Question

A solenoid has an inductance of $10$ henry and a resistance of $2\Omega$. It is connected to a $10$ battery. How long will it take for the magnetic energy to reach $1_{/4}$ of its maximum value?

Answer 1

The L-R circuit is shown in the figure. Let the current through the circuit be i after a time t. Then, $L\frac{di}{dt}+Ri=V$
or $\frac{di}{V-Ri}=\frac{dt}{L}$
Integrating this expression and keeping $i=0$ when $t=0$, we get
$i=\frac{V}{R}(1-e^{-Rt/L})=i_0(1-e^{-Rt/L})$, where $i_0=V/R=$ maximum current
Magnetic energy stored $=\frac{1}{2}L{i}^2$
Here, $\frac{1}{2}L{i}^2=\frac{1}{4}\left(\frac{1}{2}L{i}_0^2\right)$ or $i=\frac{i_0}{2}=\frac{V}{2R}$
Substituting the values in equation $\left(1\right)$, we get
$\frac{V}{2R}=\frac{V}{R}(1-e^{-2t/10})$ or $0.5=1-e^{-0/2t}$
or $e^{-0.2t}=0.5$, which gives $t=3.478$ sec.