Magnetic flux linked with a stationary loop of resistance $4 Ω$ varies with respect to time during the time period 2 second, as follows: $ϕ = 3 t (2 - t) .$ Find the amount of heat generated ( in joules) during that time.

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answer is 6.

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Detailed Solution

$\begin{array}{l} dH = i^{2} Rdt; ρ = \frac{dϕ}{dt} = \frac{d}{dt} (6 t - 3 t^{2}) V = 6 - 6 t; i = \frac{V}{R}; dH = \frac{V^{2}}{R^{2}} Rdt \\ dH = \frac{ρ^{2}}{R} dt; \int dH = \frac{1}{4} \int_{0}^{2} (6 - 6 t)^{2} dt = 6 J \end{array}$

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