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 `= epsilon/R`

At the steady state, current in the coil, `i =epsilon/R.`

The magnetic field energy stored at the steady state is given by

\[U = \frac{1}{2}L i^2\text{ or } U\]

\[= \frac{\epsilon^2}{2 R^2}L\]

One-fourth of the steady-state value of the magnetic energy is given by

\[U' = \frac{1}{8}L\frac{E^2}{R^2}\]

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,

\[\frac{1}{8}L\frac{E^2}{R^2} = \frac{1}{2}L\frac{E^2}{R^2}(1 - e^{- t_1 R/L} )^2 \]

\[ \Rightarrow 1 - e^{- t_1 R/L} = \frac{1}{2}\]

\[ \Rightarrow t_1 \frac{R}{L} = \ln 2\]

And,

\[\frac{1}{4}L\frac{E^2}{R^2} = \frac{1}{2}L\frac{E^2}{R^2}(1 - e^{- t_2 R/L} )^2 \]

\[ \Rightarrow e^{- t_2 R/L} = \frac{\sqrt{2} - 1}{\sqrt{2}} = \frac{2 - \sqrt{2}}{2}\]

\[ \Rightarrow t_1 = \tau \ln\left( \frac{1}{2 - \sqrt{2}} \right) + \ln 2\]

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

\[t_2  -  t_1  = \tau  \ln\frac{1}{2 - \sqrt{2}}\]