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# Electric potential

## Definition of electric potential

The electric potential $(V)$ is a property of points in an electric field. The magnitude of the potential depends on how close the point is to the source of the field.

The electric potential is formally defined as the work done by a field in bringing a unit positive charge from infinity to a point.

This is equivalent to the work done by the field per unit charge (i.e. per coulomb) in moving a positive charge from a point in space where the electric field strength is zero to its current position.

The electric potential can also be defined as the electric potential energy per unit charge at a point. This is equivalent to the potential difference between a point at infinity (where the potential is zero) and a particular point in space.

The electric potential at $r=a$ is the work done in bringing a unit positive charge from infinity to $r=a$ (indicated by the area under the force-distance graph for a unit positive charge).

## Formula and units of electric potential

The electric potential $V$ at a point is given by: $$V=\frac{W}{q}$$ $W$ is the work done in bringing a charge $q$ to that point from infinity (or potential energy at that point). Electric potential is a scalar quantity.

The SI unit for potential is the volt $(\text{V})$. One volt is equivalent to one joule per coulomb (i.e. $1\text{ V}=1\text{ J C}^{-1}$).

## Potential difference

The difference in potential between two points in an electric field is called the potential difference. It is the work done when a unit charge (i.e. one coulomb) moves from one point to another in an electric field.

The work done $W$ in bringing a charge from one point to another in an electric field is: \begin{align*}W&=\Delta U\\&=Q\Delta V\\&=Q(V_{1}-V_{2})\end{align*} $\Delta U$ is the change in potential, $Q$ is the charge and $V_{1}$ and $V_{2}$ represent the potential at two different points.

The potential difference between $r=a$ and $r=b$ is the work done in bring a unit positive charge from $r=b$ to $r=a$ (indicated by the area under the force-distance graph for a unit positive charge).

## Equipotential lines

Equipotential lines link the points with the same potential in an electric field. These lines are always perpendicular to the direction of the electric field.

The equipotential lines of a point charge are concentric circles and the equipotential lines of a uniform electric field are equally spaced straight lines.

Recall that potential decreases linearly away from the source of a uniform force field, so equally spaced straight lines will show equipotentials with a constant difference.

Equipotential lines (red coloured) of a uniform (left) and a radial (right) electric field. The arrows represent the direction of the electric fields.

## Electric field strength and electric potential of a point charge

The electric field strength $E$ at a point is equal to the negative of the potential gradient (rate of change of potential with distance) at that point.

The electric potential $V$ at a distance $r$ from a point charge can be calculated using: $$V=\frac{Q}{4\pi\epsilon_{0}r}$$ The potential at infinity ($r=\infty$) is zero and the potential at the location of the charge ($r=0$) is infinite.

$Q=$charge; $\epsilon_{0}=$permittivity of free space.