Discussion
A field is given as \( \mathbf{D} = k\mathbf{a}_r/r^2 \) in spherical coordinates. The charge density away from the origin is
Answer: B
Correct answer: zero, all the charge being at the origin
\[ \nabla\cdot\mathbf{D} = \frac{1}{r^2}\frac{d}{dr}\left(r^2 \cdot \frac{k}{r^2}\right) = \frac{1}{r^2}\frac{d}{dr}(k) = 0. \] This is the point charge field, and the divergence vanishes everywhere except at the origin, where the expression is undefined and all the charge in fact resides.
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