Answer
$t = 0.520~ms$
Work Step by Step
The emf across the inductor will be equal to the potential difference across the resistor when the current $i$ is half the value of the steady state current $i_0$
We can find $t$:
$i = i_0~(1-e^{-tR/L})$
$0.500~i_0 = i_0~(1-e^{-tR/L})$
$0.500 =1-e^{-tR/L}$
$e^{-tR/L} = 0.500$
$e^{tR/L} = 2.00$
$\frac{tR}{L} = ln(2.00)$
$t = \frac{ln(2.00)~L}{R}$
$t = \frac{ln(2.00)~(15.0\times 10^{-3}~H)}{20.0~\Omega}$
$t = 0.520~ms$
