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Is it possible for an electric field $\vec{E}$ to exist such that its electric field intensity increases continuously, something like $E=kx$, while all the $\vec{E}$ pointing in the same direction?

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Yes, that is possible, but only in a space filled with charge (of such a concentration as to produce the effect). Such a field has divergence (which means it cannot happen in empty space) but no curl (which would necessitate currents as well as charges).

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  • $\begingroup$ Interestingly the divergence of this field is $k$ which is the same as the divergence of a field $E = ky$. Meaning that they both give the same charge distributions. Presumably it is boundary conditions that establishes the difference in the two case, but just now I'm having trouble visualizing it. $\endgroup$ – dmckee Mar 11 at 17:44

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