases
• B. Decreases
• C. Remains the same
• D. None of these

Hint:

In a balanced Wheatstone bridge, removing a resistor from one arm does not affect the total equivalent resistance, as no current flows through that branch when the bridge is balanced.

Answer 1

The Correct Option is C

Step 1: {Wheatstone Bridge Identification}
The provided network constitutes a balanced Wheatstone bridge. A Wheatstone bridge is characterized by four resistors arranged in a diamond configuration, with two resistors per arm, and a galvanometer positioned across the bridge's center. In this scenario, the bridge maintains its balance even after the \( 3 \, \Omega \) resistor in the BD arm is omitted. Key Concept: A Wheatstone bridge is balanced when the resistance ratio in one pair of opposite arms equals the ratio in the other pair. The balance condition is expressed as: \[ \frac{R_1}{R_2} = \frac{R_3}{R_4} \] Here, \( R_1, R_2, R_3, \) and \( R_4 \) represent the resistances of the four bridge arms. The removal of the \( 3 \, \Omega \) resistor from the BD arm does not alter the total resistance, as the bridge is balanced, resulting in zero current flow through the galvanometer. Consequently, no current passes through the branch containing the \( 3 \, \Omega \) resistor, and the bridge's equivalent resistance remains unchanged. Step 2: {Equivalent Resistance Determination}
Due to the sustained balance of the bridge, the network's equivalent resistance is unaffected. For a balanced Wheatstone bridge, the total equivalent resistance is determined by the resistances of the remaining arms. Let \( R_1, R_2, R_3, \) and \( R_4 \) be the resistances of the other arms. When the bridge is balanced, the equivalent resistance of the two parallel arms (AC and BD) is calculated as: \[ R_{{eq}} = \frac{R_1 R_2}{R_1 + R_2} \] This formula applies to both the upper and lower sections of the Wheatstone bridge, contributing equally to the overall equivalent resistance. The removal of the \( 3 \, \Omega \) resistor from the BD arm does not modify the network's equivalent resistance because the Wheatstone bridge was balanced, and no current flowed through the removed resistor. Therefore, the equivalent resistance remains as previously calculated. Step 3: {Conclusion}
Consequently, the total equivalent resistance of the bridge remains constant upon the removal of the \( 3 \, \Omega \) resistor.