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.

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answer is 15.

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Detailed Solution

$\begin{array}{l} \frac{dQ}{dt} = i^{2} R = \frac{V^{2}}{R} (we know) \\ \Rightarrow In 't' time, ΔQ = (\frac{V^{2}}{R}) t \end{array}$
Given that, (for same source, v = same)
$\begin{array}{l} Q_{0} = \frac{v^{2}}{R_{1}} \times 20 = \frac{V^{2}}{R_{2}} \times 60 . . . (1) \\ \Rightarrow R_{2} = 3 R_{1} . . . . . . (ii) \end{array}$
If they are connected in parallel then
$Req = \frac{R_{2} R_{1}}{R_{1} + R_{2}} = \frac{3 R_{1} \cdot R_{1}}{3 R_{1} + R_{1}} = (\frac{3 R_{1}}{4})$
To produce same heat, using equation …(1)
$\begin{array}{l} Q_{0} = \frac{V^{2}}{R_{1}} \times 20 = \frac{v^{2}}{(\frac{3 R_{1}}{4})} \times t \\ t = \frac{3 \times 20}{4} = 15 \min \end{array}$