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

$\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 \ldots \ldots$ (1)
$\Rightarrow R _{2}=3 R _{1} \ldots \ldots $ (ii)
If they are connected in parallel then
Req $=\frac{ R _{2} R _{1}}{ R _{1}+ R _{2}}=\frac{3 R _{1} \cdot R _{1}}{3 R _{1}+ R _{1}}=\left(\frac{3 R _{1}}{4}\right)$
To produce same heat, using equation ...(1)
$Q _{0}=\frac{ V ^{2}}{ R _{1}} \times 20=\frac{ v ^{2}}{\left(\frac{3 R _{1}}{4}\right)} \times t$
$t =\frac{3 \times 20}{4}=15 \,\min$