Q.A spherical shell of radius b with charge Q is expanded to a radius a. Find the work done by the electrical forces in the process.

[!TLDR] Self-energy of a charged shell is U=Q^2/(8πε0R); work done by the electric forces as it expands from b to a eq …

The electrostatic self-energy (potential energy) of a uniformly charged spherical shell of radius R and charge Q is $U(R)=\tfrac{1}{2}QV_{surface}=\tfrac{1}{2}Q\times\dfrac{1}{4\pi\epsilon_0}\dfrac{Q}{R}=\dfrac{Q^2}{8\pi\epsilon_0R}$ (built the same way as the capacitor energy $U=\tfrac{1}{2}QV$ in section 8.12, applied to a self-charged sphere instead of a capacitor). As the shell EXPANDS from radius b to a LARGER radius a (a > b), its own charge Q spreads over a bigger surface, and its stored self-energy correspondingly FALLS, from $U(b)=\dfrac{Q^2}{8\pi\epsilon_0b}$ down to $U(a)=\dfrac{Q^2}{8\pi\epsilon_0a}$. Since the shell's own mutual repulsion is what drives this expansion (just like a charged balloon inflating under its own charge's repulsion), the …

Use the self-energy of a charged sphere U=Q^2/(8πε0R) at both radii, and equate the work don …

A common error is to instead use a two-point-charge PE formula (section 8.6.1), which does not apply here since this is a single continuous shell's o …