Answer
a. 1.8 ms.
b. $17\Omega $.
Work Step by Step
a. The current through an LR circuit is given by the following equation.
$$I=\frac{V_o}{R}(1-e^{-t/\tau})= I_{max}(1-e^{-t/\tau})$$
$$\frac{I}{ I_{max}}=(1-e^{-t/\tau}) $$
Set I equal to 0.75 of its maximum value at t = 2.56 ms, and solve for the time constant.
$$0.75=(1-e^{-(2.56ms)/\tau}) $$
$$\tau=\frac{-2.56ms}{ln0.25}$$
$$\tau = 1.8\;ms$$
b. Find the resistance by using $\tau=L/R$.
$$R=\frac{L}{\tau}=\frac{31.0mH}{1.846ms}=17\Omega $$
