Question

The time taken for the magnetic energy to reach $25\%$ of its maximum value, when a solenoid of resistance $R$ and inductance $L$ is connected to a battery, is -

• A. $\frac{L}{R}\ln 2$
• B. $\frac{L}{R}\ln 10$
• C. Infinite
• D. $\frac{L}{R}\ln 5$

Answer 1

The correct option is A $\frac{L}{R}\ln 2$

We know that, magnetic energy,
$U=\frac{1}{2}L{i}^2$,

When circuit has maximum current:

Maximum value of magnetic energy,
$U_o=\frac{1}{2}L{i}_o^2$

Given,

$U=25\%$ of $U_o$

$\Rightarrow \frac{1}{2}L{i}^2=\frac{1}{4}\times \frac{1}{2}L{i}_o^2$

$\Rightarrow i^2=\frac{i_o^2}{4}\Rightarrow i=\frac{i_o}{2}$

We know that, current in the circuit at any time $t$ is given by :

$i=i_o(1-e^{-Rt/L})$

$\Rightarrow \frac{i_o}{2}=i_o(1-e^{-Rt/L})$

$\Rightarrow \frac{1}{2}=e^{-Rt/L}$

$\Rightarrow e^{Rt/L}=2$

$\Rightarrow \frac{Rt}{L}=\ln 2$

$\Rightarrow t=\frac{L}{R}\ln 2$