How is the speed of light derived in Maxwell's equations?

[geordief]

I know we end up with
$c=\sqrt{\frac{1}{μ _0.ε _0}}$

The reason I would like a bit of help is that I understand that the value of c as deduced from Maxwell's equations is independent of any frame of reference.

I can see that this is the case from the above equation involving the permittivity and the permeability of the vacuum but how is this formula arrived at and where in the mathematics does it become apparent that a value for the speed of the radiation is being used without any reference to any particular reference frame (and so invariant,if I understand correctly)?

I am anticipating that the mathematics may be hard for me to attempt to undertake but would like to be shown what this mathematics looks like and if there is perhaps any discussion of it elsewhere that might help me understand it in as detailed a way as possible (I do have some understanding of Calculus and differential equations ...)

 

[Heikki Tuuri]

The invariance of c is an empirical observation which was made in the late 19th century. Maxwell did not know if light propagates in a medium, the aether, or not.

Special relativity is the theory where c is independent of the reference frame.

 

[Vanadium 50]

how is this formula arrived at

Start with $\nabla \times \nabla \times E$

 

[geordief]

The invariance of c is an empirical observation which was made in the late 19th century. Maxwell did not know if light propagates in a medium, the aether, or not.

Special relativity is the theory where c is independent of the reference frame.

I am simply referring to the value c as used in Maxwell's equations(making no connection with c as it was used later in Special Relativity)

C ,in Maxwell's equations describes the speed of em radiation (I believe) and I have been told that it is shown in his calculations as not depending on any frame of reference,

I would like to look at the maths behind that and would like to know at what point in his maths it becomes apparent that no frame of reference is used but a speed is calculated nonetheless.

 

[geordief]

It seems to me that ,prior to this formula equating Maxwell's c with the square root of the inverse of the product of the permittivity and the permeability of the vacuum that any speed would have implied a frame of reference,

Here ,it seems that was not the case.

I am wondering how this absence of a frame of reference got ,so to speak embedded in his equations.

 

[Ibix]

I'm not quite sure of the point you are trying to ask about, so I don't know if the following answers you.

The problem with light in Maxwell's equations is that if you find a traveling wave solution (i.e., light) and Galilean transform it, the result is not a traveling wave (trivially, since the electromagnetic wave equation gives a speed of $c$ and no finite velocity is invariant under Galilean transforms so the transformed wave cannot travel at $c$). The traveling wave solution remains a traveling wave under Lorentz transforms. Thus Maxwell's equations imply the Lorentz transforms.

 

[Heikki Tuuri]

I am wondering how this absence of a frame of reference got ,so to speak embedded in his equations.

If Maxwell would have known that the permittivity and the permeability of vacuum are independent of the reference frame, then he would have known that c is independent of the reference frame.

But the aether hypothesis was well and alive in 1862. Maxwell did not know if they are independent of the reference frame.

Michelson and Morley in 1887 measured c to be constant.

 

[vanhees71]

The math is not that hard, but you need standard vector calculus, but only div, grad, and curl, no integrals :-)).

Start with the Maxwell equations in a vacuum with no currents and charges (i.e., free em. fields). I use SI units. Then the Maxwell equations read
$$\begin{equation}
\label{1}
\vec{\nabla} \times \vec{E}+\partial_t \vec{B}=0
\end{equation}$$
$$\begin{equation}
\label{2}
\vec{\nabla} \times \vec{B}-\mu_0 \epsilon_0 \partial_t \vec{E}=0,
\end{equation}
$$
$$\begin{equation}
\label{3}
\vec{\nabla} \cdot \vec{B}=0,
\end{equation}
$$
$$\begin{equation}
\label{4}
\vec{\nabla} \cdot \vec{E}=0.
\end{equation}
$$
To get rid of $\vec{B}$ we take the curl of Eq. (\ref{1}) and use (\ref{2}):
$$\vec{\nabla} \times (\vec{\nabla} \times \vec{E})-\mu_0 \epsilon_0 \partial_t^2 \vec{E}=0.$$
Now with (\ref{4})
$$\vec{\nabla} \times (\vec{\nabla} \times \vec{E})=\vec{\nabla} (\vec{\nabla} \cdot \vec{E})-\Delta \vec{E}=-\Delta \vec{E}$$
and thus
$$(\mu_0 \epsilon_0 \partial_t^2 + \Delta) \vec{E}=0,$$
and this is a wave equation for waves with (phase) velocity $c=1/\sqrt{\epsilon_0 \mu_0}$.

