A metal sphere with a radius R is charged to a potential V . Determine the electrostatic energy stored in the electric field inside a concentric sphere of radius 2R .

Step-by-step solution

The potential of the metal sphere is V = Q 4 ₀ R . The charge on the sphere is Q = 4 ₀ R V . The electric field at a distance r from the center ( r > R ) is E = Q 4 ₀ r^2 . Since the electric field inside the metal sphere ( r The energy density of the electric field is u = 1 2 ₀ E^2 . The electrostatic energy stored in the region between r = R and r = 2R is given by integrating the energy density over the volume: U = _ R ^ 2R u , dV = _ R ^ 2R 1 2 ₀ ( Q 4 ₀ r^2 )^2 4 r^2 , dr U = Q^2 8 ₀ _ R ^ 2R 1 r^2 , dr = Q^2 8