Problem — RL current at switch-on and switch-off

Problem

In an $RL$ circuit the switch is closed at $t=0$ and then reopened to disconnect the battery at $t=3\tau$ (short-circuiting the inductor across $R$). Sketch qualitatively $i(t)$ and indicate the characteristic values.

Solution

Closing ($0\le t<3\tau$). The inductor opposes the change in current: it grows exponentially towards the steady-state value $i_\infty=E/R$ with time constant $\tau=L/R$: $i(t)=i_\infty\left(1-e^{-t/\tau}\right)$ At $t=3\tau$ we have $i=i_\infty(1-e^{-3})\approx 0{,}95\,i_\infty$.

Opening ($t\ge 3\tau$). With the battery disconnected, the inductor is closed across $R$: the current cannot jump to zero and decays exponentially with the same $\tau$: $i(t)=i(3\tau)\,e^{-(t-3\tau)/\tau}\ \longrightarrow\ 0$

Rise $i_\infty(1-e^{-t/\tau})$ up to $\approx 0{,}95\,i_\infty$ at $t=3\tau$, then decay $e^{-(t-3\tau)/\tau}$ towards $0$.

$\ev{i(t)=i_\infty\!\left(1-e^{-t/\tau}\right)\ (t<3\tau),\quad i(t)=i(3\tau)\,e^{-(t-3\tau)/\tau}\ (t\ge 3\tau),\quad i_\infty=\tfrac{E}{R},\ \tau=\tfrac{L}{R}}$

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