In a similar way, starting by taking the curl of (\ref{2}), you get the same equation for $\vec{B}$.

 

[PeroK]

I am simply referring to the value c as used in Maxwell's equations(making no connection with c as it was used later in Special Relativity)

C ,in Maxwell's equations describes the speed of em radiation (I believe) and I have been told that it is shown in his calculations as not depending on any frame of reference,

I would like to look at the maths behind that and would like to know at what point in his maths it becomes apparent that no frame of reference is used but a speed is calculated nonetheless.

To take a step back. Maxwell obtained his equations from known "laws" of electromagnetism (EM): Gauss's Law, Faraday's law. He converted these to a new mathematical context (his equations) and by solving these equations showed that EM radiation (waves) was possible. He calculated the speed of that radiation and, extraordinarily, got the known speed of light.

This is presented in modern vector calculus notation in post #8.

But, this left a conundrum. In what frame of reference were the laws of EM valid? If these laws (Gauss's law etc.) were universally valid (valid in every inertial reference frame), then EM radiation would have the same speed $c$ in every inertial reference frame. This was clearly at odds with classical physics.

Note that classical EM is at odds with classical physics in other ways. E.g. in a frame of reference in which a charge is at rest, there is only an electric field. But, in a reference frame where this charge is moving there is the same electric field and also a magnetic field. It cannot be both.

Most people, including Maxwell himself, assumed that the laws would only be exactly true in the universal "aether" frame. The alternative - that Maxwell's equations were universally valid - was too radical at the time.

Then, thru the Michelson-Morley experiment, came experimental evidence that the speed of light was indeed frame invariant.

It was, of course, Einstein in 1905 who finally dared to write down as a postulate that the speed of light is invariant and to follow the line of inexorable logic that led to Special Relativity.

In summary, there is nothing in Maxwell's work to say whether his laws are universal or only valid in an aether frame. The postulate of SR is effectively that they are universally valid and the foundations of classical physics, in terms of absolute time and space, must give way.

 

[anorlunda]

Indeed, before SR there were some unresolved puzzles implicit in Maxwell's Equations. Here is the opening part of Einstein's 1905 paper on SR.

It is known that Maxwell’s electrodynamics—as usually understood at the present time—when applied to moving bodies, leads to asymmetries which do not appear to be inherent in the phenomena. Take, for example, the reciprocal electrodynamic action of a magnet and a conductor. The observable phenomenon here depends only on the relative motion of the conductor and the magnet, whereas the customary view draws a sharp distinction between the two cases in which either the one or the other of these bodies is in motion. For if the magnet is in motion and the conductor at rest, there arises in the neighbourhood of the magnet an electric field with a certain definite energy, producing a current at the places where parts of the conductor are situated. But if the magnet is stationary and the conductor in motion, no electric field arises in the neighbourhood of the magnet. In the conductor, however, we find an electromotive force, to which in itself there is no corresponding energy, but which gives rise—assuming equality of relative motion in the two cases discussed—to electric currents of the same path and intensity as those produced by the electric forces in the former case.

Examples of this sort, together with the unsuccessful attempts to discover any motion of the Earth relatively to the “light medium,” suggest that the phenomena of electrodynamics as well as of mechanics possesses no properties corresponding to the idea of absolute rest.

 

[Mister T]

I am wondering how this absence of a frame of reference got ,so to speak embedded in his equations.

It didn't. It was quite possible that his equations were valid only in the rest frame of the ether. Scientists set out to modify them so that they would take the motion of the ether into account. Nature, however, revealed that there was no ether and that the equations didn't need to be modified.

Any college-level calculus-based introductory physics textbook will have a derivation of what you asked for, that is, the derivation of the speed at which Maxwell's waves would have to travel if his equations of electromagnetism are valid.