Two coils require 20 minutes and 60 minutes respectively to produce same amount of heat energy when connected separately to the same source. If they are connected in parallel arrangement to the same source; the time required to produce same amount of heat by the combination of coils, will be ______ min.

Two coils produce the same amount of heat when connected separately to a voltage source $$V$$, taking 20 minutes and 60 minutes respectively. The heat generated by a coil of resistance $$R$$ in time $$t$$ is given by $$H = \frac{V^2}{R} \times t$$, and since both coils produce the same heat, we have

$$\frac{V^2}{R_1} \times 20 = \frac{V^2}{R_2} \times 60$$

which simplifies to

$$\frac{20}{R_1} = \frac{60}{R_2}$$

and hence

$$R_2 = 3R_1\,. $$

When these two coils are connected in parallel, the equivalent resistance $$R_{eq}$$ satisfies

$$\frac{1}{R_{eq}} = \frac{1}{R_1} + \frac{1}{R_2} = \frac{1}{R_1} + \frac{1}{3R_1} = \frac{4}{3R_1}$$

so that

$$R_{eq} = \frac{3R_1}{4}\,. $$

To produce the same heat $$H$$ with this parallel combination, we write

$$H = \frac{V^2}{R_{eq}} \times t$$

and use the fact that $$H = \frac{V^2}{R_1} \times 20$$, giving

$$\frac{V^2}{R_1} \times 20 = \frac{V^2}{\frac{3R_1}{4}} \times t$$

which leads to

$$\frac{20}{R_1} = \frac{4t}{3R_1}\,,\quad 20 = \frac{4t}{3}\,,\quad t = 15 \text{ min}\,. $$

Hence, the time required is 15 minutes.