There are two point charges on the $x$ axis and $x'$ is a place where the potential, relative to infinity, is the biggest. Why is the electric field zero at this point?
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$\begingroup$ The only point with a zero field in your charge distribution is the middle point between them, and at this point the potential is minimum on the $x$ axis, not maximum. So your statement is wrong. $\endgroup$– MostafaOct 26, 2014 at 22:49
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$\begingroup$ Desperado, the strength of the electric field is proportional to the how quickly the electric potential changes with position (the slope of the potential). So, for example, one can have a constant potential and zero electric field. Now, at the top of a hill or the bottom of a valley, what is the slope? $\endgroup$– Alfred CentauriOct 26, 2014 at 23:57
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$\begingroup$ I do not see how your statement is correct. The closer you get to any of the point particles the biggest the potential, and the strongest the electric field $\endgroup$– user65081Oct 27, 2014 at 0:18
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The statement that $V$ is at an extremum can be written $\partial_i V=0$. Now think about the mathematical relation between $V$ and $E$ in electrostatics.