First we recapitulate some of the vector notations due to Gibbs and still commonly taught. We denote 3D vectors by upper case latin letters or triples of 3 reals: (3, 4, 5).

(p, q, r)×(x, y, z) = (qz−ry, rx−pz, py−qx)

(p, q, r)∙(x, y, z) = px + qy + rz

In these definitions p, q, and r might be all reals, or all differential operators, such as ∇ = (∂/∂x, ∂/∂y, ∂/∂z).
With this notation we can write Maxwell’s equations, in suitable units, thus:

∇×B = ∂E/∂t + J

∇∙E = ρ

∇×E = −∂B/∂t

∇∙B = 0

In these equations:

- E is the electric field, which is a vector,
- B is the magnetic field, (vector too),
- ρ is the density of electric charge.
- J is the vector current density.

Consider the varying electromagnetic field where t is time:

E = (sin(z − ct), 0, 0)

B = (0, −sin(z − ct), 0).

We claim that this represents a beam of polarized light moving with velocity 1, in the z direction, with intensity = 1, wave length = 2π, spread over all space, and lasting forever.

We compute ∇×B for this field:

∇×B = (∂0/∂y+∂sin(z − ct)/∂z, ∂0/∂z−∂0/∂x, −∂sin(z − ct)/∂x−∂0/∂y)

= (cos(z − ct), 0, 0).

Note that this is exactly −∂E/∂t as required by Maxwell’s equations, considering that we have chosen units where c=1.

∇×E = (∂0/∂y − ∂0/∂z, ∂sin(z − ct)/∂z − ∂0/∂x, ∂sin(z − ct)/∂x − ∂0/∂y)

= (0, cos(z − ct), 0) = ∂B/∂t

That this proposed field solves the differential equations shows that the waves will persist as described.

Maxwell’s equations explained the observed properties of light in many ways. Polarization was a direct outcome. Marconi’s radio waves now had a quantitative theory explaining the apparatus that produced and detected them. The energy of the lightwave arose from theory and matched the observed warmth of the Sun. The emerging technology of diffraction gratings confirmed data about light wave lengths. Vibrating charged particles in mass seemed to be perfect sources and sinks for light. Refraction of light could be explained by the limited motion that electric charges in a transparent solid could undergo. This also explained why refractive index depended on wave-length. Even birefringence was explained by anisotropic limitations on the motion of these charges. Like Newton’s theories before, there seemed to be no loose ends to this theory.

Yet there was something that would have annoyed Newton or even Galileo in Maxwell’s equations: They required agreeing upon a fixed framework or coördinate system. Maxwell’s equations predicted a particular light speed, but in what reference frame? Galilean relativity had long deprecated the notion of standing still in some special framework. The luminiferous ether was invented but not much liked as it merely ensconced the special framework. The Michelson-Morley experiment confounded even the ether. Special relativity swept away these cobwebs; Maxwell’s equations were slightly transformed in the same revolution that fixed many other niggling problems.

No sooner had Einstein solved this class of problems (1905) than he inaugurated a new class by suggesting that light was particulate. He was never comfortable with the ramifications of photons, but it seems necessary to think about them to understand the world.