Question #93104

A solid transparent sphere of radius $R$, uniform mass density $\rho$, and refractive index $n_s = 2.5$ is fixed in an infinite, uncharged transparent fluid. The sphere carries a uniform surface charge density $\sigma$. Due to a non-linear electro-optic effect, the fluid behaves as a gradient-index (GRIN) medium. Its local refractive index at a distance $r \ge R$ from the center is given by $n(r) = \sqrt{1 + \kappa |\vec{E}(r)|}$, where $\vec{E}(r)$ is the local electrostatic field and $\kappa = \frac{16 \epsilon_0}{\sigma}$ is a constant. Assume the fluid's relative permittivity for electrostatic fields is $\epsilon_r \approx 1$. A monochromatic light ray is incident from infinity with an impact parameter $b = 5R$. What is the distance of closest approach of this ray to the center of the sphere?