Question

The current in a coil of self inductance $2.0H$ 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 $2A$ is _____$J$.

Answer 1

Answer: 4

$I=2\sin \left(t^2\right)\Rightarrow dI=4t\sin \left(t^2\right)dt$
If $I=0\Rightarrow t=0$
and $I=2\Rightarrow 2=2\sin t^2$
$\Rightarrow t=\sqrt{\frac{\pi }{2}}$
$E=\int LIdI$
$=\int 2\times 2\sin \left(t^2\right)\times 4t\cos \left(t^2\right)dt$
$=8\underset{0}{\overset{\sqrt{\pi /2}}{\int }}t\sin \left(2t^2\right)dt$
$=2{\left[-\cos \left(2t^2\right)\right]}_0^{\sqrt{\pi /2}}$
$=2[-\cos \pi +\cos 0]=4$