Current Electricity Ques 5
5. The charge flowing through a resistance $\mathrm{R}$ varies with time $t$ as $Q=a t-b t^2$, where $a$ and $b$ are positive constants. The total heat produced in $\mathrm{R}$ is:
[2016]
(a) $\frac{a^3 R}{6 b}$
(b) $\frac{a^3 R}{3 b}$
(c) $\frac{a^3 R}{2 b}$
(d) $\frac{a^3 R}{b}$
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Answer:
Correct Answer: 5.(a)
Solution: (a) Given: Charge $Q=a t-b t^2$
$\therefore$ Current $i=\frac{\partial Q}{\partial t}=a-2 b t$
$\left\{\right.$ for $\left.\mathrm{i}=0 \Rightarrow t=\frac{a}{2 b}\right\}$
From joule’s law of heating, heat produced $\mathrm{dH}=\mathrm{i}^2 \mathrm{Rdt}$
$ \begin{aligned} & H=\int_0^{a / 2 b}(a-2 b t)^2 R d t \\ & H=\left.\frac{(a-2 b t)^3 R}{-3 \times 2 b}\right|_0 ^{\frac{a}{2 b}}=\frac{a^3 R}{6 b} \end{aligned} $
BETA
