First slide

Faraday's law of Electromagnetic induction

Question

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

Easy

A

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

$\frac{{aT}^{2}}{3 R}$
C

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

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

Solution

Given that, $Φ = at (T - t) = atT - {at}^{2}$
$∴ e = \frac{dΦ}{dt} = aT - 2 at = a (T - 2 t)$
Let heat generated in time dt be dH Then
$dH = i^{2} Rdt = \frac{e^{2}}{R^{2}} \times Rdt = \frac{e^{2} dt}{R}$
Total heat generated can be obtained by integrating above expression, i,e.,
$\begin{array}{l} \int_{0}^{T} dH = \int_{0}^{T} \frac{e^{2} dt}{R} \\ H = \frac{1}{R} \int_{0}^{T} e^{2} dt = \frac{1}{R} \int_{0}^{T} a^{2} (T - 2 t)^{2} dt \\ = \frac{a^{2}}{R} {[(T^{2} t + \frac{4 t^{3}}{3} - 2 t^{2} T)]}_{0}^{T} \\ = \frac{a^{2}}{R} [T^{3} + \frac{4 T^{3}}{3} - 2 T^{3}] = \frac{a^{2} T^{3}}{3 R} \end{array}$

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