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Is keeping $dV$ constant a necessary condition in the formula $$E=-\frac{dV}{dr}$$ for equipotential surfaces ?

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Equipotential literally means "the same potential". So if you pick any point on one equipotential surface and pick any point on a different equipotential surface, if you know the potential on each surface then you must know the potential difference between those two points.

The equation you give $\mathbf E=-\frac{\text dV}{\text dr}\hat r$ is true in general for potentials with only a radial component.

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Yes, because equipotential means that the potential is the same, so it's necessary to keep dV the same (constant) for both surfaces.

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