All Problems Conductors and Dielectrics in an Electric Field
Problem 3.89
A point charge \(q\) is located in vacuum at a distance \(l\) from the plane surface of a uniform isotropic dielectric filling up all the half-space. The permittivity of the dielectric equals \(\varepsilon\). Find: (a) the surface density of the bound charges as a function of distance \(r\) from the point charge \(q\); analyse the obtained result at \(l \rightarrow 0\); (b) the total bound charge on the surface of the dielectric.
3.89 . (a) Since the normal component of the vector \(D\) is continuous at the dielectric interface, we obtain \(\sigma^{\prime}=-q l(\varepsilon-1) / 2 \pi r^{3}(\varepsilon+1),\) for \(l \rightarrow 0\) and \(\sigma^{\prime} \rightarrow 0\) (b) \(q^{\prime}=-q(\varepsilon-1) /(\varepsilon+1)\)