Question #42650

A fixed non-conducting sphere of radius $R$ has a spherically symmetric, variable volume charge density given by $\rho(r) = \frac{\rho_0 R}{r}$ for $0 < r \le R$, where $\rho_0$ is a positive constant and $r$ is the distance from the center. A narrow, smooth diametric tunnel is drilled through the sphere. A particle of mass $m$ and charge $-q$ ($q > 0$) is released from rest from the surface of the sphere into the tunnel. Let $T_1$ be the time period of its resulting oscillatory motion. Another identical particle of mass $m$ and charge $-q$ is placed in a circular orbit of radius $R$ around the sphere, grazing its surface. Let $T_2$ be the time period of this circular orbit. Assuming no gravitational forces are present, what is the ratio $\frac{T_1}{T_2}$?