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Electromagnetic Waves

Turn off every charge and every current and ask what Maxwell's equations still have left to say. The answer is not silence — it's light, deriving its own existence and its own speed from four equations that were never about light in the first place.

Before this, you should know:

Take Maxwell's four equations and switch off the universe: no charges anywhere, ρ=0\rho=0; no currents anywhere, J⃗=0\vec J=0. In empty space, with nothing left to source anything, it would be reasonable to expect the equations to just say E⃗=0\vec E=0 and B⃗=0\vec B=0 and stop talking. That is one solution. It is very much not the only one — and finding out what else is hiding in these four equations, with absolutely nothing put in by hand to make it happen, is one of the great moments in the history of physics.

Maxwell's equations with nothing left to source them

Set ρ=0\rho=0 and J⃗=0\vec J=0 in Maxwell's equations:

∇⋅E⃗=0,∇⋅B⃗=0,∇×E⃗=−∂B⃗∂t,∇×B⃗=μ0ϵ0∂E⃗∂t.\begin{aligned} \nabla\cdot\vec E &= 0, \\ \nabla\cdot\vec B &= 0, \\ \nabla\times\vec E &= -\frac{\partial\vec B}{\partial t}, \\ \nabla\times\vec B &= \mu_0\epsilon_0\frac{\partial\vec E}{\partial t}. \end{aligned}

Look at what's left: no sources, but the two curl equations still couple E⃗\vec E and B⃗\vec B tightly together, each one's time derivative feeding directly into the other's circulation. A changing E⃗\vec E still generates a circulating B⃗\vec B (the displacement-current term, alone now that J⃗\vec J is gone); a changing B⃗\vec B still generates a circulating E⃗\vec E, exactly as Faraday found. Picture it as a hand-off: a bit of changing E⃗\vec E creates a B⃗\vec B; that new B⃗\vec B, if it's changing too, creates a new E⃗\vec E next door; that E⃗\vec E creates the next bit of B⃗\vec B, and so on. Nothing here has to decay — this can, in principle, sustain itself indefinitely, propagating from each point in space to its neighbor, needing no charge and no current to keep it going once it's started. That self-sustaining hand-off is worth making completely precise, and it's exactly the kind of thing the wave equation from classical field theory was built to recognize.

A transverse electromagnetic wave traveling to the right along a central axis. A blue sine curve oscillates vertically above and below the axis, representing the electric field E, with short vertical arrows from the axis to the curve at several points. A purple sine curve, drawn narrower and tilted to suggest a perpendicular plane, oscillates through the same axis in phase with the blue curve, representing the magnetic field B, with short tilted arrows from the axis to that curve. An amber arrow at the right end of the axis, labeled c, shows the direction of propagation, and a labeled double-headed arrow between two adjacent crests marks one wavelength.

E and B, always perpendicular to each other and to the direction of travel, always in phase, each one's change in time feeding the other's circulation — this pattern is what the wave equation below proves must travel at exactly one over the square root of the vacuum permeability times the vacuum permittivity, which is the speed of light.

Where this is heading

Special relativity opened with an emergency: Maxwell's equations produced one definite number, c=299,792,458 m/sc=299{,}792{,}458\ \text{m/s}, as the speed of electromagnetic waves — not a speed measured relative to some medium, the way sound moves relative to air, but a number that came out of the field equations themselves, with no medium mentioned anywhere in their derivation. That mystery — a speed relative to what? — is precisely what forced Einstein to rebuild space and time from the ground up. The derivation below is the other half of that story: showing exactly how, and exactly where, that number cc actually comes from. It's sitting inside μ0\mu_0 and ϵ0\epsilon_0 the entire time — two constants originally measured in completely unrelated experiments, one from the force between current-carrying wires, the other from the force between static charges — and Maxwell's equations lock them together into a speed.

Worked example

Starting from Maxwell's equations in vacuum, derive the wave equation for E⃗\vec E by direct substitution, showing every step, and identify the wave's speed. (click to reveal the solution)

Setting up: start from Faraday's law in vacuum,

∇×E⃗=−∂B⃗∂t,\nabla\times\vec E = -\frac{\partial\vec B}{\partial t},

and take the curl of both sides:

∇×(∇×E⃗)=−∇×(∂B⃗∂t).\nabla\times(\nabla\times\vec E) = -\nabla\times\left(\frac{\partial\vec B}{\partial t}\right).

