## Electrostatic potential

Glossary Definition for 16-19

#### Description

The electrostatic potential at a point is *V* = *W**q*, where *W* is the work done in bringing a positive test charge *q* to that point from a defined position of zero potential.

Given a point, *P* in an electric field and a reference point A of zero potential

, the electrostatic potential at *P* is equivalent to the electrostatic potential difference between *A* and *P*.

If the field is due to an isolated charged object then one convention is to define the potential to be zero at an infinite distance from the object. If the charge of the object is positive, then the potential is positive everywhere in the field; if the charge is negative, the potential is negative everywhere in the field.

Electrostatic potential is usually represented by the symbol *V*

#### Discussion

The electrostatic potential at a point is often defined as the ratio between the work done in bringing a test charge from infinity to that point, and the charge itself. While this definition is not incorrect, there are two observations worth making. First, strictly speaking, only *differences* in electrostatic potential are meaningful, which means that, to define electrostatic potential, it is necessary to choose some point that acts as the zero of potential. By choosing infinity

as that point, we really mean a point that is a very long way from the source of the field we are considering, where no force is exerted by the field. This is only a convenience, however, and in some situations it is more convenient to choose a different point, such as one on the Earth’s surface.

The second observation is the use of test charge

in this definition (and in this glossary entry). It is assumed that the test charge is sufficiently small to have a negligible effect on the field through which it is moving. For the purposes of the definition we use

*V* = *W**q*

where *W* is the work done externally, i.e. by a force acting against the field, in bringing the test charge *q* to the specified point at a constant velocity.

Electrostatic potential is closely related to electrostatic potential energy. If a charge *q* is at a point that has electrostatic potential *V*, then the potential energy associated with the charge and whatever is giving rise to the potential is simply *q**V*.

#### SI unit

volt, V

#### Expressed in SI base units

A^{–1} kg m^{2} s^{–3}

#### Other commonly used unit(s)

J C^{-1}

#### Mathematical expressions

*V*=*Q*4π*ε*_{0}*r*

is the electrostatic potential a distance*r*from a point charge of magnitude*Q*, where*ε*_{0}is the permittivity of free space (8.85 × 10^{-12}F m^{-1}).*V*=*W**q*

is the electrostatic potential at a point, where*W*is the work done in bringing a test charge*q*to that point from a defined position of zero potential

#### Related entries

- Charge
- Electric field
- Potential difference, electrostatic
- Work

#### In context

In a hydrogen atom, the electron is on average at a distance 5.3 × 10^{-11} m from the proton. If we want to ionise the atom, that is remove a single electron, the work we have to do is equal to the electrostatic potential due to the proton, at this distance, multiplied by the electron’s charge. Expressed in terms of volts, the electrostatic potential is 13.6 V.