Electric Field Lines

Electric Field Lines of Two Different Charges

The definitions and rules only help our understanding to an extent but visualizing the field lines would be much more helpful. A positively charged particle can often be represented as a point charge in free space. We could use the rules above to aid in constructing the field lines, as in the example below.

Note that spherical, charged objects can be treated as point charges with the charge concentrated at the center of the object.

Equipotential Lines and the Electric Field

The electric potential \(V\) due to a point charge at a distance \(r\) from it is given by \[V=\frac{1}{4\pi \varepsilon_0}\frac{q}{r},\] where the permittivity of free space \(\varepsilon_0=8.85\times 10^{-12}\,\mathrm{F\,m^{-1}}.\) The SI unit of measurement of potential is the \(\text{joule per coulomb,}\) \(\mathrm{J\,C^{-1}},\) which is equivalent to the \(\text{volt},\) \(\mathrm{V}.\) Conceptually, the potential is the work done per unit charge in the field. For a uniform electric field, the electric field lines will be parallel to each other and point in the same direction. This shows that the field strength is constant, and the direction is the same at any point in the region containing the field. That direction will be determined by the sign of the charge on the surface of the object generating the potential.

The equation for electric potential tells us that at different distances \(r\) from the surface containing the charge, there will be different potentials. However, along a line that is parallel to the surface, the potential will be constant, as all points on that line are equidistant from the surface. These lines of constant potential are called isolines, and for a uniform field, they appear as in Fig. 3 below.

Fig. 3 - The field lines for a uniform electric are parallel to each other. The isolines of equipotential are also parallel to each other but are perpendicular to the field lines at all points.

Note that the isolines are always perpendicular to the field lines. This is always necessary since any component of the electric field along the direction of an isoline will cause an electric force on a charge along that line. Work would be done along that isoline and potential would not remain constant, which cannot occur.

The scenario is different for a point charge. The field lines would be radial, but we would require that the isolines always be perpendicular to them. The isolines would therefore form concentric circles centered on the point charge \(q.\) Fig. 4 below shows the field lines and isolines due to a positive point charge.

Fig. 4 - The electric field lines of a positive charge point radially outward and the lines of equipotential are always perpendicular to them and so form concentric circles centered on the charge.

The circular isolines mean that the potential is constant along a circular path of radius \(r\) surrounding the point charge. If we think classically and assume that electrons orbit the nucleus of an atom in a circular path, this would be why the nucleus does not work on electrons. The magnitude of the average electric field is given by \[\left|\vec{E}\right|=\left|\frac{\Delta V}{\Delta r}\right|,\] in a region between two points separated by a distance \(\Delta r\) and having a potential difference \(\Delta V\) between them. The SI unit of measurement of the electric field strength is \(\text{volts per meter},\) \(\mathrm{V\,m^{-1}}.\)

Drawing Isolines for a Point Charge

There are a few things to consider when drawing an isoline of equipotential. Firstly, the isolines are circular rather than polygonal because there are many field lines not drawn in the diagram. The only way for the isolines to be perpendicular to them all is if they are circular. The following steps can be taken when asked to draw isolines:

• Consider the sign of the charge and draw the electric field lines, as in Fig. 5 below.

Fig. 5 - The first step in drawing isolines of equipotential is drawing the electric field lines which are radially outward for a positive charge.

• Construct tiny line segments that are perpendicular to all of the field lines and at equal distances from the charge, as in Fig. 6 below. Only a few a drawn but the picture can be completed by drawing the rest.

Fig. 6 - The second step in drawing isolines is to draw short line segments that are parallel to the field lines.

• Join the segments as smoothly as possible to create the electric lines of equipotential, as in Fig. 7 below.

Fig. 7 - The final step in drawing isolines is to join the segments together to form smooth curves. In the case of a positive point charge, this would result in concentric circles.

Note that there are many field lines, but the number is not definite; it is only used to compare the strengths of two or more fields. There are infinitely many isolines since there should be one for every value of the energy.

Differences between Electric Field and Gravitational Field

Electric fields are not the only type of field in physics, so it would be difficult to believe that electric field lines would be the only type of field lines. In fact, gravitational fields are quite similar to electric fields. The field lines are radial for point masses, and the equipotential lines are always perpendicular to the field lines. The equipotential lines are lines of constant potential energy per unit mass rather than per unit charge as in the case of electric fields.

On a large scale, we can consider the Earth to be a point mass with its mass being concentrated at its center (called the center of mass). The field lines, as viewed from afar, would be radially inward. Unlike charge, mass can only be positive, and so field lines can only ever point inward to represent the attractive force of gravity. The field lines represent the direction in which another mass would move when entering the field. Fig. 8 below shows the gravitational field lines and gravitational equipotential lines for an isolated mass \(m.\)

Fig. 8 - The gravitational field lines for a point mass point radially inward and the lines of equipotential form concentric circles centered on the mass.

The field lines and equipotential lines may look the same, and even Coulomb's law shares similarities with Newton's law of gravitation, \(F\propto \frac{1}{r^2},\) but there are many significant differences between the two fields. The table below describes some of the differences between the electric field and the gravitational field.

Table 1 - Differences between Electric Fields and Gravitational Fields

Field lines are used to represent lines of force for all types of fields and are not restricted to only electric and gravitational fields.

Electric Field Lines Example

Now that we have seen illustrations of the field lines and equipotential isolines for the electric field, we can test our knowledge on the following example.