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

2
Sol: Magnetic energy in a solenoid is given by
$U=\frac{1}{2}L{I}^2$  where $I=I_0\left(1-e^{-\frac{Rt}{L}}\right)$   in R – L circuit 
   $U=\frac{1}{2}L{I}_0^2{\left(1-e^{\frac{-Rt}{L}}\right)}^2$$U=U_0{\left(1-e^{\frac{-Rt}{L}}\right)}^2\left(\because U_0=\frac{1}{2}L{I}_0^2\right)$
   $\frac{25U_0}{100}=U_0{\left(1-e^{\frac{-RT}{L}}\right)}^2$$\frac{1}{2}=1-e^{\frac{-Rt}{L}}$
Applying logarithm on both sides   $\ln 2=\frac{Rt}{L}ort=\frac{L}{R}\ln 2$