Question

In a Wheatstone bridge arrangement (P, Q, R and S), the resistors P and Q are nearly equal. The bridge is balanced when R = 500 \(\Omega\). On interchanging P and Q, the value of R for balancing is 505 \(\Omega\). The value of S will be:

• A. 500 \(\Omega\)
• B. 502.5 \(\Omega\)
• C. 505 \(\Omega\)
• D. 5 \(\Omega\)

Hint:

Use \(S = \sqrt{R_1 R_2}\) for the two balancing values.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
A Wheatstone bridge is balanced when the ratio of the two resistors in one arm equals the ratio in the other arm.

Step 2: First balance:
With P and Q in place, balance gives \[ \frac{P}{Q} = \frac{R_1}{S} = \frac{500}{S} \]

Step 3: Second balance:
After P and Q are interchanged, the ratio flips. So \[ \frac{Q}{P} = \frac{R_2}{S} = \frac{505}{S} \]

Step 4: Multiply the two results:
Multiplying gives \[ 1 = \frac{500 \times 505}{S^2} \] \[ S = \sqrt{500 \times 505} = \sqrt{252500} \approx 502.49\ \Omega \]

Step 5: Compare with the options:
The value is about 502.5 ohm. 500 and 505 are the two balance values, and S is between them. 5 ohm is the difference and has no meaning here.

Final Answer:
S is the geometric mean of 500 and 505, about 502.5 ohm. \[ \boxed{S \approx 502.5\ \Omega} \]