An infinitely long conducting cylinder (radius a) oriented along the z-axis is exposed to a uniform electric field E₀ ˆy.

(a) Consider the conformal map g(w) = w + a²/w, where g = u + iv and w = x + iy. Show that the circle |w| = a and the parts of the x-axis that lie outside the circle map onto the entire u-axis.

(b) Let the potential on the cylinder be zero. What is the potential on the x-axis? Use this potential and the mapping in part (a) to solve the corresponding electrostatic problem in the g-plane. Find a complex potential f (u, v) which satisfies the boundary conditions.

(c) Map the complex potential from part (b) back into the w-plane. Find the physical electrostatic potential ϕ(x, y) and the electric field E(x, y). Sketch the electric field and the equipotentials.