A point charge `q` is located at a distance `l` from an infinite conduting plane. Determine the surface density of charges induced on the plane as a function of separaction `r` from the base of the perpendicular drawn to the plane from the chagre.

Explanation

To find the surface charge density, we must know the electric field at a point `P` (fig), which is at a distance `r` from the point `O`.
Using the image mirror method, the field at `P`,
`E = 2E cso alpha = 2 (q)/(4pi epsilon_(0) x^(2)) (1)/(x) = (q l)/(2pi epsilon_(0) (l^(2) + r^(2))^(3//2))`
Now from Gauss theorem the surface charge density on conductor is connected with the electric field near its surface (in vacumm) through teh relation `sigma = epsilon_(0) E_(n)`, where `E_(n)` is the projection of `vec(E)` onto the outward normal `vec(n)` (with respect to the conductor).
As our field strength `vec(E) uarr darr vec(n)`, so
`sigma = - epsilon_(0) E = (ql)/(2pi (l^(2) + r^(2))^(3//2))`