Question #42789

This question tests the misapplication of the 'effective capacitance' shortcut and the superposition principle in infinite systems.

A common trap is assuming the concentric shells act purely as a series combination of capacitors with an equivalent capacitance $C_{eq} = (\sum C_n^{-1})^{-1} = 4\pi\epsilon_0 R$, leading to an incorrect energy of $U = \frac{1}{2} C_{eq} V_0^2 = 2\pi\epsilon_0 R V_0^2$.

This shortcut implicitly assumes the intermediate shells carry zero net charge, which would result in a natural potential decay of $V_n = V_0 2^{-n}$.

However, the forced potential decay $V_n = V_0 3^{-n}$ dictates that external work was done to place net charges on the intermediate shells.

The correct physical approach requires abandoning the global equivalent circuit and instead integrating the self-energy locally by summing the energy stored in the electric field of each individual spherical capacitor region.