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Reference: Introduction to ELectrodynamics (D.J Griffiths)

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Divergence of Electric field is zero iff the electric field is a pure faraday one?

What if it isn't?

Why does this condition have to be imposed on the field for its divergence to be zero?

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  • $\begingroup$ You do not have to impose it to get a divergence free field, but if you are sure that the source is a curl and only a curl you can be sure that it is divergence free. It is sufficient but not necessary. $\endgroup$ Apr 26, 2017 at 15:32

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Gauss's law says $$ \nabla \cdot \vec{E} = 4\pi\rho$$ where $\rho$ is the charge density (CGS units). If there are no charges around then $\rho = 0$ and you have a divergence-less electric field. That's the case for a pure Faraday field: it's produced by a changing $\vec{B}$ without any charges being anywhere in the vicinity. But it's not always true: if there are charges around then the divergence is not zero.

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