What’s so Special about Maxwell’s Equations?
By: Colton Kawamura
Tags: STEM, Physics, Education, Mathematics, Science

Maxwell’s equations are some of the hallmark equations of physics culture, but what makes these equations so special that people are willing to get tattoos of them?

A Quick Summary

Maxwell’s equations are typically introduced in college-level electrodynamics courses. One thing undergraduate physics students quickly discover is that you can start almost any problem in those courses with at least one of the four equations. So, unless you’re already very familiar with them, I think it’s important to have an idea of what these four short, but potent equations do:

Gauss’ Law for Electricity: Captures the concept that charges are sources of electric fields. The concentration of the electric field passing through a closed area (flux) is proportional to the charge density contained within that closed area with a proportionality constant.

Gauss’ Law for Electricity

Gauss’ Law for Magnetism: States that all magnetic field lines from a source return to the source.

Gauss’ Law for Magnetism

Maxwell- Faraday’s Law: Mathematically describes the phenomena of the induction of an electrical current from a changing magnetic field or vice versa: how a circulating electrical current can induce a change in the magnetic field with time.

Maxwell- Faraday’s Law

Ampere’s Law: A circulating magnetic field is created around electric currents and changing electrical fields.

Ampere’s Law

What Makes Them Special

Merging of Fields. You might notice that each one of Maxwell’s equations is also another physicist’s law. That’s because Maxwell took other existing independent laws and modified them in a way that allowed them to be coupled in the set we see today. In manipulating the equations, he was able to discover that these electromagnetic fields are described by the same equations for waves. When he used these relations to calculate the speed of the electromagnetic waves in a vacuum, he found it was equal to the speed of light — it is an electromagnetic wave. In doing so, Maxwell unified theories of light, magnetism, electricity into a single field: electromagnetism.

Elegancy. At first glance, they look like four relatively short equations. But, despite their aesthetically minimal appearance, these four coupled equations describe how electromagnetic radiation behaves through a given medium at all times. The fact that so much information is packed into four very short lines showcases the elegance of nature and the language of mathematics. But to give credit where it’s due, it was physicist Oliver Heaviside that condensed Maxwell’s original twenty equations into four equations in vector notation.

Maxwell’s original equations in his own notation — thank you Heaviside!

Impact. Having the recipe for electromagnetic radiation is really useful. Virtually all of our technologies communicate through electromagnetic radiation using these principles. Furthermore, if you combine Maxwell’s equations with the Lorentz force law for given conditions, you know everything that a given electromagnetic system will do as it responds to its environment. This is perfect for how determining how electromagnetic radiation will affect a chemical or biological system for medical purposes.

In addition to the in-your-face applications we see every day, Maxwell’s work was a source of inspiration for Albert Einstein to develop the theory of relativity. This influence was due to the equations’ Lorentz invariance — these relations hold for all observers traveling at any speed. What’s more, these Maxwell’s equations lack any evidence of a “luminiferous aether” which was, at the time, believed to facilitate the propagation of light.

The Legacy

Maxwell’s equations are, without a doubt, some of the most iconic and useful relations ever developed in physics. They represent a turning point in physics away from old concepts and towards a new frontier of our understanding of the universe. But are they worth a tattoo? Maybe not, but perhaps a shirt will do.