But why the particular speed c? James Clerk Maxwell answered this question in 1862. In our modern SI system of units, the interactions of fields with charges and fields with each other depend on two constants, ε0, the so-called permittivity of space, and μ0, the so-called permeability of space. Maxwell did not use these units, but I shall give his answer as if he did. When studying all the then-known equations governing charge and fields, he discovered that mutually interacting electric and magnetic fields should be able to propagate through space, far removed from any charge, at a fixed speed given by
(It works out that the combination μ0ε0 has the dimension of (time/distance)2.) He calculated this inverse square root (or the equivalent quantity he used at the time) and found it to have the magnitude 3.11 x 108 m/s, within a few percent of the then-known speed of light. He then wrote, “We can scarcely avoid the conclusion that light consists in the transverse undulations of the same medium which is the cause of electric and magnetic phenomena.” As you can see from this quote, Maxwell still believed in a space-filling ether that was doing the vibrating. He apparently did not comment on the fact that the speed of his predicted waves did not depend on their wavelength or intensity.
The energy per unit volume in an electric field is given in terms of the field ℰ and the constant ε0 by
The overbar designates a mean-square average. Averaging is necessary because the electric field is oscillating.
The energy per unit volume in a magnetic field is given in terms of the field ℬ and the constant μ0 by
Again the overbar designates a mean-square average. In a propagating electromagnetic wave the vibrating electric and magnetic fields possess (and carry) equal energy. This is reflection of the perfect balance (the perfect pas de deux) between the two fields—the changing electric field creating and reinforcing the magnetic field, and the changing magnetic field creating and reinforcing the electric field.
Another way to phrase the question at the beginning of this Essay is: Why is the speed of light determined by basic electric and magnetic constants that can be measured in the laboratory quite independently of any wave? One answer is “Well, that’s what the mathematics predicts.” But if we want to let our fancy run free for a moment, recalling that those electric and magnetic constants appear in the equations that link the two kinds of field, we can look at it in this way. Suppose, hypothetically, that a pair of linked oscillating electric and magnetic fields propagated at less than the speed c. Their mutual reinforcement would be insufficient to maintain their strength, and they would quickly die out. Where would the energy go? On the other hand, if this hypothetical pair of fields propagated at more than the speed c, their mutual reinforcement would be so powerful that their energy content would increase, again violating the law of energy conservation. Only at the speed c does the propagating wave maintain its balanced electric and magnetic energy. And only at this speed is the energy in the wave conserved. (Recall that we are imagining the impossible in order to nail down the possible, in this case the necessary.)1
All of this is easier to explain—and to comprehend—in a no-longer-popular set of units called the cgs system (for centimeter gram second). In these units, electric and magnetic fields have the same physical dimension, and, in an electromagnetic wave carrying equal energy in its electric and magnetic fields, the mean-square values of the two fields are equal:
In cgs units, where the constants ε0 and μ0 do not appear, the quantity c makes its appearance initially not as a speed but as a constant in the laws relating magnetic and electric effects. For instance, if a charge q moves in a straight line at constant speed v, the magnetic field it generates at a distance d off to one side is, in these units,
The electric field generated by that charge at the same point (Coulomb’s law) is
Before relativity and before the realization that no material particle can move at speed c or faster, this pair of equations would have suggested that equal strengths of electric and magnetic fields could be produced by a charged particle moving at speed v = c. Although Maxwell didn’t know that producing equal-strength fields in this way is impossible, he discovered another way to do it, taking advantage of the fact that, even without charge, changing magnetic fields can produce electric fields (so-called magnetoelectric induction), just as changing electric fields can produce magnetic fields (electromagnetic induction).2 It turns out that if an oscillating electric field of average magnitude3 ℰ in otherwise empty space has frequency f and wavelength λ, it creates a magnetic field with average magnitude given by (in cgs units)
This is a form of the law of magnetoelectric induction in free space. Pretend that it is 1862 and you know only that c is a constant appearing in the laws of electromagnetism, such as the next-to-last equation above. You don’t yet know that c is the speed of light. You can quickly figure out that it must be. The only way for an oscillating electric field to generate an also-oscillating magnetic field of the same average strength is for the multiplier in the above equation to be equal to unity. With this condition, the magnetic field will in turn generate (induce) an electric field of the same average strength. Since the product of frequency and wavelength of a propagating wave is the wave’s speed, you can write
and conclude that the speed of the wave must be c.
What I present here is more a rationale than a derivation, but it illustrates two main points of Maxwell’s discovery. (1) Propagating waves require electric and magnetic fields of equal strength.4 (2) To maintain equal strength, the fields must change at such a rate that their speed of propagation is equal to c. Right at the heart of electromagnetic radiation is the balanced coupling of electricity and magnetism, and it is the speed of light that measures the strength of this coupling.
1 Relativity approaches this speed question in quite a different way. According to Einstein’s theory there is a maximum speed in nature, a speed achieved by any particle with zero mass—notably by the photon (and also by the graviton).
2 This discovery of Maxwell survived intact in the theory of relativity.
3 This is a root-mean-square average.
4 Strictly, it is the energy contents of the two fields that are equal. In cgs units, the field strengths are also equal.