Question #43842

A solid semiconductor sphere of radius $R$ and relative permittivity $\epsilon_r$ is in thermal equilibrium at temperature $T$. The sphere contains a fixed positive background charge distribution with volume charge density $\rho_f(r)$ and mobile electrons of charge $-e$. Due to internal redistribution, the mobile electron number density is observed to be $n(r) = n_0 e^{-\alpha r^2}$, where $n_0$ and $\alpha$ are positive constants, and $r$ is the radial distance from the center ($r < R$). Assuming the Einstein relation $D = \mu k_B T / e$ holds (where $D$ is the diffusion coefficient, $\mu$ is the mobility, and $k_B$ is the Boltzmann constant), determine the fixed positive background charge density $\rho_f(r)$. Let $\epsilon_0$ be the permittivity of free space.