Question

The current in a coil of inductance $2$H is changing according to varying current $I=\sin \left(2t\right)$. The amount of magnetic energy in the inductor during $t=0$ to $t=\frac{\pi }{4}$s is:

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

Answer 1

The correct option is C $1$J

self inductance of coil is 2H current flow through the coil 2 ampere of time t2 $2=2sin{t}^2$ $t=\sqrt{\frac{\pi }{2}}$ energy spent current charges 0 to 2A self inductance of coil $L\frac{di}{dt}$ i is instantaneous coil dq is amount of charge $dw=L\frac{di}{dt}dt$ $L\frac{di}{dt}idt$ $Lidi$

$w={\int }_0^{T}Lidi$ $w={\int }_0^{T}L2sin{t}^{2}d\left(sin{t}^2\right)$

$w={\int }_0^{T}8L2sin{t}^2\left(cos{t}^2\right)tdt$

$w=4L{\int }_0^{T}sint2t^{2}tdt$ $\theta =2{\theta }^2$ $dt=\frac{d\theta }{4t}$

$W=4L\int \frac{sin\theta d\theta }{4}$ $L(-cos\theta )$

$-Lcos2t^2$ $2L2X24J$

$1J$