Circuits with Resistors and Inductors (LR Circuits)

An inductor holds its current across any switching instant, so at $t = 0^+$ in a fresh circuit it is an open branch and long after, with $dI/dt = 0$, it is a wire: the exact mirror of the capacitor's (wire, open). The loop rule $\varepsilon - IR - L\,dI/dt = 0$ integrates to $I = (\varepsilon/R)(1 - e^{-t/\tau})$ with $\tau = L/R$, so more resistance settles the current faster, toward a smaller final value. Building the current banks $\tfrac12 LI^2$ in the field on top of the resistor's heat, and at shutdown that energy is paid out through whatever path exists, $\int I^2R\,dt$ recovering it exactly.

Four errors do most of the damage. Reversing the endpoints, so full current flows at switch-on and the coil blocks it at steady state, usually by importing the capacitor's roles unmirrored. Building the time constant as $LR$ or scaling it the wrong way with $R$, which the units and the $RC$ contrast both catch. Letting the inductor current jump when a switch opens, when in fact it continues and, offered only an air gap, arcs across it. And treating the field's $\tfrac12 LI^2$ as free at start-up or as vanishing at shutdown, when the ledger has to close at both ends.