The x > 0 half of a conducting plane at z = 0 is held at zero potential. The x < 0 half of the plane is held at potential V . A tiny gap at x = 0 prevents electrical contact between the two halves.

(a) Use a change-of-scale argument to conclude that the z > 0 potential ϕ(ρ,φ) in plane polar coordinates cannot depend on the radial variable ρ.
(b) Find the electrostatic potential in the z > 0 half-space.
(c) Make a semi-quantitative sketch of the electric field lines and use words to describe the most important features.

P = V N Vo = 0 X