The magnetic flux through a stationary loop w | vedclass

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

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

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(A) The magnetic flux is given by $\phi = a t(T - t) = a T t - a t^2$.According to Faraday's law,the induced electromotive force $(EMF)$ is $\varepsilon = -\frac{d\phi}{dt}$.$\varepsilon = -\frac{d}{dt}(a T t - a t^2) = -(a T - 2 a t) = 2 a t - a T$.The power dissipated in the loop is $P = \frac{\varepsilon^2}{R} = \frac{(2 a t - a T)^2}{R}$.The total heat generated $H$ in the interval $t = 0$ to $t = T$ is given by $H = \int_{0}^{T} P dt$.$H = \int_{0}^{T} \frac{(2 a t - a T)^2}{R} dt = \frac{1}{R} \int_{0}^{T} (4 a^2 t^2 - 4 a^2 T t + a^2 T^2) dt$.$H = \frac{a^2}{R} [\frac{4 t^3}{3} - 2 T t^2 + T^2 t]_{0}^{T}$.$H = \frac{a^2}{R} [\frac{4 T^3}{3} - 2 T^3 + T^3] = \frac{a^2}{R} [\frac{4 T^3 - 6 T^3 + 3 T^3}{3}] = \frac{a^2 T^3}{3 R}$.

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

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