Equipotential Lines

Equipotential Lines Rules

Given a set of equipotential lines for a field, there are a number of properties of that field that can be ascertained by analyzing the equipotential lines. For example, consider the definition of the electric field as the negative gradient of the potential

\[E(x)=-\frac{\mathrm{d}V(x)}{\mathrm{d}x}.\]

As the potential is constant everywhere along an equipotential line, \(\frac{\mathrm{d}V(x)}{\mathrm{d}x}=0\) everywhere along the line, implying that there is no component of the electric field along an equipotential line. This can only be the case if the equipotential lines are perpendicular to the electric field lines at all points. Knowing this allows us to construct the field lines if given the equipotential lines and vice versa, as we shall when taking a closer look at some examples later on.

This relationship between field lines and equipotential lines gives us a definition for electric field lines in terms of potential. The electric field lines are always directed such that they are perpendicular to the equipotential lines and point in the direction of reducing potential. We can also say something about the magnitude of an electric field based solely on its equipotential lines. Each equipotential indicates a value of the potential in a region, so the separation of equipotential lines is a visual representation of the gradient of the potential. The closer the equipotential lines are to each other, the higher the gradient will be. As the magnitude of the electric field is equivalent to the gradient of the potential, we see that the separation of equipotential lines indicates the strength of the electric field in a region.

Let's use these rules that we have laid out, to analyze the equipotential line of a few key electric fields.

Equipotential Lines for a Single Positive Charge

The fundamental example of an electric field is that of a single charge, in this case, we will consider a positive charge. The electric field strength of a field emanating from a single charge \(q\) follows Coulomb's Law\[E(r)=\frac{kQ}{r^2},\]

where \(k=9\times10^9\,\mathrm{N}\,\mathrm{m}^2\,\mathrm{C}^{-1}\) is a fundamental constant known as Coulomb's constant. Coulomb's law tells us that the electric field around a point charge varies with the radial distance from the charge. The electric field lines of a radial field, as seen in Figure 2, extend out from a single point, and are much closer together at small distances from the charge and much less dense further away. This shows how the strength of the field decreases away from the charge.

Fig. 2 - A single positive charge produces field lines that extend radially outwards from the source charge. Note that the field lines are closer together nearer the charge, as the field is the strongest and closest to the source.

We can construct the equipotential lines for this field by using the rules we laid out in the previous section. As the equipotential lines must be perpendicular to the field lines everywhere, they must be circles of a fixed radius, with this radius increasing the further one gets from the charge. This can be seen as a consequence of the radial symmetry of the field, points an equal radial distance from the charge have the same field and potential.

These equipotential lines can be seen in Figure 3.

Fig. 3 - The equipotential lines, or isolines, of a radial electric field, are concentric circles of increasing radii. These circles are perpendicular everywhere to the field lines.

This can be seen mathematically by integrating the electric field to find\[\begin{align}V&=-\int \frac{kQ}{r^2}\mathrm{d}r\\&=\frac{kQ}{r}+C.\end{align}\]

where \(C\) is just the choice of reference potential and can be taken to be zero. So the potential also varies radially, and so equipotential lines are paths of a fixed radius from the charge, as we saw in figure 3.

Equipotential Lines of a Dipole

An electric dipole refers to a system of two opposite charges separated by some distance. For example, a simple hydrogen atom is an electric dipole consisting of a negatively charged electron orbiting a positively charged proton. The force between a dipole is again defined by Coulomb's Law as\[F=\frac{kq_1q_2}{r^2}.\]

However, we can also consider the effect of this dipole on a third test charge by establishing the field lines around a dipole. Note that the electric field is an additive quantity. This means that the total electric field of a dipole is given by the sum of the electric fields for each charge. Using what we know about single charges from the previous section, the electric field around a dipole is

Fig. 4 - Two charges \(q_1,q_2\) combine to form an electric dipole. The force between them is determined by the distance \(r\) whilst the electric field around them is determined by both \(r_1,r_2\).

\[E(r_1,r_2)=\frac{kq_1}{r_1^2}+\frac{kq_2}{r_2^2}.\]

Where \(r_1\) is the distance from \(q_1\) and \(r_2\) is the distance from \(q_2\). If we consider the simple example of two charges of equal magnitude \(q\) but opposite signs, like the hydrogen atom, then this simplifies to\[E(r_1,r_2)=kq\left(\frac{1}{r_1^2}-\frac{1}{r_2^2}\right). \]

This electric field produces the field lines shown in figure 5.

Fig. 5 - The electric field lines around a dipole are shown in black, whilst the equipotential lines are shown in green.

Note how the electric field lines are no longer radially symmetric but are in a closed loop such that every field line begins at the positive charge and ends at the negative charge. In this way, the positive charge is a source of field lines, and the negative charge is a sink.

The equipotential lines of the dipole can also be seen in figure 5, shown in green. This time, they are elliptical paths around each charge, no longer circular due to the influence of the other charge. There is also an equipotential line of zero charges along the line \(r_1=r_2\) as we can see from the fact that the potential follows the equation\[V(r_1,r_2)=kq\left(\frac{1}{r_1}-\frac{1}{r_2}\right).\]

Equipotential Lines for Parallel Plates

For the final electric field example, we'll look at the equipotential lines between parallel plates. The field between parallel plates is quite different from that of a single charge or dipole as it is a Uniform Electric Field.

The fact that electric field strength is constant everywhere is represented by field lines that are parallel to one another and evenly spaced. The reason for this uniformity can be seen in figure 6. By considering the parallel plates to be formed of a row of individual point charges, it follows that any vertical components of field lines, i.e., pointing parallel to the surface of the plate, will be canceled out by the field lines of the neighboring point charges. Hence all that is left is a set of parallel, evenly spaced field lines from the positive plate to the other.

Fig. 6 - The electric field between parallel plates is made up of evenly spaced parallel field lines, as can be seen by considering the plates as a column of point charges.

The equipotential lines in a uniform field are particularly simple to find, as given that they must be perpendicular to the field lines, it follows that they are also a set of mutually parallel lines equivalent to the field lines but rotated by \(90^{\circ}\). As the electric field strength is constant so too is the gradient of the potential, hence the separation of the equipotential lines is constant. Using these properties, we can easily construct the equipotential lines, as shown in figure 7.

Fig. 7 - The equipotential lines in a uniform field are evenly spaced parallel lines, much like the field lines but simply rotated by \(90^{\circ}\,\mathrm{deg}\).