The current in a coil of self inductance 2.0 henry is increasing according to `i= 2 sin t^(2)` ampere. Find the amount of energy spent during the period when the current changes from 0 to 2 amp.

Explanation

Let the current be 2 amp at `t=tau`
Then `2=2sintau^(2)impliestau=sqrt((pi)/(2))`
When the instantaneous current is i, the self induced emf is `L(di)/(dt)`. . If the amount of charge that is displaced in time dt is dq, then the elementary work done
`=L.((di)/(dt))dq=L.(di)/(dt)idt=Lidi`
`W=underset(0)overset(tau)intLidi=underset(0)overset(tau)intL(2sint^(2))d(2sint^(2))`
`W=underset(0)overset(tau)int8Lsint^(2)cost^(2)(tdt)=4Lunderset(0)overset(tau)intsin2t^(2)(tdt)`
Let `theta=2t^(2)`
Differentiating `d""theta=4tdt`
`:.W=4Lint(sinthetad""theta)/(4)`
`=L(-costheta)=-Lcos2t^(2)`
`W=-L[cos2t^(2)]_(0)^(sqrt(pi//2))`
`=2L=2xx2=4` joule