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What is the relationship that connects potential difference between two points and the strength of the electric field between those two points?

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In the absence of time dependent magnetic fields $$ V({\bf r}_1)-V({\bf r}_2)= \int_{{\bf r}_1}^{{\bf r}_2} {\bf E}\cdot d{\bf r}. $$

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  • $\begingroup$ Would it to be correct to say then that if the voltage between two points is decreased, then the electric field in the region is smaller as well? $\endgroup$ – daljit97 Feb 15 '17 at 15:36
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To pick up on @mike stone, in general there is none.

For instance, in a region where the electric field is constant in magnitude and direction, the potential difference between two points is $V(\vec r_1)-V(r_2)=-\vec E\cdot (\vec r_1-\vec r_2)$. This can be $0$ if $\vec E$ is perpendicular to $\vec r_1-\vec r_2$.

More generally there is a range of possible values depending on the angle between $\vec E$ and the difference $\vec r_1- \vec r_2$. Note here that I have assumed above $\vec E$ is constant, meaning no difference in $\vec E$ between $\vec r_1$ and $\vec r_2$. This is enough to show there is no connection between potential difference and field difference in this special case.

This holds even more strongly when the field varies with position; the key observation is that the scalar product $\vec E\cdot d\vec \ell$ produces a geometrical factor that makes it impossible to link potential and fields differences in general.

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