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The magnetic flux through a stationary loop having resistance $R$ varies with time $t,$ during an interval of time $T$ as $ϕ = a t (T - t)$ where $a$ is a constant of appropriate dimensions. The heat generated during this time $T$ (neglecting the inductance of loop) will be

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$\frac{a^{2} T^{3}}{3 R}$

$\frac{a^{2} T^{2}}{3 R}$

$\frac{a^{2} T}{3 R}$

$\frac{a^{3} T^{2}}{3 R}$

answer is A.

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Detailed Solution

$ϕ = a t (T - t) = a T t - a t^{2} ε = - \frac{d ϕ}{d t} = - (a T - 2 a t) P = \frac{ε^{2}}{R} = \frac{{(a T - 2 a t)}^{2}}{R} H = \int_{0}^{T} P d t = \int_{0}^{T} \frac{{(a T - 2 a t)}^{2}}{R} d t = \frac{a^{2} T^{3}}{3 R}$

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