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Confusion over Gauss' law for an ideal electric dipole
To find the charge density of an ideal electric dipole centred at the origin, I can evaluate the divergence of $\vec{E}(\vec{x})$ which equates to evaluating the laplacian of the potential.
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Would this be divergenceless?
If I were to put two charges of opposing polarity next to each other (a finite distance away but not superimposed on each oter) and would then take the divergence of the whole $E$-field, would that be ...