Electric potential (2.0) (AQA A-Level Physics): Revision Notes

Electric potential (2.0)

Definition and concept

Electric potential at a point is defined as the work done per unit charge when bringing a small positive test charge from infinity to that point.

When external work is done against the repulsive force to bring a small positive test charge from infinity towards an isolated positive charge, the system gains electric potential energy. Consequently:

• The electric potential around an isolated positive charge is always positive
• The electric potential around an isolated negative charge is always negative

For a negative charge, no external work is required to bring a positive test charge closer. The electric field accelerates the test charge towards the negative charge, resulting in a loss of electric potential energy.

Relationship between electric field strength and potential

The change in potential $\Delta V$ when moving a small distance $\Delta r$ in an electric field is related to the electric field strength $E$ by:

$\Delta V = E \Delta r$

Rearranging this expression gives:

$E = \frac{\Delta V}{\Delta r}$

This shows that the electric field strength equals the gradient of the potential with respect to distance. In other words, the field strength represents the rate of change of potential with position.

Electric potential of a point charge

The electric potential $V$ at distance $r$ from an isolated point charge $Q$ is given by:

$V = \frac{Q}{4\pi\varepsilon_0 r}$

where $\varepsilon_0$ is the permittivity of free space ($8.85 \times 10^{-12}$ F m$^{-1}$).

Superposition of potentials

Electric potential is a scalar quantity. When multiple charges are present, the total potential at any point is found by adding the individual potentials algebraically. Unlike electric field, which is a vector requiring component addition, potentials simply add or subtract according to their signs.

Equipotentials

An equipotential surface is a surface on which the electric potential has the same value at every point. No work is done when moving a charge along an equipotential surface, since there is no potential difference to overcome.

Properties of equipotentials

For a point charge:

• Equipotential surfaces are concentric spheres centred on the charge
• In two dimensions, these appear as concentric circles
• The potential difference between adjacent equipotentials is constant

For a uniform electric field (between parallel plates):

• Equipotential surfaces are equally spaced planes parallel to the plates
• The field strength $E = V/d$, where $V$ is the potential difference between plates and $d$ is their separation

Relationship with field lines

When both a positive and negative charge are present, their equipotentials and field lines interact. Equipotentials from opposite charges meet field lines perpendicularly, creating a more complex pattern than for a single charge.

Electric fields and potentials in conductors

Inside an isolated conductor, charge distributes on the surface such that:

• The electric field strength inside is zero
• The potential is constant throughout the entire conductor

Since there is no potential difference between any two points inside the conductor, no work is done moving charges within it. The surface of a conductor is therefore an equipotential surface.

Earth as a reference point

In practical electrical work, it is convenient to connect a point in a circuit to earth, giving that point a potential of 0 V. This is acceptable because measurements involve potential differences, not absolute potentials. The Earth acts as a large conductor and forms an equipotential surface that can reasonably be assigned the value zero.