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________ $min$
JEE MAIN 2022, Diffcult
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$\frac{d Q}{d t}=i^{2} R=\frac{V^{2}}{R}$ (we know)
$\Rightarrow \text { In 't' time , } \Delta Q =\left(\frac{ V ^{2}}{ R }\right) t$
Given that, (for same source, $v=$ same)
$Q _{0}=\frac{ v ^{2}}{ R _{1}} \times 20=\frac{ V ^{2}}{ R _{2}} \times 60 \ldots$ $(i)$
$\Rightarrow R _{2}=3 R _{1} \ldots.$ $(ii)$
If they are connected in parallel then
$\operatorname{Req}=\frac{ R _{2} R _{1}}{ R _{1}+ R _{2}}=\frac{3 R _{1} \cdot R _{1}}{3 R _{1}+ R _{1}}=\left(\frac{3 R _{1}}{4}\right)$
To produce same heat, using equation ...$(i)$
$Q _{0}=\frac{ V ^{2}}{ R _{1}} \times 20=\frac{ v ^{2}}{\left(\frac{3 R _{1}}{4}\right)} \times t$
$t =\frac{3 \times 20}{4}=15 \,min$
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