Electrostatic potential, electric potential difference

The electric potential or electrostatic potential is the electrostatic potential energy that a unit test charge would have in an electric field.

The study of the interaction between two charges can be approached in two different ways: electrostatic force or electric field.

With the first approach, the mutual force undergone by the charges is given by Coulomb’s law, and as this force is conservative, it is associated with a potential electrostatic energy.

With the second approach, an electric field is created by one of the charges called the source charge. The other charge, called the test charge, undergoes a force because it is located in a region of space where an electric field exists.

The electrostatic force is associated with a potential energy, and in the same way, an electrostatic potential V is associated with the electric field in such a way that the negative of the gradient of the electrostatic potential is the electric field:

And conversely from the electric field we can obtain the potential difference between two points A and B:

Point charges

Let’s consider a positive source charge q₁ at rest such as the one represented in the next figure:

If a 1 coulomb test charge is located at point P at a distance r from q₁, its electrostatic potential energy is given by:

This situation can be interpreted as a perturbation caused by the charge q₁ in its surroundings such as that a test charge located at point P of space would have an electrostatic potential energy. The electrostatic potential V is the quantity that describes this perturbation:

The previous expression is the electrostatic potential due to a point charge.

Where k is the Coulomb constant and its value in SI units in vacuum is:

Or, if we express it as a product of the vacuum permittivity or permittivity of free space ε₀, it is equal to:

The SI unit of the electric potential is the volt (V).

The electric potential due to a point charge can be deduced from the electric field due to the charge. The source charge q₁ and the points A and B between which we are going to calculate the potential difference are represented in the next figure.

As you can see the in the figure, the displacement vector dr is parallel to the unit vector  u_r in the radial direction, therefore it can be expressed as:

After substituting in the potential difference expression, we have:

And after integration we obtain:

The previous expression can be used only to calculate the potential difference between two points located at distances r_A and r_B from the source charge. However, if the zero of the potential is chosen at infinity r = ∞, we can define a value for the potential at a distance r from the source charge:

Therefore, the electrostatic potential at a distance r from the source charge is:

We can draw equipotential surfaces to represent the electric potential. An equipotential surface is a the region of space whose every point has the same potential.

In the case of point charges, the equipotential surfaces are concentric spheres all centered on the charge, because every point at a certain distance r from the source charge has the same potential.

When the charge is positive (in green in the upper figure), the potential decreases when we move away from the charge, because when r increases, V decreases. On the contrary, when the charge is negative (in blue in the figure), the potential decreases when we move closer to the charge. In both cases the potential decreases in the direction of the electric field lines.

As you can see in the previous figure, the equipotential surfaces can never intersect because if they would the potential would have two different values at the same point. Furthermore, the field lines are always perpendicular to the equipotential surfaces.

The electrostatic potential due to a set of N point charges at a point P of space is equal to the sum of the potentials due to each individual source point charge at this point:

Where each charge is used with its sign.

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