User:Inconspicuum/Physics (A Level)/Electric Potential

Relationship to Electric Potential Energy

You will probably remember from AS (or even GCSE) that the energy U which flows along a wire is given by:

$U=Vq$,

where V is the potential difference between either end of the wire, and q is the amount of charge which flows. A simple rearrangement shows that:

$V={\frac {U}{q}}$

This potential difference is the same thing as electric potential. In a wire, the electric field is very simple. There are other electric fields, and in these fields as well, the electric potential is the electric potential energy per. unit charge. Electric potential energy between two point charges Q and q is given by:

$U={\frac {Qq}{4\pi \epsilon _{0}r}}$

So, the electric potential at a distance r from any point charge Q (ignoring other charges) is:

$V={\frac {Q}{4\pi \epsilon _{0}r}}$

Relationship to Electric Field Strength

Electric potential is also the integral of electric field strength. This is why it is often called potential difference - it is an integral between two limits (two points in space) with respect to distance. So, the potential difference between two points a and b is:

$V_{ab}=\int _{a}^{b}E\;dx=\int _{a}^{b}{\frac {Q}{4\pi \epsilon _{0}x^{2}}}\;dx=\left[-{\frac {Q}{4\pi \epsilon _{0}x}}\right]_{a}^{b}$

But, if we define b as infinity and a as r:

$V=\left[-{\frac {Q}{4\pi \epsilon _{0}x}}\right]_{r}^{\infty }=-{\frac {Q}{4\pi \epsilon _{0}\infty }}-\left(-{\frac {Q}{4\pi \epsilon _{0}r}}\right)={\frac {Q}{4\pi \epsilon _{0}r}}$

So, the area under a graph of electric field strength against distance, between two points, is the potential difference between those two points.

For a uniform electric field, E is constant, so:

$V=\int _{r}^{\infty }E\;dx=Er$

In other words, V is proportional to r. If we double the distance between us and a point, the potential difference between us and that point will also double in a uniform electric field.

Equipotentials

Equipotentials are a bit like contours on a map. Contours are lines which join up all the points which have the same height. Equipotentials join up all the points which have the same electric potential. They always run perpendicular to electric field lines. As the field lines get closer together, the equipotentials get closer together.