Double-Exponential LR Circuit

S imple LR and RC circuits are familiar to generations of physics students as examples of single-exponential growth and decay in the relevant voltages, currents, and charges. An element of novelty can be introduced by connecting two (instead of one) LR coils in parallel with a battery. The resulting circuit can still be treated using little more than the basic tools (Kirchhoff ’s rules plus a trial exponential solution) employed in the standard LR analysis. But the solution is now a double exponential, as can be verified by constructing such a circuit. Consider the circuit shown in Fig. 1, which is adapted from Ref. 1. The resistors include the internal resistances of the coils and battery.2 (This is the reason for the addition of the resistor R2, which was absent from the original circuit in Ref. 1.) Assume the currents in the circuit have reached their steady-state values with switch S open. The switch is then closed at t = 0. The problem is to find the subsequent currents in the circuit as a function of time.