Answer
$t = 1.23~\tau_L$
Work Step by Step
We can write an expression for the energy stored in the inductor's magnetic field:
$U_B = \frac{L~i^2}{2}$
Let $~~i_0~~$ be the steady-state value of the current.
If the energy stored in the magnetic field is 0.500 of the steady state value, we can find an expression for the current $i$ at that time:
$\frac{L~i^2}{2} = 0.500~\frac{L~i_0^2}{2}$
$i^2 = 0.500~i_0^2$
$i = 0.707~i_0$
We can find $t$:
$i = i_0~(1-e^{-t/\tau_L})$
$0.707 =(1-e^{-t/\tau_L})$
$e^{-t/\tau_L} = 0.293$
$e^{t/\tau_L} = 3.413$
$\frac{t}{\tau_L} = ln(3.413)$
$t = ln(3.413)~\tau_L$
$t = 1.23~\tau_L$
