Question #43942

An ideal battery of EMF $\varepsilon$, a switch $S$, an inductor $L$, a resistor $R$, and a special semiconductor device are connected in series. The semiconductor device is designed such that its macroscopic voltage drop $V_D$ is dynamically governed by the relation $V_D = r_d(I) I$, where $I$ is the instantaneous current and $r_d(I) = \frac{dV}{dI}$ is the small-signal dynamic resistance of an ideal P-N junction diode in forward bias ($I \gg I_s$). The ideal diode follows the Shockley equation $I = I_s e^{eV/kT}$, where $e$ is the elementary charge, $k$ is the Boltzmann constant, and $T$ is the absolute temperature. If the switch $S$ is closed at $t=0$, which of the following expressions correctly represents the transient current $I(t)$ in the circuit for $t > 0$? (Assume $\varepsilon > \frac{kT}{e}$ and the initial current is zero).