Electromagnetism & Special Relativity

In 1865, James Clerk Maxwell sat down and put together something that should not, by any reasonable expectation, have worked. He took Faraday's experimental observations about electric and magnetic fields, noticed a mathematical inconsistency in Ampere's law, added a correction term he called the "displacement current," and wrote down four equations.

The result was a complete theory of electromagnetism. But more than that — he discovered, almost as a side effect, that his equations predicted the existence of transverse waves propagating at a speed $c = 1/\sqrt{\mu_0 \varepsilon_0}$. When he calculated that number, it came out to roughly $3 \times 10^8$ m/s — the measured speed of light. Maxwell wrote, with admirable restraint, that this was "scarcely possible to avoid the inference that light consists in the transverse undulations of the same medium which is the cause of electric and magnetic phenomena."

With one stroke, he had unified electricity, magnetism, and optics. Three apparently separate phenomena turned out to be facets of a single theory.

  ┌──────────────────────────────────────────────────────────────┐
  │  ∇·E  = ρ/ε₀            Gauss — charges source E fields      │
  │  ∇·B  = 0               No magnetic monopoles exist           │
  │  ∇×E  = −∂B/∂t          Faraday — changing B induces E       │
  │  ∇×B  = μ₀J + μ₀ε₀∂E/∂t  Ampère-Maxwell — J or ∂E/∂t → B  │
  └──────────────────────────────────────────────────────────────┘

Each equation tells you something profound. The first says electric charges create diverging electric field lines — they act as sources and sinks. The second says there are no magnetic charges, so magnetic field lines always form closed loops. The third is Faraday's induction law — a changing magnetic field creates a curling electric field. The fourth is the key one Maxwell modified: a changing electric field creates a curling magnetic field, just as a real current does.

That last term — $\mu_0 \varepsilon_0 \partial\mathbf{E}/\partial t$ — is the displacement current. It was not observed experimentally when Maxwell added it; he added it because without it, the equations were mathematically inconsistent. And it is that term that allows electromagnetic waves to exist and propagate through a vacuum.

The wave equation Maxwell derived predicts transverse waves — oscillations perpendicular to the direction of travel. In an electromagnetic wave, the electric and magnetic fields oscillate perpendicular to each other and to the direction of propagation, always in phase and always satisfying $|\mathbf{E}| = c|\mathbf{B}|$.

                     E (vertical)
                          ╱╲        ╱╲
                         ╱  ╲      ╱  ╲
           ──────────── ╱    ╲────╱    ╲──────────────▶ x (propagation)
                               ╲  ╱        
                                ╲╱ 
                     B (horizontal, into/out of page)
                     
          E ⊥ B ⊥ propagation direction
          Speed in vacuum: c = 1/√(μ₀ε₀) ≈ 3×10⁸ m/s

The entire electromagnetic spectrum — from radio waves with kilometre-scale wavelengths to gamma rays with wavelengths smaller than an atomic nucleus — is the same phenomenon at different frequencies. Visible light is just the narrow band that evolution happened to tune our eyes to detect.

Maxwell's equations contained a constant: the speed of light. But speed relative to what? In the late 1800s, physicists assumed there must be a medium — the "luminiferous ether" — through which light propagated, just as sound propagates through air. The speed of light would be $c$ relative to the ether, and you could measure the Earth's motion through it.

Michelson and Morley tried to detect the ether in 1887. They found nothing. The speed of light was the same regardless of which direction they measured it. This was deeply confusing.

Einstein, in 1905, resolved the confusion by refusing to be confused by it. He said: take the result seriously. The speed of light is the same for all observers, in all inertial frames. Accept that as a postulate and see what follows.

What follows is special relativity. And the consequences are startling.

If the speed of light is constant for all observers, then time and length cannot be. Consider two people with perfectly synchronised clocks. One gets on a rocket and flies away at near the speed of light. When they reunite, the travelling twin is younger than the one who stayed. This is not a metaphor. It is a measurement.

The factor $\gamma$, the Lorentz factor, governs everything. At everyday speeds, $\gamma \approx 1$ and the effects are negligible. But GPS satellites orbit at speeds where relativistic corrections matter at the nanosecond level — without accounting for special (and general) relativity, GPS would accumulate errors of kilometres per day.

   γ
   ▲
10 │                                           ╭──
   │                                        ╭─╯
 5 │                                     ╭─╯
   │                                  ╭─╯
 2 │                             ╭───╯
   │                     ╭──────╯
 1 │─────────────────────╯
   └──────────────────────────────────────────────▶ v/c
   0      0.3     0.6      0.9   0.99  0.999  1.0
   
   At v = 0.9c: γ ≈ 2.3   At v = 0.99c: γ ≈ 7   At v = 0.999c: γ ≈ 22

The deepest insight of special relativity is that space and time are not separate things. They are two aspects of a single four-dimensional structure: spacetime. Different observers disagree about the spatial and temporal separations between events, but they all agree on the spacetime interval between them.

When $ds^2 < 0$, the interval is timelike — the events can be causally related, and a clock can travel between them. When $ds^2 > 0$, it is spacelike — no signal can connect the events. When $ds^2 = 0$, it is null — only light can travel between them.

And at the end of all this, Einstein's most famous equation sits like a punchline: mass is just a form of stored energy. Every kilogram of matter contains$mc^2 \approx 9 \times 10^{16}$ joules — roughly the energy output of a large power plant running for three years. From two postulates about the speed of light, we derived that matter itself is a kind of concentrated energy. That is not a bad return.