Question

In a coil of resistance $8 \Omega$, the magnetic flux due to an external magnetic field varies with time as $\phi=\frac{2}{3}\left(9- t ^2\right)$. The value of total heat produced in the coil, till the flux becomes zero, will be _____$J$.

Answer 1

$\phi=\frac{2}{3}\left(9- t ^2\right)=0 $
$ t =3 sec$
$ e =\frac{- d \phi}{ dt }=-\frac{2}{3}(0-2 t )=\frac{4 t }{3}$
Heat produced in $3 sec =\int \frac{ e ^2}{ r } dt =\int\limits_0^3 \frac{16 t ^2}{9 \times 8} dt$ $=2 J$