# How does c pop out of Maxwell's equations

1. Sep 18, 2011

### YummyFur

Not knowing enough math to be able to understand the equations I nevertheless wonder how c is able to magically materialise. Is it because of other numbers that are inserted, numbers that are measured quantities, like the mass of an electron for example.

Also when the speed of an electromagnetic wave did pop out, would that not already imply that c was a constant and not relative? Otherwise how could c pop out and still be a relative number?

2. Sep 19, 2011

### Ken G

c is just a parameter of Maxwell's equations, no aspect of the theory tells us what c is except agreement with experiment. And you are quite right-- c being constant in Maxwell's equations means c would not be relative to the observer frame. But when Maxwell's equations first came out, the observations were not precise enough to be able to see distinctions between frames, so in effect all observations were in the same frame. It was assumed that was the frame of the "aether", so it was assumed that Maxwell's equations (like other wave equations such as for sound) would only hold in that one frame. The big surprise was that it was Newton's equations that only held in the rest frame of the object, whereas Maxwell's worked in all frames.

3. Sep 19, 2011

### YummyFur

I was under the impression that because the equations said that the Maxwell's electromagnetic waves travels at c, that led to the deduction that light must therefore be an electromagnetic wave. Why is it often said that the speed of light pops out of Maxwell's equations, in what sense does the speed of the EM wave 'pop out'.

4. Sep 19, 2011

### JeffKoch

I don't know about popping out, but the speed of an electromagnetic wave can be derived from Maxell's equations (http://en.wikipedia.org/wiki/Electromagnetic_wave_equation) and related to the values of the vacuum permittivity and permeability - experimental data can provide values for these constants in the appropriate units, and the result matches experimental data on the speed of light.

5. Sep 19, 2011

### vanhees71

The use of the SI is somewhat misleading here. The speed of light doesn't pop out of the equations, but is inherently part of them. The SI has the strange splitting of this inherent dependence in terms of $c=1/\sqrt{\mu_0 \epsilon_0}$ for practical reasons. It just gives nice numerical values to everyday currents used in electrical engineering by introducing a special unit for the electric current, the Ampere, in addition to the three mechanical base units (metre, kilogram, second).

The physics is somwhat different: Electromagnetism is a relativistic field theory with a massless spin-1 field (and thus necessarily a gauge field to avoid a continuum of spin-like intrinsic degrees of freedom). The empirical input is thus the masslessness of the em. field and thus the identity between the speed of light (i.e., the phase velocity of electromagnetic waves in the vacuum) and the limit velocity for signal propagation, which is a basic quantity for the Minkowski geometry of space-time. It's value is arbitrary and thus just defined to reproduce the old unit metre from the definition of the basic unit of time, the second. It's much more convenient to set $c=1$, and thus this is done in high-energy physics all the time.

6. Sep 19, 2011

Staff Emeritus
Does anyone else see a mismatch between those two excerpts?

YummyFur, what kind of answer are you looking for? If you don't understand the equations, what kind of answer about the features of the equations would satisfy you?

7. Sep 19, 2011

### YummyFur

You've put me on the spot a bit so I'll do my best to answer you.

I'm still digesting the above posts. It seems that I may have been misled into asking the question, and the above answers will necessarily force me to have a new look at it, which I will do. I've read many times over the years that c just 'popped out' of the equations and I felt it was time I understood how it happened because it must have been a surprise to Maxwell.

I wondered what numbers Maxwell started with whereby he describes EM waves and as a by product there is this speed of light constant. When I study science I try to picture how it would be for the people at the time who discover these things.

Now that I think about it more I guess that what I'm trying to do is imagine where Maxwell's mind was as he was manipulating some numbers and then see a connection to light as his EM waves have the same speed.

I don't know how to do differential calculus nevertheless after listening a dozen times to Feynman's lectures using the Stern Gerlach 'apparati' I am able to get a picture in my mind as to what is going on.

8. Sep 19, 2011

### ZapperZ

Staff Emeritus
You really need to know the mathematics to be able to understand how c "pops out" of the equations. This is because, by the nature of your own question, you are asking the mathematics of the derivation.

There are several resources you need to read. The first link gives you the general form of the differential equation for a wave:

http://mathworld.wolfram.com/WaveEquation.html

Note the form of the equation, and that the velocity of the wave is explicitly written in that equation.

Now, without going into what Maxwell equations are and how they arrived at the different wave equation for both E and B fields, look at the last 2 equations for E and B on this page:

http://planetmath.org/encyclopedia/DerivationOfWaveEquationFromMaxwellsEquations.html [Broken]

This has the SAME FORM as the wave equation, but this is for the propagation of E and B fields of the electromagnetic wave. The permittivity and permeability of free space ($\mu_0$ and $\epsilon_0$) is the coefficient in front of the time derivative. If you make the connection that

$$v = \frac{1}{\sqrt{\mu_0 \epsilon_0}}$$

then the value of c=v pops right out since the the value of permittivity and permeability of free space can be measured separately.

Zz.

Last edited by a moderator: May 5, 2017
9. Sep 19, 2011

### vanhees71

Now this I don't understand. How do you measure permittivity and permeability of free space? These quantities are "artifacts" of a somewhat unnatural (but useful for everyday life of electricians) choice of units.

In the SI of units, the basic electromagnetic unit is that of electric current. It's defined by the magnitude of the force, acting between two parallel straight infinitely long wires of 0 thickness at a distance of 1 m. The current through this wires is by definition 1 A (1 Ampere) if this force per unit length of the wires is $2 \cdot 10^{-7} \; \mathrm{N}$).

Since this force is, in the SI unit system, given by

$$F=\mu_0 \frac{I_1 I_2 l}{2 \pi r}$$

from the definition of the Ampere it follows

$$2 \cdot 10^{-7} \mathrm{N} = \frac{\mu_0}{2 \pi} {\mathrm{A}^2}$$

or

$$\mu_0=4 \pi 10^{-7} \frac{\mathrm{N}}{\mathrm{A}^2}.$$

Then one defines the unit of charge by: "1C is the amount of electric charge flowing through a unit surface within 1 second, if the particles run perpendicular to that unit surface and together make a current of 1 A."

The Coulomb force law then reads, in SI units, by definition

$$F=\frac{q_1 q_2}{4 \pi \epsilon_0 r^2}.$$

From this the Maxwell equations turn out to be consistent only, if

$$\epsilon_0=\frac{1}{c^2 \mu_0}.$$

Through high-precision measurements it turns out that, to an amazing accuracy, the velocity of light is identical with the upper boundary for signal velocities in relativisic space-time. Thus, nowadays within the SI the speed of light has a defined exact value, defining that light travels a certain distance within 1 second. In this way, the SI of units uses very fundamental properties of space time to define the unit of length (metre) from the unit of time by fixing arbitrarily the value of the fundamental velocity in Minkowski space. Since $\mu_0$ is as arbitrarily defined to set the unit of electric current to a certain value, also $\epsilon_0$ is arbitrarily set to the above given value. All these "fundamental constants" are thus just used to define arbitrary units for length, time, mass (inherent in the definition of the unit of force, Newton), and electric current.

From the point of view of fundamental laws of nature, the quantities $c$, $\epsilon_0$, and $\mu_0$ are just arbitrary settings for convenient use of men in everyday life. For fundamental physics, it's much better to set all of the to $1$ and just use one arbitrary unit of length (or time, mass, energy,...) to define everything else.

If you take into account also gravity, you can also eliminate this last arbitrariness and measure lengths in Planck units.