Question

A coil of inductance $1$ henry and resistance $10\Omega$ is connected to an ideal battery of emf $50$ V at time $t=0$. Calculate the ratio of the rate at which magnetic energy is stored in the coil to the rate at which energy is supplied by the battery at $t=0.1$ sec.

• A. $e$
• B. $\frac{1}{e-1}$
• C. $1-\frac{1}{e}$
• D. $\frac{1}{e}$

Answer 1

The correct option is D $\frac{1}{e}$

D. $\frac{1}{e}$

Given,

$L=1H$

$R=10\Omega$

$V_0=50V$

$t=0.1sec$

The current in $L-R$ circuit.

$I=I_0(1-e^{-t/\tau })$. . . . . .(1 )

where, $\tau =\frac{L}{R}$

$\tau =\frac{1}{10}=0.1sec$

The rate at which magnetic energy is stored in the coil,

$K=\frac{dE}{dt}=\frac{d\left(\frac{1}{2}L{I}^2\right)}{dt}$

$K=\frac{dE}{dt}=250\left(\frac{e-1}{e^2}\right)$ by putting the value of $I$ at $t=0.1sec$. . . . . . .(2)

The energy supplied by battery,

$E^{\prime }=V_{0}It$

$E^{\prime }=250\left(\frac{e-1}{e}\right)$ By puttting the value of $I$ at $t=0.1sec$. . . . . .(3)

Dividing euation (2) by (1), we get

$\frac{K}{E^{\prime }}=\frac{1}{e}$

Thus, the correct option D