All Problems Conductors and Dielectrics in an Electric Field
Problem 3.60
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 modulus of the vector of the force acting on a unit length of the thread; (b) the distribution of surface charge density \(\sigma(x)\) over the plane, where \(x\) is the distance from the plane perpendicular to the conducting surface and passing through the thread.
(a) \(F_{1}=\frac{\lambda^{2}}{4 \pi \varepsilon_{0} l}\); (b) \(\sigma=\frac{l \lambda}{\pi\left(l^{2}+x^{2}\right)}\)