Maxwell’s Equations and the Wave Equation

Rhett Allain
11 min readApr 12
Photo: Rhett Allain. Maxwell’s Equations and the Wave Equation

There’s a bunch of math here, but it’s kind of a big deal. In short, it shows something really important — that light is an electromagnetic wave.

Well, let’s just get to it.

Maxwell’s Equations

I’m going to give my very brief review of Maxwell’s equations. Really, you spend one whole semester of physics just getting to this point — but we are just going to do it in a blog post.

If you need an “elevator pitch” for Maxwell’s equations — they are the four equations that show the nature of electric and magnetic field. This includes the “shape” of the fields and how to create them. Oh, I’m going to start off with integral versions of Maxwell’s Equations and then we can talk about the differential versions.

Gauss’s Law

Here is the version you see in your introductory physics course.

The left side of the equation is the electric flux over a closed surface. The flux is basically the component of the electric field that’s perpendicular to the surface integrated over the whole surface. In this equation, the integral with the circle means that it’s a “closed” integral over a surface area (we know it’s a surface integral because of dA (the area element). Finally, n-hat is the unit vector normal (perpendicular) to the area.

On the right hand side is the total (net) charge inside of the volume enclosed by that surface. The value of the constant (ε-0) is usually written as:

But what this law really says is that electric charges make electric fields. These electric fields point away from positive charges and towards negative charges.

Illustration: Rhett Allain. Electric fields due to positive and negative charges.

Gauss’s Law for Magnetism

Rhett Allain

Physics faculty, science blogger of all things geek. Technical Consultant for CBS MacGyver and MythBusters. WIRED blogger.