In a coil of resistance 8Ω , the magnetic flux due to an external...

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

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

According to Faraday's law of electromagnetic induction,

$ε = - \dfrac{dϕ}{dt} = - \dfrac{2}{3}\left(0 - 2t\right) = \dfrac{4t}{3}$

Heat produced from $0$ to $3s$,

$H = {∫}_{0}^{3}\dfrac{{ε}^{2}}{R}dt = {∫}_{0}^{3}\dfrac{16{t}^{2}}{9 \times 8}dt = 2J$