Two conducting wires of the same material and of equal lengths and equal diameters are first connected in series and then parallel in a circuit across the same potential difference. The ratio of heat produced in series and parallel combinations would be −

(a) 1:2

(b) 2:1

(c) 1:4

(d) 4:1

(c) The Joule heating is given by, H = i²Rt

Let, R be the resistance of the two wires.

The equivalent resistance of the series connection is R_S = R + R = 2R

If V is the applied potential difference, then it is the voltage across the equivalent resistance.

The heat dissipated in time t is,

The equivalent resistance of the parallel connection is R_P =

V is the applied potential difference across this R_P.

The heat dissipated in time t is,

So, the ratio of heat produced is,

Note: H R also H i² and H t. In this question, t is same for both the circuit. But the current through the equivalent resistance of both the circuit is different. We could have solved the question directly using H R if in case the current was also same. As we know the voltage and resistance of the circuits, we have calculated i in terms of voltage and resistance and used in the equation H = i²Rt to find the ratio.