Question #43943

Consider a P-N junction diode operating under a forward bias voltage $V$ (in Volts) and carrying a current $I$ (in Amperes). The charge carriers in the conduction band are modeled as an ideal gas of electrons following a Maxwell-Boltzmann distribution, with an adiabatic index $\gamma$. The diode's current-voltage relationship is given by $I = I_0 \exp\left(\frac{eV}{kT}\right)$, where $e$ is the elementary charge, $k$ is the Boltzmann constant, $T$ is the instantaneous temperature of the electron gas (in Kelvin), and $I_0$ is a constant reverse saturation current.

The junction is subjected to a rapid transient voltage fluctuation $dV$. The timescale of this fluctuation is much shorter than the lattice thermal diffusion time, meaning the electron gas undergoes an adiabatic process. During this process, the differential work done on each electron by the external voltage source is $dW = e dV$.

Which of the following expressions gives the adiabatic dynamic resistance $r_{d, ad} = \frac{dV}{dI}$ of the diode under these conditions?