The left side, using the standard vector identity for a double curl, ∇×(∇×F⃗)=∇(∇⋅F⃗)−∇2F⃗\nabla\times(\nabla\times\vec F) = \nabla(\nabla\cdot\vec F) - \nabla^2\vec F, applied to E⃗\vec E:

∇×(∇×E⃗)=∇(∇⋅E⃗)−∇2E⃗.\nabla\times(\nabla\times\vec E) = \nabla(\nabla\cdot\vec E) - \nabla^2\vec E.

In vacuum, Gauss's law gives ∇⋅E⃗=0\nabla\cdot\vec E=0, so the first term vanishes entirely, leaving

∇×(∇×E⃗)=−∇2E⃗.\nabla\times(\nabla\times\vec E) = -\nabla^2\vec E.

The right side: space and time derivatives commute, so the curl can be pulled inside the time derivative:

−∇×(∂B⃗∂t)=−∂∂t(∇×B⃗).-\nabla\times\left(\frac{\partial\vec B}{\partial t}\right) = -\frac{\partial}{\partial t}(\nabla\times\vec B).

Now substitute the vacuum Ampère-Maxwell law, ∇×B⃗=μ0ϵ0 ∂E⃗/∂t\nabla\times\vec B = \mu_0\epsilon_0\,\partial\vec E/\partial t:

−∂∂t(∇×B⃗)=−∂∂t(μ0ϵ0∂E⃗∂t)=−μ0ϵ0∂2E⃗∂t2.-\frac{\partial}{\partial t}(\nabla\times\vec B) = -\frac{\partial}{\partial t}\left(\mu_0\epsilon_0\frac{\partial\vec E}{\partial t}\right) = -\mu_0\epsilon_0\frac{\partial^2\vec E}{\partial t^2}.

Equating the two sides:

−∇2E⃗=−μ0ϵ0∂2E⃗∂t2.-\nabla^2\vec E = -\mu_0\epsilon_0\frac{\partial^2\vec E}{\partial t^2}.

Multiply both sides by −1-1:

∇2E⃗=μ0ϵ0∂2E⃗∂t2.\boxed{\nabla^2\vec E = \mu_0\epsilon_0\frac{\partial^2\vec E}{\partial t^2}.}

Recognizing the result: this is exactly the wave equation, in the same form derived for a scalar field, ∂t2ϕ=v2∂x2ϕ\partial_t^2\phi = v^2\partial_x^2\phi, generalized from one spatial dimension to three (where ∂x2\partial_x^2 is replaced by the full Laplacian ∇2\nabla^2) and matched term by term with 1/v2↔μ0ϵ01/v^2 \leftrightarrow \mu_0\epsilon_0. Reading off the wave speed:

v2=1μ0ϵ0⟹c=1μ0ϵ0.v^2 = \frac{1}{\mu_0\epsilon_0} \quad\Longrightarrow\quad \boxed{c = \frac{1}{\sqrt{\mu_0\epsilon_0}}.}

(An identical calculation, starting instead from the curl of the Ampère-Maxwell law and substituting Faraday's law, gives ∇2B⃗=μ0ϵ0 ∂2B⃗/∂t2\nabla^2\vec B = \mu_0\epsilon_0\,\partial^2\vec B/\partial t^2 — the same equation, same speed, for B⃗\vec B.)

Checking the number: using μ0=4π×10−7 T⋅m/A\mu_0 = 4\pi\times10^{-7}\ \text{T}\cdot\text{m/A} and ϵ0≈8.85×10−12 C2/(N⋅m2)\epsilon_0\approx 8.85\times10^{-12}\ \text{C}^2/(\text{N}\cdot\text{m}^2),

μ0ϵ0≈(1.2566×10−6)(8.85×10−12)≈1.112×10−17 s2/m2,\mu_0\epsilon_0 \approx (1.2566\times10^{-6})(8.85\times10^{-12}) \approx 1.112\times10^{-17}\ \text{s}^2/\text{m}^2, c=11.112×10−17≈13.34×10−9≈3.00×108 m/s.c = \frac{1}{\sqrt{1.112\times10^{-17}}} \approx \frac{1}{3.34\times10^{-9}} \approx 3.00\times10^8\ \text{m/s}.

