Maxwell's Equations & the Birth of the EM Wave

Four laws, one sentence each

By the 1860s physicists had a drawer full of experimental laws — Coulomb's, Gauss's, Faraday's, Ampère's — each describing one corner of electricity or magnetism. They worked, but they sat there like four unrelated recipes. Maxwell's genius was to see that they were four verses of a single poem, and that the poem, read carefully, predicted something nobody had measured: a wave. Before we get to the wave, let's hear the four verses plainly, in words, with no vector calculus.

[[ee-gauss-law|Gauss's law for electricity]]: electric charge is the source of the electric field. Field lines begin on positive charges and end on negative ones. Wrap any closed surface around a lump of charge and you can count exactly how much field pokes out — it's proportional to the charge inside.
Gauss's law for magnetism: there are no magnetic charges. Magnetic field lines never start or stop — they always close into loops. Saw a bar magnet in half and you get two complete magnets, never a lonely north pole. This is the shortest, strangest law: magnetic monopoles simply don't exist.
[[ee-faradays-law|Faraday's law of induction]]: a *changing* magnetic field creates a swirling electric field. Move a magnet through a coil and you generate a voltage out of nothing but motion. The faster the magnetic flux changes, the stronger the induced electric loop. This is the law that runs every generator on Earth.
[[ee-amperes-law|Ampère's law]] (the original version): an electric current creates a circling magnetic field. Run current up a wire and a magnetic field wraps around it in rings — the right-hand rule. This was the partner of Faraday's law, but as we'll see, it was secretly incomplete.

The missing term: displacement current

Maxwell found the crack by thinking about a charging capacitor. Picture two metal plates with a gap between them, wired into a circuit. Current flows in the wires, piling charge onto the plates. Ampère's law says: pick any loop, and the magnetic field around it is set by the current passing through *any surface* you stretch across that loop. Usually it doesn't matter which surface you choose — same current, same answer.

But now stretch the surface so it bulges *between the capacitor plates*, into the gap. No charge crosses that empty gap — the conduction current through this surface is zero. Yet a bulging surface through the wire still carries the full current. Same loop, two surfaces, two different answers. Ampère's law contradicts itself. Something physical must be happening in that gap.

Here is Maxwell's leap. As charge piles onto the plates, the electric field in the gap grows — the field is *changing*. Maxwell proposed that a changing electric field acts just like a current for the purpose of making a magnetic field. He called it the displacement current, and adding it patched the contradiction perfectly: through the wire it's real current; through the gap it's the changing-E term; both give the identical magnetic field. The law became consistent for *any* surface.

Charging capacitor — same Ampère loop, two surfaces:

      wire (real current I)            wire
   ----------+        +----------||----------
             |        ||          ||
   ----[loop encircles wire here]----        ||
             |        ||          ||
         Surface A    ||      Surface B
     (cuts the wire)  ||   (bulges into the gap)
                      ||
              +plate  ||  -plate
              ========||========   E field growing -->

   Surface A:  conduction current  I        -> B field
   Surface B:  conduction current  0
               + displacement current  eps0 * dE/dt  -> SAME B field

   Maxwell's fix:  total = I_conduction + eps0 * (dPhi_E/dt)

The thought experiment that forced the displacement-current term into existence: both surfaces must give the same magnetic field, so the changing E in the gap must count as a current.

How a wave lifts off into empty space

Now run the two symmetric laws as a relay race. Imagine you wiggle a charge — say an electron jiggling up and down in an antenna. The jiggle makes a changing electric field. By Maxwell's new term, that changing E creates a magnetic field a little further out. But that B field is *also* changing (it just appeared, then grows and fades). By Faraday's law, a changing B creates an E field further out still. That new E is changing too, so it makes a B beyond it — and on, and on.

Each field is constantly recreating the other a step ahead of itself. The disturbance doesn't need wires, charges, or any medium to keep going — once launched, it is entirely self-sustaining. This is an electromagnetic wave: a propagating handoff between E and B, marching outward through perfectly empty space. The source (the antenna) only has to start it; after that the wave carries on by itself, forever, until it's absorbed.

The speed that gave it away

When Maxwell did the algebra — combining the changing-B-makes-E and changing-E-makes-B laws — they collapsed into a single wave equation. And a wave equation always tells you the wave's speed, written from two constants you already measured in the lab: ε₀ (how strongly empty space responds to electric fields) and μ₀ (how strongly it responds to magnetic fields). The speed comes out as one over the square root of their product.

