# Will electric field lines begin or end at a point other than a charge? Feynman Lectures Vol II Chapter 1

In The Feynman Lectures on Physics Vol. II Ch. 1: Electromagnetism, the following is stated:

There have been various inventions to help the mind visualize the behavior of fields. The most correct is also the most abstract: we simply consider the fields as mathematical functions of position and time. We can also attempt to get a mental picture of the field by drawing vectors at many points in space, each of which gives the field strength and direction at that point. Such a representation is shown in Fig. 1–1. We can go further, however, and draw lines which are everywhere tangent to the vectors—which, so to speak, follow the arrows and keep track of the direction of the field. When we do this we lose track of the lengths of the vectors, but we can keep track of the strength of the field by drawing the lines far apart when the field is weak and close together when it is strong. We adopt the convention that the number of lines per unit area at right angles to the lines is proportional to the field strength. This is, of course, only an approximation, and it will require, in general, that new lines sometimes start up in order to keep the number up to the strength of the field. The field of Fig. 1–1 is represented by field lines in Fig. 1–2.

That contradicts what I have read elsewhere. The model I originally learned is very adamant that electric field lines only originate or terminate at charges.

I guess if we are to broaden the scope of application of field lines to other situations, there may be situations, for example the flow of a wet fluid, where field lines might appear "out of nowhere".

So, I ask, in the mathematical sense, will electric field lines ever begin or terminate at points other than charges?

• When drawing in two dimensions, one would need some adjustments. – Pieter Apr 28 '18 at 21:40
• – Emilio Pisanty Apr 28 '18 at 22:46
• There are also electric field lines that form closed loops and therefore do not begin or end on electrical charges like electric fields produced by induction or in electromagnetic waves. – freecharly Apr 29 '18 at 2:22
• I guess it might be reasonable to add that field lines may also "terminate at infinity". At least for practical purposes. – Steven Thomas Hatton Apr 29 '18 at 13:16

Electrical field lines can only begin and terminate at charges.

This is because of Gauss's law, stating that

$$\Phi_{electric} = \frac{ Q_{enclosed} }{ \epsilon_{0} }$$

Here, $\Phi$ is the flux through a closed surface, and Q is the amount of charge enclosed by that surface.

So why would we have to add lines? It happens because we are drawing a 3 dimensional phenomenon in 2 dimensions.

Looking at Fig. 1.2 from the Feynman lecture, if I draw a closed surface (blue cylinder in the picture), the flux through it in the plane of the page is 0, because there is as many field lines entering it as there are exiting it.

However, if I draw another closed surface (red cylinder in the picture) there are more field lines exiting it then entering it in the plane of the page.

This could be explained for example by a positive charge just above the plane of the page (just above the red x). The positive charge will have electric field lines emanating from it. On the left border, this flux will cancel out with some of the incoming flux. On the right border, the flux from the charge will add to the other field lines. There will therefore be more field lines at the right border than the left border, since there is a stronger magnetic field there. This is represented by having a higher concentration of electric field lines. Now, if the charge is enclosed by the red surface, by Gauss's law there will be a net flux through the surface. This makes sense. If the charge is not enclosed by the red surface, all we know is that the net flux through the red surface is 0. We only see a part of the flux through the red cylinder in the depicted plane.

The image below shows how a charge outside the plane of the page will create an electric field line appearing "out of nowhere" on the page. The charge causes an electric field at the tip of the red arrow. This field has two components: one that is perpendicular to the plane (green) and therefore invisible in our plane, and one that is in our plane (black) and therefore visible in the diagram. So in conclusion, electric field lines only begin or terminate at charges, but if we want to depict a 3D field in 2 dimensions, we might not always be able to see the charge the field started from in the plane of the page.

• Apparently Feynman didn't (like to) think of field lines as "geometric forms"; that is, "mathematically real". Historically, the concept of a field line as representing some measurable physical quantity was essential to the development of electromagnetic theory. I'm not sure why Feynman was so eager to dismiss them as misleading. In general 3-D or 4-D structure will defy exact representation in a 2-D depiction, so the argument that field lines will appear out of nowhere when projecting a 3-D field onto a page 2-D page seems frivolous. I know he prefers EM in terms of vector potentials. – Steven Thomas Hatton Apr 29 '18 at 12:31

If there is a changing magnetic field, then there are electric field lines that are loops, in the sense that they don't begin or terminate at electric charges. So, one can have electric fields without electric charges. This is a "curly-electric field" associated with Faraday's Law.

If there are charges, then there are field lines which begin or terminate at electric charges. This is associated with Coulomb/Gauss's Law.

I think the strength of the electric field is better represented by "flux tubes" rather than the "density of field lines". (I feel uncomfortable with Fig. 1-2.)

@Pieter might be referring to issues discussed here https://engfac.cooper.edu/wolf/273
"Electric Field Line Diagrams Don't Work" Am. J. Phys, 64(6) , June 1996.

• I agree that induced field lines must be loops. But wouldn't that mean that induced fields do not have any termination points or origin points? – Martin J Apr 28 '18 at 23:50
• Yes, I would agree with that. – robphy Apr 29 '18 at 0:30
• Aren't field lines synonymous with the edges of the tubes of flux? See figure 4.5 on page 109: books.google.com/books?id=w4Gigq3tY1kC&lpg=PP1&pg=PA109 of Misner, Thorne and Wheeler. – Steven Thomas Hatton Apr 29 '18 at 12:52
• I think a better interpretation is that there is one field line (in the center) per tube of flux... with units of $Coulomb/m^2$. – robphy Apr 29 '18 at 19:04