Equipotential Surface

Imagine you are at a campfire with your friends during winter. It's warm enough right next to the fire, however, the further away you sit, the less heat reaches you. So, you decided to make a seating arrangement near the fire. You make everyone sit at the same distance from the fire in a circle such that everyone feels the same temperature.

Fig. 1 - The figure shows people sitting in a circle around a campfire to feel the same warmth.

Now, let's replace the fire with a charge, then all the points on a circle around the charge are at the same potential. Instead of 2-dimensional space, let's raise the charge above the ground, such that all the points on the surface of a sphere around this charge are at the same distance from the charge. In other words, all the points on the sphere's surface are at the same electric potential. This surface is called an equipotential surface. In this article, we will discuss equipotential surfaces and their properties, an electric field in terms of the gradient of electric potential, work done on an equipotential surface, and how to draw an equipotential surface on a map of electric field lines.

Equipotential Surface Definition

First, let's define what exactly is an equipotential surface.

Before we can explain it any further, it is essential to know about electric potential and an electric potential difference.

Mathematically, electric potential \(V\) can be expressed as

\[V=\frac{U_{\mathrm{E}}}{q},\]

where \(U_{\mathrm{E}}\) is the electric potential energy, and \(q\) is the unit charge.

As electric potential is always defined in terms of some chosen reference point, the absolute value of electric potential is physically meaningless. What is important however is the change in potential when moving between two points. So if we consider a point charge moving between two points, we now consider the electric potential difference \(\Delta V\) :

\[\Delta V=\frac{\Delta U_{\mathrm{E}}}{q}\].

Keeping all of that in mind, we can now use electric field vector maps and equipotential lines to describe the field produced by charges, hence predicting the motion of charged objects within said electric field.

Equipotential Surface and Electric Field

Now, let's say a positive electric charge \(q_\mathrm{a}\) is moved from an initial point at an electric potential \(V_1\) to a final point at an electric potential \(V_2\) towards another positive charge \(q_\mathrm{b}\).

Fig. 2 - The movement of a positive charge \(q_\mathrm{a}\) towards another positive charge \(q_\mathrm{b}\) from point A to B against an electrostatic force of repulsion.

Let \(W\) be the amount of work done by an electrostatic force of repulsion in moving \(q_\mathrm{a}\) from B to A. Using all the parameters, the potential difference in the above case is,

\[V_2-V_1=-\frac{W}{q_a}\]

or

\[V_2-V_1=-\frac{F\,\left(r_1-r_2\right)}{q_a}\tag{1}\]

An electric field in terms of an electric force acting on a charge \(q_1\) is,

\[E=\frac{F}{q_1}\tag{2}\]

From equations (1) and (2),

\[\begin{align*}V_2-V_1&=-\frac{E\,q_\mathrm{a}\,\left(r_1-r_2\right)}{q_\mathrm{a}}\\V_2-V_1&=-E\,\left(r_1-r_2\right)\end{align*}\]

So, the change in electric potential between two points can be determined by integrating the dot product of the electric field with the displacement along the path connecting the points

\[\Delta V = V_\mathrm{2}-V_\mathrm{1}= -\int_{a}^{b} \vec{E}\cdot \mathrm{d}\vec{r}. \]

Electric field, on the other hand can be found using\[E=-\frac{V_2-V_1}{r_1-r_2}.\]

In three-dimensional motion of an electric charge, the above equation can be written as,

\[E=-\nabla V\tag{3}\]

Equipotential lines represent lines of equal electric potential. This field can be defined in any direction at a given location, for example, in the \(x\) direction it looks like

\[E_x=-\frac{\mathrm{d} V}{\mathrm{d}x}.\]

Examples of an Equipotential Surface

The most suitable example to understand equipotential surfaces is a charge distribution in a hollow charged conductor. Let's suppose we supply some charge to a conductor. The mobility of an electric charge is large inside a conductor. Due to an electrostatic force of repulsion, the electric charges distribute on the surface of a conductor.

Fig. 3 - The figure shows the electric charge distribution on the surface of a hollow charged conductor due to its high mobility and electrostatic force of repulsion.

According to Gauss's law, an electric field inside the hollow charged conductor is,

\[E=\frac{Q}{\epsilon_0}\]

where \(Q\) is the net charge enclosed in the conductor. The diagram shows that the net charge inside the conductor is zero due to the distribution of electric charges on the surface of a conductor.

Therefore, an electric field inside a conductor is,

\[E=0\,\mathrm{N\,C^{-1}}\tag{4}\]

From equations (3) and (4),

\[\nabla V=0\]

or

\[V=\mathrm{constant}\]

This indicates that the electric potential remains constant inside and on the conductor's surface.

The surface of a hollow charged conductor represents an equipotential surface. Similarly, outside the conductor, the surface of spheres of different radii around the conductors represents equipotential surfaces.

Fig. 4 - The figure shows the surface of spheres around a conductor as equipotential surfaces.

Electric Field Lines and Equipotential Surface

Electric field lines is one of the methods to represent the direction of an electric field graphically around an electric charge. The direction of these electric field lines is radially outward from a positive charge and radially inwards towards a negative charge. The graphical representation of these electric field lines around an electric charge is called an electric field map. The electric potential at a point around an electric charge in a free space/vacuum is

\[V=\frac{1}{4\pi\epsilon_0}\frac{q}{r^2},\]

where \(\epsilon_0\) is the electrical permittivity of free space, \(q\) is an electric charge around which an electric potential is determined, and \(r\) is a distance of test point (where the value of an electric potential is determined) from an electric charge.

The above equation shows that an electric potential varies with the distance. So, if we consider a sphere around an electric charge, then each point on the sphere is at the same distance from the electric charge. In other words, \(V\) is constant on the surface of the sphere. This surface is called an equipotential surface.

The graphical representation of this equipotential surface on an electric field map around a negative charge \(-Q\) is as follows.

Fig. 5 - The figure shows equipotential surfaces around negative charge on the electric field map.

The above diagram shows that the direction of an electric field is perpendicular to the equipotential surface.

Equipotential Surfaces Properties

In the sections above, we studied equipotential surfaces in detail. Let us further learn some main properties of these surfaces.

1. Two equipotential surfaces never intersect each other (otherwise, at the point of intersection, there will be two vectors perpendicular to them, so there will be two different electric fields, which is impossible).

2. The direction of an electric field is always perpendicular to an equipotential surface.

3. For a point charge, the equipotential surfaces are concentric spheres.

4. The surface of a hollow charged conductor is an equipotential surface.

5. The closeness of equipotential surfaces represents the greater strength of an electric field in the region between these surfaces.