Question #43607

A spherical metallic cathode of radius $R$ is kept in a vacuum with surroundings at $0 \text{ K}$. It has an initial work function $\Phi_0$ and Fermi energy $E_{F0}$ at $0 \text{ K}$. The work function $\Phi$ is related to the Fermi energy $E_F$ by $\Phi = W_0 - E_F$, where $W_0$ is a temperature-independent constant representing the depth of the potential well. The Fermi energy depends on the free electron density $n$ as $E_F \propto n^{2/3}$. The cathode is illuminated by a light source of wavelength $\lambda$, from which it continuously absorbs a constant power $P$. It loses heat only by thermal radiation (assume emissivity $\epsilon = 1$). If the coefficient of linear thermal expansion of the metal is $\alpha$ (where $\alpha \ll 1$), which of the following expressions gives the stopping potential $V_s$ for the emitted photoelectrons when the cathode reaches a steady-state temperature? (Assume $\frac{hc}{\lambda} > \Phi_0$, $e$ is the elementary charge, and $\sigma$ is the Stefan-Boltzmann constant.)