# The Unapologetic Mathematician

## The Propagation Velocity of Electromagnetic Waves

Now we’ve derived the wave equation from Maxwell’s equations, and we have worked out the plane-wave solutions. But there’s more to Maxwell’s equations than just the wave equation. Still, let’s take some plane-waves and see what we get.

First and foremost, what’s the propagation velocity of our plane-wave solutions? Well, it’s $c$ for the generic wave equation

$\displaystyle\frac{\partial^2F}{\partial t^2}-c^2\nabla^2F=0$

while our electromagnetic wave equation is

\displaystyle\begin{aligned}\frac{\partial^2E}{\partial t^2}-\frac{1}{\epsilon_0\mu_0}\nabla^2E&=0\\\frac{\partial^2B}{\partial t^2}-\frac{1}{\epsilon_0\mu_0}\nabla^2B&=0\end{aligned}

so we find the propagation velocity of waves in both electric and magnetic fields is

$\displaystyle c=\frac{1}{\sqrt{\epsilon_0\mu_0}}$

Hm.

Conveniently, I already gave values for both $\epsilon_0$ and $\mu_0$:

\displaystyle\begin{aligned}\epsilon_0&=8.85418782\times10^{-12}\frac{\mathrm{F}}{\mathrm{m}}&=8.85418782\times10^{-12}\frac{\mathrm{s}^2\cdot\mathrm{C}^2}{\mathrm{m}^3\cdot\mathrm{kg}}\\\mu_0&=1.2566370614\times10^{-6}\frac{\mathrm{H}}{\mathrm{m}}&=1.2566370614\times10^{-6}\frac{\mathrm{m}\cdot\mathrm{kg}}{\mathrm{C}^2}\end{aligned}

Multiplying, we find:

$\displaystyle\epsilon_0\mu_0=8.85418782\times1.2566370614\times10^{-18}\frac{\mathrm{s}^2}{\mathrm{m}^2}=11.1265006\times10^{-18}\frac{\mathrm{s}^2}{\mathrm{m}^2}$

which means that

$\displaystyle c=\frac{1}{\sqrt{\epsilon_0\mu_0}}=0.299792457\times10^9\frac{\mathrm{m}}{\mathrm{s}}=299\,792\,457\frac{\mathrm{m}}{\mathrm{s}}$

And this is a number which should look very familiar: it’s the speed of light. In an 1864 paper, Maxwell himself noted:

The agreement of the results seems to show that light and magnetism are affections of the same substance, and that light is an electromagnetic disturbance propagated through the field according to electromagnetic laws.

Indeed, this supposition has been borne out in experiment after experiment over the last century and a half: light is an electromagnetic wave.

February 9, 2012 -

1. You have done a remarkable work so far to expound the principles of electrostatics. I have to re-read all the posts and absorb.
Here you have found the speed of light. That is based on two other constants. I have to read the older posts to see if those constants were calculated from from principles. Otherwise, there is some circularity here.
I am still not sure whether you proved in all these posts Maxwell’s equations from vector calculus alone (generalized Stokes’ theorem). If that is the case, it proves the power of vector calculus.
No matter the questions that linger in mind, what you done relating calculus and electrostatics is truly wonderful. Congratulations! I hope you continue this effort.

Comment by Soma Murthy | February 9, 2012 | Reply

2. I didn’t get into how $\epsilon_0$ and $\mu_0$ were calculated, but in fact they are determined from laboratory experiments which are specifically concerned with electric or magnetic phenomena, and not with light as such.

Comment by John Armstrong | February 9, 2012 | Reply

3. […] The Propagation Velocity of Electromagnetic Waves (unapologetic.wordpress.com) […]

Pingback by Magnetic fields, again | cartesian product | February 26, 2012 | Reply

4. I need a formula to calculate the question below or the answer.
A 5 GHz plane electromagnetic wave is propagating in a polystyrene block(E=2.5). Find the velocity of propagation. Take E0=8.85×10^-12
U0=4pix10^-7

Comment by peter | February 21, 2014 | Reply