  Wave speed from Maxwell's equations:

            1
  c  =  -----------
        sqrt(eps0 * mu0)

  eps0 = 8.854e-12  F/m     (electric constant)
  mu0  = 4*pi*1e-7  H/m     (magnetic constant, 1.2566e-6)

  c = 1 / sqrt( 8.854e-12 * 1.2566e-6 )
    = 1 / sqrt( 1.1127e-17 )
    = 2.998e8  m/s

  Measured speed of light (Fizeau, ~1850s):  ~3.0e8 m/s
                                              ^^^ identical

Two constants measured with capacitors and coils — nothing to do with light — predict 3×10⁸ m/s. That was the giveaway.

Maxwell plugged in the numbers and got about 3×10⁸ metres per second. That figure was already famous — it was the measured speed of light. The coincidence was too exact to be coincidence. In his own restrained words, the agreement showed that 'light itself is an electromagnetic disturbance.' Light, radio, the glow of a fire — all the same wave, differing only in how fast E and B are wagging.

The shape of the wave: transverse, E ⊥ B ⊥ travel

The equations don't just give the speed — they pin down the wave's geometry exactly. An electromagnetic wave is transverse: the E field and the B field both point *sideways* to the direction the wave travels, never along it. And they're mutually perpendicular too. Hold up your right hand: if E points along your fingers and you curl them toward B, your thumb points the way the wave is going. The three directions form a tidy mutually-perpendicular trio.

  Snapshot of a plane EM wave (frozen in time):

     E (up/down)
      ^
      |    .-.        .-.        .-.
      |   /   \      /   \      /   \
  ----+--/-----\----/-----\----/-----\----> direction
      | /       \  /       \  /       \    of travel (z)
      |/         \/         \/         \

     B (in/out of page) oscillates in step, 90 deg around

   * E and B peak together (in phase), cross zero together
   * E _|_ B _|_ travel       (all three mutually perpendicular)
   * wavelength (lambda) = distance between two crests
   * frequency (f)       = crests passing a point per second
   * always:  c = lambda * f

E and B oscillate in lockstep, at right angles to each other and to the line of travel. Their crests line up; the wave moves at c.

Two numbers describe any such wave. The wavelength λ is the distance between neighbouring crests; the frequency f is how many crests sweep past a fixed point each second. They're locked together by the speed: c = λ·f. Since c is fixed, a shorter wavelength always means a higher frequency. Stretch the wave out and it's radio; squeeze it down and it's light, then X-rays. Same physics, wildly different λ.

One spectrum, from radio to gamma rays

Because λ and f can take any value, electromagnetic waves form a continuous ladder called the electromagnetic spectrum. Our eyes are tuned to a hilariously thin slice of it — visible light, wavelengths from about 400 nm (violet) to 700 nm (red). Everything else is invisible to us but governed by the exact same four equations. Your Wi-Fi router and a distant quasar are both just shaking E and B fields, only at different tempos.

  THE ELECTROMAGNETIC SPECTRUM (low f / long lambda  ->  high f / short lambda)

  Radio      ~ km to m       AM/FM, TV, Wi-Fi, mobile        (kHz - GHz)
  Microwave  ~ cm            radar, ovens, 5G, satellite     (~ GHz)
  Infrared   ~ um            heat, night vision, fibre        (THz)
  VISIBLE    400-700 nm      what your eyes can see           (~500 THz)
  Ultraviolet~ 10-400 nm     sunburn, sterilising lamps
  X-ray      ~ 0.01-10 nm    medical imaging
  Gamma      < 0.01 nm       nuclear, cosmic, most energetic

  All of it:  same c,  same E _|_ B,  same Maxwell's equations.
  Only lambda and f change.

One continuous family. The only thing changing across the whole ladder is wavelength and frequency — the underlying physics is identical.

This is why one course in electromagnetics quietly underwrites half of modern technology. Designing an antenna, beaming data through an optical fibre, imaging a broken bone, cooking dinner, listening to FM — every one is an exercise in launching, guiding, and absorbing the wave Maxwell predicted on paper before anyone had ever transmitted one. Heinrich Hertz finally generated and detected radio waves in 1887, twenty-some years after the equations said they must exist.