# Deriving the speed of light from Maxwell's equations?

Relationship between speed of light and EM force?

Can it be said that Maxwell used measurements of the "strength of electric force and strength of magnetic force", to derive the value for the speed of light?

Explicitly, is Maxwells work fundamentally based upon measurement of "forces?

I'm investigating "force" for the fundamental role it plays in the worlds operation. If Maxwell derived or "founded" his work based upon measures of force, then I think that is something important to keep in mind. From where and how and in what order we derive our understandings of the world is important context, if a fundemental understanding of the worlds operation is the ultimate goal. I think thats a fair assumption

It is true that Maxwell was able to derive an extra equation from his four equations of electromagnetism which allowed him to calculate the speed of electromagnetic waves, which turned out to be equal to the known speed of light.

That equation contained two physical constants (which had previously been measured in the lab) which expressed how easily 1) a magnetic field and 2) an electric field can propagate through empty space.

Regarding your question "Explicitly, is Maxwells work fundamentally based upon measurement of "forces"?", I am not sure I understand the question.

• Recently I heard it stated this way, I think while watching one of Richard Feynman's lectures. In effect he said "Maxwell derived the speed of light from the measured value of electric force and magnetic force". I hadnt heard it stated like that before and I would like to understand the particulars of it. I cant find the video – Steve Oct 6 '19 at 12:39
• the 2 physical constants are the electric permittivity of free space and the magnetic permeability of free space. – niels nielsen Oct 7 '19 at 5:02
• Notice how those measurements are derived from electricity and magnetism, and therefore represent properties of Elec and Mag, but then they are labeled as being properties of free space? A measurement placed on one entity then inferred to be a property of another thing entirely. Everybody attempting to learn about these properties is given the impression they are properties of space. This is a fundamental problem in science, for it gives misleading impressions. A measurement should honor and be assigned to where and how it is taken. The context being so important. – Steve Oct 7 '19 at 5:44
• I believe the property of time has suffered the same fate, due to misappropriated inferences. While measures of time are derived from cyclic rates of physical systems, but then inferred to be a property of space. If time is a property of space then that would require an explanation that justifies "why the cyclic rates of physical systems is a property of space". The absence of such an explanation spells "incomplete science". The concept of spacetime does not necessarily say that space and time are a one entity, but rather that space and time are correlated. – Steve Oct 7 '19 at 5:57
• The concept of spacetime does not necessarily attest to space and time being of a same entity. Like the term "fabric of spacetime infers". But rather, it can be representative of a correlation that exists between properties of space and properties of time. That's a fundamentally different distinction, and in fundamental science context is everything. How is one to distinguish what is cause and effect if properties of systems are misappropriated. Unjustified inferences lead students toward conceptual mistakes – Steve Oct 7 '19 at 6:07

There is a parallel in the history of physics that I think is helpful in illustrating wat Maxwell recognized.

In the Principia Newton posited that it is possible to derive the speed of sound from first principles.

Sound is oscillating compression and rarefaction of air mass. Oscillation requires presence of the following two properties:

• There is a state of lowest strain, we can call that the equilibrium state, and when away from the equilibrium state there is a restoring force, towards the equilibrium state.

• Something that has been put into motion will keep going, unless counteracted by a force. (In the case of mass referred to as 'inertia')

In the case of the speed of sound:
Newton argued: given the value of the density of air, and the value of the elasticity of air, both of which we can measure, we can derive the speed of sound.

For an example: a youtube video created by the University of New South Wales (Australia) physics department a presenting a derivation of the speed of sound

The bigger the restoring force, the faster the propagation speed.
The smaller the inertia, the faster the propagation speed.

Maxwell

In the years leading up to formulating the equations we refer to as 'Maxwell's equations' Maxwell had used a mechanical model of the Luminiferous Ether. Maxwell didn't assume that mechanical model was true, but he did use it as a guide for ideas on how to proceed. I won't go into the details of that mechanical model here.

One of the properties of that mechanical model is that it supported the concept that Maxwell called 'displacement current'.

If you have a capacitor in the form of two separated plates the substance that is in between the plates is called the dielectric. The composition of the dielectric has an effect on the physical properties of the capacitor as a whole. In particular, if the dielectric has ions that have some leeway to move relative to each other then the capacitor can store more electric energy.
If all the negative charge in the dielectric moves a bit to the right and all the positive ions move a bit to the left the overall effect is the same as that of overall current flow.

Maxwell pointed out that the known properties of capacitors indicate that even in the absence of any physical dielectric, that is, even when the space between the capacitor plates was depleted to a vacuum, measurement of the physical properties was consistent with a displacement current in the vacuum. (For more information see the wikipedia article about permittivity

Maxwell noticed that the properties that he had to craft into his mechanical model (of the Luminiferous Ether) in order to account for all known phenomena gave rise to the necessary conditions for oscillation.

When you strain a dielectric away from equilibrum there is a force that forces back towards equilibrium.

Maxwell's model also had a counterpart of inertia.

In the case of Maxwell's mechanical model of the Luminiferous Ether the restoring force is very, very large, leading to a very, very large speed of propagation. See also the wikipedia article about the Electromagnetic wave equation

Maxwell's mechanical model of the Luminiferous Ether cannot be physical reality, but clearly Maxwell was able to use it to great advantage

Wikisource has a transcript of the 1861 paper On physical lines of force

Proposition XVI states:

To find the rate of propagation of transverse vibrations through the elastic medium of which the cells are composed, on the supposition that its elasticity is due entirely to forces acting between pairs of particles.

Maxwell finds a velocity of propagation of the transversal vibrations in the medium that is close to the known speed of light (314858 km/s, as measured by Fizeau). The 'Treatise on Electricity and Magnetism' was published in 1873, but the recognition of transversal vibrations propagating at the speed of light was already in the 1861 paper.

• Thank you, most helpful and fascinating. So it is believed that Maxwell employed an analogous model "propagation of sound waves" to achieve a model that successfully predicts the speed of light. But despite the similarities and success of this approach, the two systems are believed to be fundamentally different. ie, light waves are not oscillations in a medium of space. His model borrowed two properties from sounds propagation. 1. Restoration force and 2. Inertia, which is the capacity to resist force. Which classifies inertia as a force. So the answer to my question is, yes? C = force – Steve Oct 7 '19 at 4:26
• So the answer to the following question is "yes"? is Maxwells work fundamentally based upon measurement of "forces? Furthermore, It can be correctly said that the speed of light was derived from two empirically derived measures of force. – Steve Oct 7 '19 at 4:50