The electric field due to a unit electric dipole oriented in the \(\mathbf{k}\)-direction is \(\mathbf{E}=abla\left(z / r^{3}ight)\), where \(r=\) \(\left(x^{2}+y^{2}+z^{2}ight)^{1 / 2}\) (Figure 20). Let \(\mathbf{e}_{r}=r^{-1}\langle x, y, zangle\).
(a) Show that \(\mathbf{E}=r^{-3} \mathbf{k}-3 z r^{-4} \mathbf{e}_{r}\).
(b) Calculate the flux of \(\mathbf{E}\) through a sphere centered at the origin.
(c) Calculate \(\operatorname{div}(\mathbf{E})\).
(d) Can we use the Divergence Theorem to compute the flux of \(\mathbf{E}\) through a sphere centered at the origin?

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