A very long straight thread is oriented at right angles to an infinite conducting plane, its end is separated from the plane by a distance `l`. The thread carries a uniform charge of linear density `lambda`. Suppose the point `O` is the trace of the thread on the plane . FInd the surface density of the induced charge on the plane
(a) at point `O`.
(b) as a function of a distance `r` from the point `O`.

Explanation

At `O`,
`E_(n) (O) = 2 int_(l)^(oo) (lambda dx)/(4pi epsilon_(0) x^(2)) = (lambda)/(2pi epsilon_(0) l)`
So `sigma (O) = epsilon_(0) E_(n) = (lambda)/(2pi l)`
(b) `E_(n) (r ) = 2 int_(l)^(oo) (lambda . dx)/(4pi epsilon_(0) (x^(2) + r^(2))) (x)/((x^(2) + r^(2))^(1//2)) = (lambda)/(2pi epsilon_(0)) int_(l)^(oo) (x dx)/((x^(2) + r^(2))^(3//2))`
`(lambda)/(4pi epsilon_(0)) int_(l^(2) + r^(2))^(oo) (dy)/(y^(3//2)`, on putting `y = x^(2) + r^(2)`,
`= (lambda)/(2pi epsilon_(0) sqrt(l^(2) + r^(2)))`
Hence `sigma (r ) = epsilon_(0) E_(n) = (lambda)/(2pi sqrt(l^(2) + r^(2)))`