Verifying with a plane-wave solution: try E⃗=E0cos⁡(kx−ωt) y^\vec E = E_0\cos(kx-\omega t)\,\hat y, propagating along xx. Substituting into the boxed wave equation exactly as the same check was carried out for the scalar field — two time derivatives bring down −ω2-\omega^2, two spatial derivatives bring down −k2-k^2 — gives

−k2E0cos⁡(kx−ωt)=μ0ϵ0[−ω2E0cos⁡(kx−ωt)]⟹ω2=k2μ0ϵ0=c2k2,-k^2E_0\cos(kx-\omega t) = \mu_0\epsilon_0\left[-\omega^2E_0\cos(kx-\omega t)\right] \quad\Longrightarrow\quad \omega^2 = \frac{k^2}{\mu_0\epsilon_0} = c^2k^2,

so ω=ck\omega = ck: the wave's frequency and wavenumber are locked together by exactly the speed cc just derived, for every wavelength, from radio waves to gamma rays.

The number that started this whole crisis — no medium required to define it, no preferred frame anywhere in this derivation — is not an extra fact bolted onto electromagnetism. It is 1/μ0ϵ01/\sqrt{\mu_0\epsilon_0}, sitting inside Maxwell's equations from the moment the displacement current was added, waiting to be read off. This is the same cc that walks into special relativity and refuses to add like an ordinary velocity — because it was never an ordinary velocity to begin with. It is a structural constant of the field equations themselves, the same in every inertial frame for the same reason μ0\mu_0 and ϵ0\epsilon_0 don't care which frame measured them.

Light is this

Once you accept that E⃗\vec E and B⃗\vec B satisfy a wave equation, the further properties of the wave follow from the same equations, not from any new assumption. Requiring ∇⋅E⃗=0\nabla\cdot\vec E=0 for a plane wave forces E⃗\vec E to be perpendicular to the direction of travel — the wave is transverse, never longitudinal. Faraday's law then locks B⃗\vec B's direction and phase to E⃗\vec E's: the two fields oscillate together, in step, each perpendicular to the direction of propagation and perpendicular to each other, exactly as drawn above. And the speed of this disturbance, c=1/μ0ϵ0≈3.00×108 m/sc=1/\sqrt{\mu_0\epsilon_0}\approx 3.00\times10^8\ \text{m/s}, matches — not approximately, not suggestively, but to every digit anyone has ever measured — the speed of light.

This is not a coincidence physics stumbled onto. It is the discovery that light is an electromagnetic wave: not a separate phenomenon that happens to travel at a suspicious speed, but literally a self-propagating disturbance of the same E⃗\vec E and B⃗\vec B fields that push on charges and deflect compass needles, oscillating too fast for the eye to see anything but a steady glow. Visible light, radio waves, X-rays, and gamma rays are all solutions of the same equation derived above, differing only in ω\omega and kk — the same ω=ck\omega=ck relation, all the way across the electromagnetic spectrum.

Where this leads

This branch of the track began with a spark jumping from a doorknob and ends with a derivation of light itself, both from the same four-letter alphabet: E⃗\vec E, B⃗\vec B, ρ\rho, J⃗\vec J. Classical field theory showed that a field's wave modes are, mathematically, an infinite collection of independent harmonic oscillators — and that quantizing even one such oscillator turns its energy ladder into a countable stack of identical quanta. Apply that same construction to the electromagnetic field derived here, and each rung of each mode's ladder is a photon — not a separate particle bolted onto the theory, but a single quantum of exactly the wave found above, described in full by quantum field theory. And because that same field equation carries the invariant speed cc discovered here into every inertial frame without alteration, it is also exactly the fact special relativity needed to get started. Two enormous branches of twentieth-century physics, both standing on the four equations assembled on this track — which is as good a place as any to stop and notice how much came from so little.