Concept of Electric Field Lines

There is an interesting congruence between the mathematical structure of electric field and the geometrical behavior of lines emanating from a point-source. In this section, we will analyze this congruence and its implications in electrostatics. Let's start by considering a simple case of a point in three-dimensional space with \(N\) number of lines emanating radially outward from the point as shown in the figure below. The lines are distributed uniformly around the point. Now, let's draw consider a spherical shell of radius \(r\) centered at this point.

The number of lines per unit surface area of the sphere, or the number density \(\rho\) of lines, is:

\[ \begin{equation*} \rho = \frac{N}{4 \pi r^{2}} \end{equation*} \]

where \(4 \pi r^{2}\) is the surface area of the sphere. Thus, it follows that as the radius of the spherical shell increases, the density of lines (\(\rho\), lines per unit area of the shell) decreases as \(\frac{1}{r^{2}}\). This is because lines diverge as we move away from the source. The density of lines at an arbitrary point \(P\) on the sphere is given by equation (1). However, at point \(P\) on the spherical shell, the flux of lines is radially outward. This allows us to attach direction to density at any point on the sphere: \[ \begin{equation} \vec{\rho} = \frac{N}{4 \pi r^{2}} \hat{r} \end{equation} \] where \(\hat{r}\) is the unit vector along the radius vector \(\vec{r}\).

Now let's compare this system of lines with electric field from a point charge. Recall that electric field strength at a point \(\vec{r}\) due to a point charge \(Q\) is inversely proportional to the square of the distance \(r\) and is directed along \(\hat{r}\): \[ \begin{equation*} \vec{E} = \frac{1}{4 \pi \epsilon_{0}} \frac{Q}{r^{2}} \hat{r} \end{equation*} \] We can rearrange above equation as \[ \begin{equation} \vec{E} = \frac{1}{4 \pi r^{2}} \frac{Q}{\epsilon_{0}} \hat{r} \end{equation} \] Equations (1) and (2) differ in factors \(N\) and \(Q/\epsilon_{0}\) only. If we set \(N = Q/\epsilon_{0}\), the two equations become identical. \[ \begin{equation} \vec{\rho} = \frac{1}{4 \pi r^{2}} N \hat{r} = \frac{1}{4 \pi r^{2}} \frac{Q}{\epsilon_{0}} \hat{r} = \vec{E} \end{equation} \]

This is remarkable! We can use this \(\textit{mathematical}\) equivalence between electric field of a point charge \(Q\) and lines emanating from a point-source to geometrically model and visualize the behavior of electric field using lines. If we place the charge \(Q\) at the point-source of lines and fix the total number of lines \(N\) such that \(N=\frac{Q}{\epsilon_{0}}\), then the magnitude of electric field at an arbitrary point due to the charge \(Q\) is equal to the density of lines at that point, and the direction of the field is simply same as the direction of lines at that point. These lines, customized to represent electric field are referred to as \(\textit{electric field lines}\).

I must stress here that this is strictly a mathematical equivalence, not a physical one. No where in this discourse have I implied that there are really some sort lines emanating out of a charge into space. Because lines radiating from a point-source and electric fields happen to have identical mathematical structure, we can use one to accurately represent another without compromising mathematical integrity of either. Thus, electric field lines are entirely a mathematical construct, a tool used to facilitate geometric representation of electric fields. It would be erroneous to conclude that field line 'real' or tangible as a chair or desk is.

Using mathematical equivalence between two otherwise unrelated objects to facilitate complex calculations (or visualization, as in the present case) is a common practice in science. Unfortunately, it is also a common practice to not state explicitly the mathematical nature of such equivalences which, inadvertently, causes students to believe that there is some sort of physical connection between the objects (which in this case are lines and electric field). I would suggest students to always be critical of what they read and question every vague remark made by the author which requires you assume too much without justification. As a student of science, it is your right, and duty, to split hairs and bug you teacher/professor with questions until all the ambiguity is removed.

Coming back to electric field, the condition \(N=Q/\epsilon_{0}\) is of critical importance in electrostatics. It is the foundation of Gauss' law of electrostatics which is the subject of my next post.