A Wheatstone's bridge is balanced with a resistance of 625Ω in the third arm, where P, Q, and S are in the 1st, 2nd, and 4th arm respectively. If P and Q are interchanged, the resistance in the third arm has to be increased by 51Ω to secure balance. The unknown resistance in the fourth arm is:
A. 625Ω
B. 650Ω
C. 676Ω
D. 600Ω

To solve this problem, let's use the principle of a Wheatstone bridge, which states that the bridge is balanced when the ratio of resistances in the two arms (P/Q) is equal to the ratio of resistances in the other two arms (R/S), where R is the unknown resistance in the fourth arm.

Initially, the bridge is balanced with a resistance of 625Ω in the third arm, so:

(P/Q) = (625/625) = 1.

Now, when P and Q are interchanged, the resistance in the third arm has to be increased by 51Ω to maintain balance. This means the new balance condition is:

(P'/Q) = (R/S),

where P' is the resistance in the first arm after the interchange, and it is equal to Q + 51Ω. So:

(P'/Q) = ((Q + 51)/Q).

We know that (P/Q) must be equal to (P'/Q) for the bridge to be balanced in both cases, so:

1 = ((Q + 51)/Q).

Now, let's solve for Q:

Q = Q + 51.

Subtract Q from both sides:

0 = 51.

This is not possible, which means there's no valid value of Q that satisfies this equation. The bridge cannot be balanced when P and Q are interchanged. Therefore, there might be an error in the question or the given information.

The correct answer should be that the bridge cannot be balanced with P and Q interchanged. So, none of the provided answer choices (A, B, C, D) is correct.