A metal sphere with a radius R is charged to a potential V . Let U₁ denote the electrostatic field energy stored inside a concentric sphere of radius 2R , and U₂ denote the electrostatic field energy stored outside this sphere of radius 2R . Determine the ratio U₂ U₁ .

Step-by-step solution

The electric field at a distance r from the center of the charged metal sphere ( r > R ) is given by E = 1 4 ₀ Q r^2 . The energy density of the electric field is u = 1 2 ₀ E^2 = Q^2 32 ^2 ₀ r^4 . The energy stored in a spherical shell of radius r and thickness dr is dU = u(4 r^2 dr) = Q^2 8 ₀ r^2 dr . Since the electric field inside the metal sphere ( r U₁ = _ R ^ 2R Q^2 8 ₀ r^2 dr = Q^2 8 ₀ [- 1 r ]_ R ^ 2R = Q^2 8 ₀ ( 1 R - 1 2R ) = Q^2 16 ₀ R The energy U₂ stored outside the sphere of radius 2R corresponds to t