Question

A set-up to control the brightness of a bulb is shown below.

X and Y are to connected to a uniform circular ring which has a constant cross-sectional area and resistivity X is connected to the left side of the ring.

Where should Y be connected for the light to be the dimmest?

Answer

The circular ring and its connection point can be seen as a connection of 6 resistors of resistance R as shown below:

When Y is connected to A, the connection is as shown below:

The effective resistance is 0.833 R.

When Y is connected to B, the connection is as shown below:

The effective resistance is 1.33 R.

When Y is connected to C, the connection is as shown below:

The effective resistance is 1.5 R.

When Y is connected to D, the connection is as shown below:

The effective resistance is 1.33 R.

Hence, connecting Y to C will result in the greatest resistance and the light bulb will be the dimmest.