Question #43777

Students often incorrectly apply Gauss's Law by assuming that any spherical shell behaves exactly like a point charge for regions outside the shell ($r > R$).

This is only true for spherically symmetric charge distributions.

In this problem, the non-uniform charge density $\sigma(\theta) = \sigma_0(1 + \cos\theta)$ creates both a monopole and a dipole moment.

Ignoring the dipole contribution leads to the incorrect potential $V(z) = \frac{\sigma_0 R^2}{\epsilon_0 z}$, which yields the trap answer $\frac{16}{15} R$.

The question rigorously tests the student's ability to derive the potential of a non-uniform distribution via integration or multipole expansion, and subsequently apply conservation of energy to find the correct turning point.