Classical Electrodynamics - Jackson Problem 1.5.pdf

PHY 5346 HW Set 1 Solutions – Kimel 4. 1.5 φr⃗ = qe−αr 1 + αr2 4π 0r ∇2φr⃗ = − ρ 0 , ∇2 = 1 r2 ∂ ∂r r2 ∂ ∂r ∇2φ = q 4π 0 1 r2 ∂ ∂r r2 ∂ ∂r e−αr r + α 2 e−αr Using ∇2 1r = −4πδr⃗ ∇2φ = q 4π 0 1 r2 ∂ ∂r −αre−αr + e−αrr2 ∂ ∂r 1 r − α2 2 r2e−αr = q 4π 0 − α r2 e−αr + α 2e−αr r + αe −αr r2 − 4πδr⃗ − α 2e−αr r + α 3e−αr 2 = − 10 qδr⃗ − qα3 8π e−αr ρr⃗ = qδr⃗ − qα 3 8π e−αr That is, the charge distribution consists of a positive point charge at the origin, plus an exponentially decreasing negatively charged cloud.