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Homework No. 05 (Fall 2014) PHYS 520A: Electromagnetic Theory I Due date: Thursday, 2014 Oct 16, 4.00pm 1. Consider a solid cylinder of radius R and infinite length with uniform permanent polarization P(r, t) = P0 θ(R − ρ), (1) where ρ2 = x2 + y 2 and P0 is perpendicular to the axis of the cylinder. We shall find the electric potential and the electric field outside the cylinder. (a) Show that the effective charge density is given by the expression ρeff (r) = −∇ · P = P0 · ρ̂ δ(ρ − R). (2) (b) Beginning from 1 φ(r) = 4πε0 Z d3 r ′ ρeff (r′ ) , |r − r′ | (3) after integrating by parts, and writing 1 P0 · ∇ φ(r) = − 4πε0 show that 1 φ(r) = 4πε0 Z Z θ(R − ρ′ ) dr , |r − r′ | 3 ′ d3 r ′ θ(R − ρ′ ) P0 · (r − r′ ) . |r − r′ |3 (4) (5) (c) Evaluate the integrals, z ′ , φ′ , and ρ′ , to show that the electric potential outside the cylinder is given by 2πR2 P0 · ρ φ(r) = . (6) 4πε0 ρ2 (d) Evaluate the gradient of the electric potential to show that the electric field outside the cylinder is given by i 2πR2 1 h 2(P0 · ρ̂)ρ̂ − P0 . E(r) = 4πε0 ρ2 1 (7)