# 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.

## 3 Comments	»

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 | February 9, 2012 | Reply

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

Pingback by | February 26, 2012 | Reply

« Previous | Next »

## About this weblog

This is mainly an expository blath, with occasional high-level excursions, humorous observations, rants, and musings. The main-line exposition should be accessible to the “Generally Interested Lay Audience”, as long as you trace the links back towards the basics. Check the sidebar for specific topics (under “Categories”).

I’m in the process of tweaking some aspects of the site to make it easier to refer back to older topics, so try to make the best of it for now.

• ## Feedback

Got something to say? Anonymous questions, comments, and suggestions at Formspring.me!
%d bloggers like this:
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 11, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "github"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.9311769008636475, "perplexity_flag": "middle"}