Electric Potential

Electric potential in a field

The electric potential of a point charge (q) in a field is proportional to the charge creating the potential, and inversely proportional to the permittivity and distance from the point charge. This is expressed mathematically in the equation below, where V is the electric potential in volts, Q is the point charge, r is the distance measured in metres and εo is the permittivity of a vacuum measured in Farad/metre, equal to 8.85 ⋅ 10^-12 F/m.

\[V = \frac{Q}{4 \pi \varepsilon_0 r}\]

From the equation, it can be summarised that for a positive charge q, the electric potential increases when the distance r decreases, as more work will be required to move the positive charge due to the repulsive force. Similarly, for a negative charge q, the distance from the charge decreases, as the positive test charge would move more easily due to the attractive force. This is illustrated below, where the interaction between a positive and a negative charge is shown.

What is electric potential energy?

Electric potential energy is the energy that is needed to move a charge (q) from one point to another in an electric field.

For example, in order to move a positive charge closer to another positive charge, work is needed to overcome the repulsive force. Similarly, when a positive charge is moved away from a negative charge, work is also needed to overcome the attractive force.

The energy transferred to the moving charge is called electric potential energy. The stronger the electric field, the larger the potential energy required to move the charge through the field.

Electric potential energy for two point charges

The electric potential of a pair of point charges is directly proportional to the magnitude of the product of the two charges as shown in the equation below.

Vacuum permittivity ε₀ is a constant that represents the tendency of the electric field to permeate in a vacuum. Its value is given below, and is measured in F/m.

\[V [V] = \frac{q_1 \cdot q_2}{4 \pi \varepsilon _0 \cdot r}\]

Change in electric potential energy can be found by utilising the respective distance of each unit charge.

\[V = \frac{q_1 \cdot q_2}{4 \pi \varepsilon _0 \cdot r} \cdot \Big( \frac{1}{r_1} - \frac{1}{r_2} \Big)\]

Electric potential and work

The electric potential can also be expressed mathematically in terms of work; remember, it is the work required to move a charge through an electric field. The work is equal to the product of the electric potential and the charge causing electric potential. This is shown below, where ΔV is the change in electric potential measured in volts, and Q is the charge measured in Coulombs.

\[W [J] = \Delta V Q\]

What is the electric potential gradient?

The electric potential gradient is the varying electric potential across an electric field. The electric field at any point is equal to the negative gradient of the potential distance at that point. The potential difference is shown below.

The gradient is defined by the equipotential lines which are represented in orange circular dotted lines and show the electric potential in an electric field. These are always perpendicular to the electric field lines which are illustrated with blue lines. The equipotential lines express the strength of the electric potential. The denser the equipotential lines the stronger the potential.

What is electric potential difference?

Electric potential difference is the work needed to move a charged particle in an electric field, from point A to point B. This is expressed in the equation below, where E is the electric field strength, V is the electric potential in volts, and r is the distance between the two points of interest in metres.

The negative sign shown below expresses the direction of the electric field. It is always outwards from the positive charge and inwards towards the negative charge (take a look at the first image).

\[E [N/C] = -\frac{\Delta V}{\Delta r}\]