In a coil of resistance 8 \,\Omega , the magnetic flux due to an external magnetic field varies with time as \phi=\frac23(9-t^2) . The value of total heat produced in the coil, till the flux becomes zero, will be \underline\hspace2cmJ .

Electromagnetic Induction PYQ — JEE Main 2022 · 26 Jul · Shift 2 · Q64 with Solution

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Correct answer: 2

We need the total heat produced till the flux becomes zero.

Set $\phi=0$ : $\frac{2}{3}(9-t^2)=0$ $9-t^2=0$ $t^2=9$ $t=3\,\text{s}$

So we calculate heat produced from $t=0$ to $t=3\,\text{s}$ .

By Faraday's law, $e=-\frac{d\phi}{dt}$

Differentiate: $\phi=\frac{2}{3}(9-t^2)=6-\frac{2}{3}t^2$ $\frac{d\phi}{dt}=-\frac{4}{3}t$

Hence magnitude of induced emf is $|e|=\frac{4}{3}t$

Using Ohm's law, $i=\frac{e}{R}$

So magnitude of current is $i=\frac{\frac{4}{3}t}{8}=\frac{t}{6}$

Joule heating power: $P=i^2R$

Substitute $i=\frac{t}{6}$ :

=\frac{t^2}{36}\cdot 8 =\frac{2t^2}{9}$$ --- 6. **Total heat produced** $$H=\int_0^3 P\,dt =\int_0^3 \frac{2t^2}{9}\,dt$$ $$H=\frac{2}{9}\int_0^3 t^2\,dt =\frac{2}{9}\left[\frac{t^3}{3}\right]_0^3$$ $$H=\frac{2}{9}\cdot \frac{27}{3} =\frac{2}{9}\cdot 9 =2\,\text{J}$$ --- 7. **Final answer** The total heat produced is $$\boxed{2\,\text{J}}$$ This matches the stored correct answer.