A thin infinitely long thread carrying a charge `lambda` per unit length is oriented parallel to the infinite conducting plane. The distance between the thread and the plane is equal to `l`. Find :
(a) the moducles of the vector of the force acting on a unit length of the thread ,
(b) the disribution of surface chagre density `sigma (x)` over the plane, where `x` is the distance form the plane perpendicular to the conducting surface and passing through the thread.

Explanation

(a) The force `F_(1)` on unit length of the thread is given by
`F_(1) = lambda E_(1)`
where `E_(1)` is the field at the thread due to image chagre :
`E_(1) = (-lambda)/(2pi epsilon_(0) (2l))`
Thus `F_(1) = (-lambda^(2))/(4pi epsilon_(0) l)`
minus signs means that the force is one of attraction.
(b) There is an image thread with charge density `-lambda` behind the conducting plane. We calculate the electric field on the counductor. It is
`E(x) = E_(n) (x) = (lambda l)/(pi epsilon_(0) (x^(2) + l^(2)))`
on counsidering the thread and its image.
Thus `sigma (x) = epsilon_(0) E_(n) = (lambda l)/(pi (x^(2) + l^(2)))`