Question #43152

An intrinsic semiconductor component has a temperature-dependent resistance given by $R(T) = R_0 \exp\left(\frac{E_g}{2k_B T}\right)$, where $E_g$ is the energy band gap, $k_B$ is the Boltzmann constant, and $T$ is the absolute temperature. The component is connected to an AC voltage source $V(t) = V_0 \cos(\omega t)$. The frequency $\omega$ is sufficiently high such that the temperature $T$ of the component remains effectively constant at a steady-state value. The component dissipates heat to the surroundings at a constant temperature $T_0$ according to Newton's law of cooling, with a heat transfer coefficient $K$ (in $\text{W/K}$). As the peak voltage $V_0$ is gradually increased, the steady-state temperature $T$ increases. However, beyond a critical voltage $V_{0,c}$, thermal runaway occurs and no steady state is possible. Assuming $E_g \gg k_B T_0$, which of the following expressions best approximates the critical peak voltage $V_{0,c}$? (Here $e$ is the base of the natural logarithm).