University of Illinois at Urbana-Champaign
The Grainger College of Engineering
Physics Van

Speed of Light From Maxwell's Equations

Most recent answer: 12/16/2015

Q:
Maxwell's equations lead him to postulate EM radiation. To calculate EM radiation speed he must've had a reference point from which to measure distance, time, and speed. That reference point I assume was the source of EM radiation, i.e., the oscillating magnet or charge, i.e., a physical device. As such, the speed c of EM radiation which he calculated was based on a physical object (EM source). If this physical object moved at speed v then the speed c must be reduced by v, i.e., c-v. However, in literature speed of EM radiation is accepted as constant regardless of the observer based on Maxwell's equations. And this result is without considering the experimental observations of Michelson-Morley or other one's of Einstein's thought experiments; purely on Maxwell's equations. In classical physics, any speed is reference-dependent. Where in Maxwell's equations speed of EM radiation becomes reference-free?
- Mehran (age 65)
Arlington Heights, IL
A:

EM radiation is implied by the four Maxwell's equations, which just describe how electric field and magnetic field should behave. In other words, Maxwell did not postulate the existence of EM wave when he was constructing the four equations. However, manipulating the equations directly results in a wave equation, which describes the propagation of EM wave. Let's see how the wave equation pops up from Maxwell's equations. 

If you consider the Maxwell's equations in vacuum, the curl of E is proportional to the time derivative of B and vice versa as shown below. (Sorry that I had to copy and paste the equations from the external source. I don't know how to directly type mathematical equations on this editor.)

\begin{align} \nabla \cdot \mathbf{E} &= 0 \ \nabla \times \mathbf{E} &= -\frac{\partial \mathbf{B}} {\partial t}\ \nabla \cdot \mathbf{B} &= 0 \ \nabla \times \mathbf{B} &= \mu_0 \varepsilon_0 \frac{ \partial \mathbf{E}} {\partial t}\\end{align}             

(from Wikipedia https://en.wikipedia.org/wiki/Electromagnetic_wave_equation)

Taking the curl to the curl of E and B gives rise to the second derivative of E and B with respect to time. 

\begin{align} \nabla \times \left(\nabla \times \mathbf{E} \right) &= -\frac{\partial}{\partial t} \nabla \times \mathbf{B} = -\mu_0 \varepsilon_0 \frac{\partial^2 \mathbf{E}}{\partial t^2} \\nabla \times \left(\nabla \times \mathbf{B} \right) &= \mu_0 \varepsilon_0 \frac{\partial}{\partial t} \nabla \times \mathbf{E} = -\mu_0 \varepsilon_0 \frac{\partial^2 \mathbf{B}}{\partial t^2} \end{align}   

(from Wikipedia https://en.wikipedia.org/wiki/Electromagnetic_wave_equation)

Using \nabla \times \left( \nabla \times \mathbf{V} \right) = \nabla \left( \nabla \cdot \mathbf{V} \right) - \nabla^2 \mathbf{V} and the fact that the divergences are zero, we get 

\begin{align} \frac{\partial^2 \mathbf{E}}{\partial t^2} - c_0^2 \cdot \nabla^2 \mathbf{E} &= 0\\frac{\partial^2 \mathbf{B}}{\partial t^2} - c_0^2 \cdot \nabla^2 \mathbf{B} &= 0 \end{align}  (<-wave equation!)

(from Wikipedia https://en.wikipedia.org/wiki/Electromagnetic_wave_equation)

c2 is 1/(mu*epsilon) where mu is the magnetic permeability and epsilon is the electric permittivity of free space. A wave equation is usually of the form that the second derivative with respect to time equals the velocity squared times the second derivative with respect to position (Laplacian in 3-dimension). The above final equations are of the exact form! So we derived wave equations from Maxwell's equations. The speed of this wave (c in above equation) is the square root of 1/(mu*epsilon). 

Now we have a set of equations for wave propagation with constant speed c. Maxwell's equations we started with do not say anything about the reference frame. (they were not even expected to describe a wave) Maxwell himself did not postulate that this speed should be constant in all inertial frames. He assumed just one reference frame. 

The speed of EM wave happened to coincide with the experimentally measured value of the speed of light. So physicists postulated that the EM wave should be light, and indeed it turned out to be so. However, physicists could not find any medium through which light propagates (Michelson-Morley experiment you mentioned), which was quite shocking because no wave propagation without media was known before. Einstein came up with special relativity to make sense out of this result, assuming that Maxwell's result was right. So he postulated that the speed of light is constant in all inertial frames and revised Newtonian mechanics to fit his new theory. Amazingly, special relativity looks right and so does this postulate of constant light speed until now. 

To sum up, a wave equation directly follows from Maxwell's equations and the implied EM wave has constant speed c. The four equations by Maxwell do not state anything about the reference frame. So we cannot get any information about the reference frame from Maxwell's equations. Einstein postulated that light (EM wave) has the same speed in all inertial frames and constructed special relativity, and the theory is working until now. 

I hope this helps!

SHC


(published on 12/16/2015)

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