Question

A spherical shell of radius R₁ with a uniform charge q has a point charge q₀ at its centre. Find the work performed by the electric forces during the shell expansion from radius R₁ to radius R₂.

Answer 1

Initially, energy of the system, U_i = W₁ + W₁₂ where, W₁ is the self energy and W₁₂ is the mutual energy.

Now, work done by the field force, A equals the decrement in the electrical energy, 

Alternate : The work of electric forces is equal to the decrease in electric energy of the system,

A = U_i - U_f 

In order to find the difference  U_i - U_f  we note that upon expansion of the shell, the electric field and hence the energy localized in it, changed only in the hatched spherical layer consequently (Fig.).

where E₁ and E₂ are the field intensities (in the hatched region at a distance r from the centre of the system) before and after the expansion of the shell. By using Gauss’ theorem, we find