Question

Current in a coil of self-inductance $2.0H$ is increasing as $i=2\sin t^2$. The amount of energy spent during the period when the current changes from $0$ to $2A$ is:

• A. $1J$
• B. $2J$
• C. $3J$
• D. $4J$

Answer 1

The correct option is D $4J$

Given that,

Self-inductance of coil = $2.0H$

Let the current flowing through the coil be 2A at time $t^2$

Now,

$2=2\sin t^2$

$t=\sqrt{\frac{\pi }{2}}$

Now, the amount of energy spent during the period when the current changes from $0to2A$

Now, Self-induced e. m. f

$E=L\frac{di}{dt}$

Now, the work done is

$dW=L\left(\frac{di}{dt}\right)dq$

$dW=L\times \frac{di}{dt}\times idt$

$dW=Lidi$

Now, on integrate

$\int dW=\underset{0}{\overset{t}{\int }}Lidi$

$W=\underset{0}{\overset{t}{\int }}L2\sin t^{2}d\left(2\sin t^2\right)$

$W=\underset{0}{\overset{t}{\int }}8L\sin t^2\cos t^{2}tdt$

$W=4L\underset{0}{\overset{t}{\int }}\sin 2t^{2}dt$

Now, let

$\theta =2t^2$

$d\theta =4tdt$

$dt=\frac{d\theta }{4t}$

Now, put the value of $2t^2$and $dt$

$W=4L\underset{0}{\overset{t}{\int }}\frac{\sin \theta d\theta }{4}$

$W=L{\left[(-\cos \theta )\right]}_0^t$

$W=-L{\left[\cos 2t^2\right]}_0^t$

$W=-L{\left[\cos 2t^2\right]}_0^{\sqrt{\frac{\pi }{2}}}$

$W=2L$

$W=2\times 2$

$W=4J$

Hence, the amount of energy is $4J$