A half-space filled with uniform istropic dielectrics with permittivity `epsilon` has the conducting boundary plane. Inside the dielectric at a distance `l` from this plane, there is a small metal ball possensing a charge `q`. Find the surface density of the bound chagres at the boundary plane as a function plane as a function of distance `r` from the ball.

Explanation

To calculate the electric field, first we note that an image charge will be needed to ensure that the electric field on the metal boundary is normal to the surface.
The image charge must have magnitude `- (q)/(epsilon)`
so that the tangential component of the electric field may vanish . Now,
`E_(n) = (1)/(4pi epsilon_(0)) ((q)/(epsilon r^(2))) 2 cso theta = (q l)/(2pi epsilon_(0) epsilon r^(3))`
Then `P_(n) = D_(n) - epsilon_(0) E_(n) = ((epsilon - 1) ql)/(2pi epsilon r^(3)) = sigma'`
This is the density of bound charge on teh surface.