Question

The current in a coil of self inductance $5H$ is increasing according to $i=2\sin t^2$ . The amount of energy spent during the period when current changes from 0 to 2 ampere is

• A. 10 joule
• B. 5 joule
• C. 100 joule
• D. 2 joule

Answer 1

Answer: (A)

10 joule

$i=2\sin \left(t^2\right)$
$e=-L\frac{di}{dt}=-4Lt\cos \left(t^2\right)$

Amount of energy spent during the period when current changes from 0 to 2 ampere,

$={\int }_{t=0}^{t=\sqrt{\pi /2}}eidt={\int }_{t=0}^{t=\sqrt{\pi /2}}-8Lt\cos \left(t^2\right)\sin \left(t^2\right)dt={\int }_{t=0}^{t=\sqrt{\pi /2}}-4Lt\sin \left(2t^2\right)dt$
$Letu=2t^2\Rightarrow du=4tdt$
${\int }_{t=0}^{t=\sqrt{\pi /2}}-4Lt\sin \left(2t^2\right)dt={\int }_{t=0}^{t=\sqrt{\pi /2}}-L\sin \left(u\right)du=2L=10joules$