All Problems Conductors and Dielectrics in an Electric Field
Problem 3.61
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 the point \(O\); (b) as a function of a distance \(r\) from the point \(O\).
(a) \(\sigma=\frac{\lambda}{2 \pi l}\) (b) \(\sigma(r)=\frac{\lambda}{2 \pi \sqrt{l^{2}+r^{2}}}\).