In a coil of resistance 8   Ω , the magnetic flux due to an external magnetic field varies with…

It is given that magnetic flux $\varphi =\frac{2}{3}\left(9-{\mathrm{t}}^2\right)$

For magnetic flux to be zero $0=\frac{2}{3}\left(9-{\mathrm{t}}^2\right)\Rightarrow t=3 \mathrm{s}$

According to Faraday's law of electromagnetic induction,

$\epsilon =-\frac{\mathrm{d}\varphi }{\mathrm{d}t}=-\frac{2}{3}\left(0-2t\right)=\frac{4t}{3}$

Heat produced from $0$ to $3 \mathrm{s}$,

$H={\int }_0^3\frac{{\epsilon }^2}{R}\mathrm{d}t={\int }_0^3\frac{16t^2}{9\times 8}\mathrm{d}t=2 \mathrm{J}$