A straight-line charge with constant linear charge density ? is located perpendicular to the x-y plane in the first quadrant at (x₀, y₀). The intersecting planes x = 0, ? ? 0 and ? = 0, x ? 0 are conducting boundary surfaces held at zero potential. Consider the potential, fields, and surface charges in the first quadrant.

(a) The well-known potential for an isolated line charge at (x₀, y₀) is ?(?, ?) = (?/4??₀)ln(R²/r²), where ?² = (x ? x₀)² + (? ? y₀)² and R is a constant. Determine the expression for the potential of the line charge in the presence of the intersecting planes. Verify explicitly that the potential and the tangential electric field vanish on the boundary surfaces.

(b) Determine the surface charge density ? on the plane ? = 0, x ? 0. Plot ?/? versus x for (x₀ = 2, y₀ = 1), (x₀ = l, y₀ = 1), and (x₀ = l, y₀ = 2).

(c) Show that the total charge (per unit length in z) on the plane ? = 0, x ? 0 is

What is the total charge on the plane x = 0?

(d) Show that far from the origin [?>>?₀, where ? = ?(x² + y²) and ?₀ = ?(x²₀ + y²₀) the leading term in the potential is

Interpret.

Xo A tan Q. -1 Yo.