Question

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.

Answer 1

Answer: 15

$\frac{dQ}{dt}=i^{2}R=\frac{V^2}{R}$ (we know)
$\Rightarrow$ In '$t$' time, $\Delta Q=\left(\frac{V^2}{R}\right)t$
Given that, (for same source, $v=$ same)
$Q_0=\frac{v^2}{R_1}\times 20=\frac{V^2}{R_2}\times 60\dots \dots$ (1)
$\Rightarrow R_2=3R_1\dots \dots$ (ii)
If they are connected in parallel then
Req $=\frac{R_2{R}_1}{R_1+R_2}=\frac{3R_1\cdot R_1}{3R_1+R_1}=\left(\frac{3R_1}{4}\right)$
To produce same heat, using equation ...(1)
$Q_0=\frac{V^2}{R_1}\times 20=\frac{v^2}{\left(\frac{3R_1}{4}\right)}\times t$
$t=\frac{3\times 20}{4}=15\min$