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 to half of the steady-state value.

Answer 1

Given:
Emf of the battery = ε
Inductance of the inductor = L
Resistance = R
Maximum current in the coil = $\frac{\epsilon }{R}$
At the steady state, current in the coil, i = $\frac{\epsilon }{R}$.
The magnetic field energy stored at the steady state is given by
$U=\frac{1}{2}L{i}^2$
or
U$=\frac{{\epsilon }^2}{2R^2}L$
One-fourth of the steady-state value of the magnetic energy is given by
$U\text{'}=\frac{1}{8}L\frac{E^2}{R^2}$
Half of the value of the steady-state energy = $\frac{1}{4}L\frac{E^2}{R^2}$
Let the magnetic energy reach one-fourth of its steady-state value in time t₁ and let it reach half of its value in time t₂.
Now,
$$\begin{gathered} \frac{1}{8}L\frac{E^2}{R^2}=\frac{1}{2}L\frac{E^2}{R^2}(1-e^{-t_{1}R/L}{)}^2 \\ \Rightarrow 1-e^{-t_{1}R/L}=\frac{1}{2} \\ \Rightarrow t_1\frac{R}{L}=\ln 2 \\ \mathrm{And}, \\ \frac{1}{4}L\frac{E^2}{R^2}=\frac{1}{2}L\frac{E^2}{R^2}(1-e^{-t_{2}R/L}{)}^2 \\ \Rightarrow e^{-t_{2}R/L}=\frac{\sqrt{2}-1}{\sqrt{2}}=\frac{2-\sqrt{2}}{2} \\ \Rightarrow t_1=\tau \ln \left(\frac{1}{2-\sqrt{2}}\right)+\ln 2 \end{gathered}$$
Thus, the time taken by the magnetic energy stored in the circuit to change from one-fourth of its steady-state value to half of its steady-state value is given by
$t_2-t_1=\tau \ln \frac{1}{2-\sqrt{2}}$