Find the energy stored in the electirc field produced by a metal sphere of radius R containing a charge Q.

Explanation

Consider a thin sherical shell of radius x (gtR) and thickness dx concentric with the given metal sphere. The energy density in the shell is `
` u = (1)/(2) epsilon_0 E^2 = (1)/(2)epsilon_0 ((Q)/(4 pi epsilon_0 x^2) )^2`
`. The volume of the shell is `
`4pi x^2`
` dx. The energy contained in the shell is , therefor, `
`dU = (1)/(2)epsilon_0 ((Q)/(4pi epsilon_0 x^2))^2 X 4 pi x^2 dx = (Q^2 dx)/(8 pi epsilon_0 x^2)`
`. The energy contained in the whole space outside the sphere is `
` U int _R ^`
`00`
` (Q^2 dx)/(8pi epsilon_0 x^2) = (Q^2)/(8pi epsilon _0 R)`
`. As the field inside the sphere is zero, this is also the total energy stored in the field. Considering a concentric shperical shell at infinity, we have a spherical capacitor. The capacitance is `
` C = 4pi epsilon_0 R`
`. The energy stored in this capacitor is the energy stored in the entire electric field. This energy is `
` U = (Q^2)/(2 C) = (Q^2)/(8pi epsilon_0 R)`.