Question #43755

A solid non-conducting sphere of radius $R$ has a positive volume charge density $\rho(r) = \rho_0 \left(\frac{r}{R}\right)^2$, where $r$ is the radial distance from the center and $\rho_0$ is a positive constant. The sphere is kept in a vacuum. The material of the sphere has a photoelectric work function $\Phi$. A monochromatic beam of light of frequency $\nu$ is incident on the surface of the sphere, causing the emission of photoelectrons. Assuming the sphere's macroscopic charge distribution remains unperturbed, what is the maximum kinetic energy of the emitted photoelectrons when they reach a distance of infinity from the sphere? (Here $e$ is the elementary charge, $h$ is Planck's constant, and $\epsilon_0$ is the permittivity of free space. Assume $h\nu$ is sufficiently large to allow electrons to reach infinity).