Question #43267

This question tests the student's ability to apply the superposition principle to continuous non-uniform dipole distributions using bound currents.

A common lethal trap is treating the magnetization as uniform or ignoring the divergence/curl of the magnetization field.

Students often incorrectly assume that only the surface bound current $\vec{K}_b$ contributes to the field, leading to the erroneous answer $\frac{2}{3}\mu_0 M_0 \hat{k}$.

Alternatively, they might incorrectly assume $\vec{B} = \mu_0 \vec{M}$ locally, ignoring the non-local nature of the magnetic field generated by the entire volume's bound current $\vec{J}_b = \nabla \times \vec{M}$.

The rigorous solution requires resolving the non-uniform magnetization into both surface and volume bound currents, and integrating their respective contributions using the Biot-Savart law for spherical shells.