Introduction:
In a Wheatstone bridge, four resistors are arranged in a diamond shape with a power source connected across one diagonal and a galvanometer connected across the other diagonal. The bridge is balanced when the galvanometer reads zero, indicating that the potential difference across the galvanometer is zero.

Given:
- The ratio of arms P and Q is approximately equal.
- R = 500 ohms when the bridge is balanced.
- On interchanging P and Q, the value of R is 505 ohms.

Analysis:
To find the value of S, we need to analyze the Wheatstone bridge circuit and the conditions for balance.

Condition for Balance:
For a Wheatstone bridge to be balanced, the following condition must be satisfied:
\( \frac{P}{Q} = \frac{R}{S} \)

First Scenario:
When the bridge is balanced, the ratio of P to Q is approximately equal. Therefore, we can assume \( \frac{P}{Q} \approx 1 \).

Applying the condition for balance, we have:
\( \frac{P}{Q} = \frac{R}{S} \)
\( \frac{1}{1} = \frac{500}{S} \)
\( S = 500 \)

Therefore, when the bridge is balanced and the ratio of P to Q is approximately equal, the value of S is 500 ohms.

Second Scenario:
When P and Q are interchanged, the value of R becomes 505 ohms.

Applying the condition for balance with the interchanged values, we have:
\( \frac{Q}{P} = \frac{R}{S} \)
\( \frac{1}{1} = \frac{505}{S} \)
\( S = 505 \)

Therefore, when P and Q are interchanged and the bridge is balanced, the value of S is 505 ohms.

Conclusion:
- When the bridge is balanced with a ratio of P to Q approximately equal, the value of S is 500 ohms.
- When P and Q are interchanged and the bridge is balanced, the value of S is 505 ohms.