Classical Electrodynamics - Jackson Problem 2.9.pdf

PHY 5346 HW Set 4 Solutions – Kimel 2. 2.9 The system is pictured below a) We have treated this probem in class. We found the charge density induced was σ = 30E0 cosθ We also note the radial force/unit area outward from the surface is σ2/20. Thus the force on the right hand hemisphere is, using x = cosθ Fz = 1 20 ∫ σ2 ẑ ⋅ da⃗ = 1 20 30E0 22πR2 ∫ 0 1 x3dx = 1 20 30E0 22πR2/4 = 94 π 0E0 2R2 An equal force acting in the opposite direction would be required to keep the hemispheres from sparating. b) Now the charge density is σ = 30E0 cosθ + Q 4πR2 = 30E0x + Q 4πR2 = 30E0 x + Q 12π 0E0R2 Fz = 1 20 ∫ σ2 ẑ ⋅ da⃗ = 1 20 30E0 22πR2 ∫ 0 1 x x + Q 12π 0E0R2 2 dx Thus Fz = 94 π 0E0 2R2 + 1 2 QE0 + 1 320πR2 Q2 An equal force acting in the opposite direction would be required to keep the himispheres from separating.