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In the circuit shown, terminals A and B are on the left and right of a bridge network. The top branch has a 20 ohm resistor followed by an unknown resistor R. The bottom branch has a 5 ohm resistor followed by a 20 ohm resistor. A switch K connects the midpoints of the two branches. When switch K is open, the equivalent resistance between A and B is 20 ohm. Which of the following statement(s) is/are correct?

1. R = 80 ohm
2. No current flows through K when it is closed
3. The powers dissipated in R and in the 5 ohm resistor are always equal
4. The powers dissipated in the two 20 ohm resistors are unequal

Correct answer: R = 80 ohm

Solution

With K open: top branch resistance = 20 + R, bottom branch = 5 + 20 = 25. These are in parallel: (20+R)*25 / (20+R+25) = 20. Solving: 25(20+R) = 20(45+R), 500 + 25R = 900 + 20R, 5R = 400, R = 80 ohm. Now check bridge balance: top ratios 20:R = 20:80 = 1:4; bottom ratios 5:20 = 1:4. The ratios are equal, so the bridge is balanced. When K is closed, no current flows through it. Powers in R (80 ohm) and 5 ohm: they carry different currents (top and bottom branches), so powers differ. The two 20 ohm resistors: one is in the top branch (current V/100) and one in the bottom (current V/25), so powers differ too.

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