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.
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Solution
Initially, energy of the system, Ui=W1+W12 where, W1 is the self energy and W12 is the mutual energy. So, Ui=12q24πϵ0R1+qq04πϵ0R1 and on expansion, energy of the system, Uf=W′1+W′12 =12q24πϵ0R2+qq04πϵ0R2 Now, work done by the field force, A equals the decrement in the electrical energy, i.e. A=(Ui−Uf)=q(q0+q/2)4πϵ0(1R1−1R2) Alternate : The work of electric forces is equal to the decrease in electric energy of the system, A=Ui−Uf In order to find the difference Ui−Uf, 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.). Ui−Uf=∫R2R1ϵ02(E21−E22)⋅4πr2dr 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 E1=14πϵ0q+q0r2 and E2=14πϵ0q0r2 As a result of integration, we obtain A=q(q0+q/2)4πϵ0(1R1−1R2).