Question

The current in a coil of self inductance $2.0 \,H$ is increasing according to $I =2 \sin \left( t ^{2}\right) A$. The amount of energy spent during the period when current changes from $0$ to $2 A$ is _____$J$.

Answer 1

$I =2 \sin \left( t ^{2}\right) \Rightarrow dI =4 t \sin \left( t ^{2}\right) dt$
If $I =0 \Rightarrow t =0$
and $I =2 \Rightarrow 2=2 \sin t ^{2}$
$\Rightarrow t =\sqrt{\frac{\pi}{2}}$
$E =\int{ LI dI }$
$=\int 2 \times 2 \sin \left( t ^{2}\right) \times 4 t \cos \left( t ^{2}\right) dt$
$=8 \int\limits_{0}^{\sqrt{\pi / 2}} t \sin \left(2 t ^{2}\right) dt$
$=2\left[-\cos \left(2 t ^{2}\right)\right]_{0}^{\sqrt{\pi / 2}}$
$=2[-\cos \pi+\cos 0]=4$