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(A) The magnetic flux is given by $\phi = at(T - t) = aTt - at^2$.According to Faraday's law,the induced electromotive force $(EMF)$ is $\xi = -\frac{

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

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(A) The magnetic flux is given by $\phi = at(T - t) = aTt - at^2$.According to Faraday's law,the induced electromotive force $(EMF)$ is $\xi = -\frac{d\phi}{dt}$.$\xi = -\frac{d}{dt}(aTt - at^2) = -(aT - 2at) = 2at - aT$.The heat generated in the loop in time $dt$ is $dQ = \frac{\xi^2}{R} dt$.Substituting the value of $\xi$,we get $dQ = \frac{(2at - aT)^2}{R} dt$.The total heat generated $Q$ from $t = 0$ to $t = T$ is:$Q = \int_{0}^{T} \frac{(2at - aT)^2}{R} dt = \frac{1}{R} \int_{0}^{T} (4a^2t^2 - 4a^2Tt + a^2T^2) dt$.$Q = \frac{a^2}{R} [\frac{4t^3}{3} - 2Tt^2 + T^2t]_{0}^{T}$.$Q = \frac{a^2}{R} [\frac{4T^3}{3} - 2T^3 + T^3] = \frac{a^2}{R} [\frac{4T^3 - 6T^3 + 3T^3}{3}] = \frac{a^2 T^3}{3R}$.

The magnetic flux through a stationary loop with resistance $R$ varies during an interval of time $T$ as $\phi = at(T - t)$. The heat generated during this time,neglecting the inductance of the loop,will be

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