Question

A coil having an inductance $L$ and a resistance $R$ is connected to a battery of emf $ϵ$. Find the time taken for the magnetic energy stored in the circuit to change from one fourth of the steady-state value of half of the steady-state value.

Answer 1

maximum current $=\frac{E}{R}$
In steady state magnetic field energy stored $=\frac{1}{2}L\frac{E^2}{R^2}$
The fourth of stady state energy $==\frac{1}{8}L\frac{E^2}{R^2}$
One half of steady energy $=\frac{1}{4}L\frac{E^2}{R^2}$
$\frac{1}{8}L\frac{E^2}{R^2}+\frac{1}{2}L\frac{E^2}{R^2}{(1-e^{-t_{1}R/L})}^2\Rightarrow 1-e^{-t_{1}R/L}=\frac{1}{2}\Rightarrow t_1\frac{R}{L}=\ln 2\Rightarrow t_1=\tau \ln 2$
$\frac{1}{4}L\frac{E^2}{R^2}=\frac{1}{2}L\frac{E^2}{R^2}{(1-e^{-t_{2}R/L})}^2$
$\Rightarrow e^{-t_{2}R/L}=\frac{\sqrt{2}-1}{\sqrt{2}}=\frac{2-\sqrt{2}}{2}\Rightarrow t_2=\tau [\ln \left(\frac{1}{2-\sqrt{2}}\right)+\ln 2]$
so $t_2-t_1=\tau \ln \left(\frac{1}{2-\sqrt{2}}\right)$