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Suppose a current density ${\bf J}({\bf r},t)$, for which $\nabla \cdot {\bf J} =0$, is compactly supported on a 3D region $R_1$ in vacuum. In general it can produce a nonzero electric field ${\bf E}({\bf r},t)$ in another 3D region $R_2$ which does not intersect $R_1$. Can a ${\bf E}({\bf r},t)$ be conservative (ie $\nabla \times {\bf E} =0$) everywhere in $R_2$? Of course, by Faraday's Law, this would require that $\partial {\bf B}/\partial t =0$ everywhere in $R_2$.

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  • $\begingroup$ Just wondering what the motivation for specifying $\vec{J}$ be divergence free is? $\endgroup$ – jacob1729 Apr 17 at 0:03
  • $\begingroup$ Initially I thought the divergence constraint might change the answer but now I think it does not. $\endgroup$ – D_J_S Apr 17 at 20:29

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