How to visualize Maxwell’s Equations

Maxwell’s Equations are a set of four differential equations that govern electricity and magnetism. If, by some bizarre twist of fate you find yourself studying electromagnetism while reading this blog, then this post may or may not be helpful. Otherwise, it simply may not. Even if you are studying electromagnetism, Wikipedia and other places also present all this, usually more clearly. But somehow I never got it, despite reading Wikipedia, and Jackson, and that guy from Reed who writes the undergrad textbooks, whose name I forgot.

The study of electricity and magnetism (E&M) has always been the most difficult branch of basic physics for me, because fields, which are hard things to imagine, are so integral to it. Specifically, pretty much everything in E&M has to do with electric and magnetic fields. What are those? Electric fields are maps of what magnitude and direction of force a certain positive charge would feel if placed in that location. For example, like charges repel. So the field very close to a positive charge is strong, and pointed away from the positive charge. This is what electric field maps around a positive and negative charge look like. Magnetic Fields are a band that has a song namechecking Ferdinand de Saussure that I like. They’re also the analogue of electric fields, but for a magnetic North pole instead of a positive charge. You may have performed an experiment that shows what a magnetic field looks like in high school: cover a bar magnet with paper, and then drop iron filings on top. The filings align with the shape of the magnetic field around a magnet (except filings don’t have tiny arrowheads on them).

One of the most surprising things in all of physics is that these fields are not just constructs humans came up with to make understanding E&M easier, but real things. The four basic laws that govern these fields are called Maxwell’s Equations. Here they are:

1. Gauss’ Law

\nabla \cdot \mathbf{E} = \frac {\rho} {\varepsilon_0}

This equation states that the divergence of the electric field is equal to the charge density. Imagine a single positive charge sitting in space, and think what would happen to charges that come near. Above you saw a picture of the electric field around positive and negative charges. Here is the same picture as a 3-d contour map. This equation is simply the statement that “bumps” in the electric field contour map are where charges are. Or on the field map, the places where everything flows in is a negative charge, and where everything flows out is a positive charge.

2. Gauss’ Law for Magnetism

\nabla \cdot \mathbf{B} = 0

Gauss’ Law’s simple brother states that the divergence of the magnetic field is zero. Once again, this statement is just that bumps in the magnetic field map are where the magnetic charges are. But there is (as far as we know) no such thing as magnetic charges. Which means, there aren’t any bumps in the magnetic field contour map if you draw a map of “southness” or “northness”. Or on the field map, there are no places where everything flows in, and no places where everything flows out.

3. Faraday’s Law

\nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}} {\partial t}

The above equation states that the curl of the electric field is equal to the negative time derivative of the magnetic field. The electric field is the force a test charge would feel, and for ease of visualization, you can imagine all of these test charges arranged in a wire loop. If the magnetic field through the loop changes, there will be a curl to the electric field. That means the charges will feel a force pushing them to circulate around the loop. Thus this statement is saying that a changing magnetic field produces current.

4. Ampère-Maxwell Law

\nabla \times \mathbf{B} = \mu_0\left(\mathbf{J} + \varepsilon_0 \frac{\partial \mathbf{E}} {\partial t} \right)

The last of the Maxwell’s Equations has two parts. The first part is just the right hand rule. The second part is Maxwell’s addition – the curl of the magnetic field is proportional (with a proportionality factor set by the speed of light) to the time derivative of the electric field. The way to figure out what is going on in the first part is to imagine a wire and the magnetic field around it. If there is current flowing in the wire (is not 0), then the magnetic field lines will circle the wire, i.e. the magnetic field will have a curl. The second part just says that a changing electric field also causes the magnetic field lines to circle.

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I don’t know if these suggestions for visualization help you remember, or make sense of, Maxwell’s Equations. Please let me know if they do – or if they don’t. I know that understanding Maxwell’s Equations was profoundly easier for me once I learned to visualize them. Unfortunately I haven’t yet found a good way to articulate that idea verbally. As you saw above, I was not consistent in the four explanations – I am still trying out different approaches to see what works best. But this is something I’m very interested in, so if you are too, then drop me a line.

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