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Q.

A magnetic flux through a stationary loop with a resistance R varies during the time interval $τ$ as $ϕ = at (τ - t)$ . What is the amount of heat generated in the loop during that time?

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a

$\frac{a^{2} τ^{3}}{6 R}$

b

$\frac{a^{2} τ^{3}}{2 R}$

c

$\frac{a^{2} τ^{3}}{4 R}$

d

$\frac{a^{2} τ^{3}}{3 R}$

answer is B.

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Detailed Solution

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$V = \frac{dϕ}{dt} = \frac{d}{dt} [at (τ - t)] ∴ V = aτ - 2 at$
Heat generated $= \frac{V^{2}}{R}$
$\begin{array}{l} ∴ P = \frac{1}{R} \int_{0}^{τ} (a^{2} τ^{2} + 4 a^{2} t^{2} - 4 a^{2} τt) dt \\ = \frac{1}{R} {(a^{2} τ^{2} t + 4 a^{2} \frac{t^{3}}{3} - 4 a^{2} τ \frac{t^{2}}{2})}_{0}^{τ} \\ = \frac{1}{R} ({aτ}^{3} + \frac{4 a^{2} τ^{3}}{3} - 2 a^{2} τ^{3}) = \frac{a^{2} τ^{3}}{3 R} \end{array}$

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