= Solution
Inside the <sphere>, the <electric field> has the form $\mathbf E=\nabla[-kr^2/(2a^3)]$, including at the origin. Thus it is a <conservative vector field>. Outside, use the <gradient of a radial function> to obtain
$$
\mathbf E=k\nabla\frac1r=-\frac{k\mathbf r}{r^3}.
$$
This is also an explicit <potential of a conservative vector field>, single-valued throughout the exterior region. Equivalently, the <curl> vanishes in both regions: for a radial coefficient $h(r)$, $\nabla\times(h\mathbf r)=\nabla h\times\mathbf r+h\nabla\times\mathbf r=0$ because $\nabla h$ is radial. The interior formula avoids any apparent $1/r$ issue at the origin. The vector-field potentials here have the opposite sign from the <electric potential> convention used in (b).
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