What is the relationship between electric field strength and potential difference?

Question

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

Answer 1

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

Answer 2

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.