By 1860 electricity and magnetism were a pile of separate rules found by separate people. Maxwell wrote them as four statements about flux and circulation, noticed that one of them was incomplete, added the missing term — and the corrected set turned out to describe a wave that carries itself along at a speed built entirely from two constants you can measure on a bench with charges and wires. That speed is 299,792,458 metres per second. Nobody had been looking for light. Take the equations one at a time below, then watch the number appear.
Draw any closed surface you like. Add up how much electric field pokes out through it minus how much pokes in, and the answer depends on one thing only: the charge sealed inside. Not where that charge sits, not what shape the surface is, not what other charges are doing outside. Move the sphere below around a set of charges and watch the flux jump the moment a charge crosses the boundary — and stay flat while it merely moves about inside. The flux number is not read off a formula: the page samples the real field at 1,200 points spread over the sphere and sums E·n̂ dA, then compares with Q/ε₀.
Equation I has a right-hand side you can change by putting charge inside. Equation II has a zero, and nothing you do can move it. Break a magnet in half and you get two magnets, not a north pole and a south pole — because a magnetic field line has no end to expose. Whatever goes in must come out. The same 1,200-point sum runs on the magnet's field below, but this time it is reported as three numbers: flux going out, flux coming in, and their total. The first two are large and change as you move the surface. The third stays at zero to the accuracy of the arithmetic. Switch on the hypothetical monopole to see what the universe would look like if it did not.
A steady magnetic field does nothing at all. It is the rate of change that drives current, which is why the coil below produces its largest voltage at the instant the flux passes through zero, not at the instant the field is strongest — the two are a quarter-cycle apart. Everything you plug in runs on this one line: a generator is a loop turning in a field, a transformer is one coil changing the flux through another. The lower panel plots flux and the induced voltage together so you can see the quarter-cycle lag; the voltage curve is a numerical derivative of the flux curve, and the tile below reports how closely it matches N·B·A·cos α·ω.
Ampère's original law said magnetic circulation equals the current threading your loop. Maxwell found the hole: charge a capacitor, and a loop drawn around the wire has current through it while an identical loop slid sideways into the gap has none — yet the magnetic field out there is plainly the same. His fix was to say that the swelling electric field between the plates counts as a current in its own right. It is not a bookkeeping trick: below, the conduction current in the wire and ε₀·dΦE/dt in the empty gap are computed separately from Q(t) and E(t), and they agree at every instant. Slide the loop out past the plate rim and the two routes give the identical field — which is exactly why you cannot tell from outside whether a real current or an empty gap is inside.
Take equations III and IV somewhere empty — no charges, no currents, nothing but space. III says a changing B makes an E. IV says a changing E makes a B. Each one is the other's cause, so nothing has to hold the pattern up: it regenerates itself as it goes. Do the algebra and the two first-order equations collapse into a single second-order one, and that equation is the wave equation — the same one that governs a plucked string, with one difference. The string's speed comes from its tension and mass. This one's speed comes from ε₀ and μ₀, two numbers already measured in the 1850s by people weighing forces between charges and between wires, who had no idea they were measuring light.
Suppose a wave of some other speed tried to exist. Give it an electric part, then let Faraday's law fix the magnetic part that must accompany it. Now check whether that same magnetic part also satisfies Ampère–Maxwell. It does not — the two laws demand different amplitudes, and the gap between the amber curves below is that disagreement. Drag the speed and the only place the two curves land on top of each other is 1/√(ε₀μ₀). Everything else is not a slower wave; it is not a wave at all.
The algebra above is a derivation; this is a measurement. The strip below is 900 cells of empty space, 1 mm apart, stepped forward 1 picosecond at a time by the two curl equations and nothing else — the only physical constants anywhere in the update rule are ε₀ and μ₀. A pulse was dropped in at the left and left alone. The page tracks its peak and divides distance by time. Neither c nor any wave formula appears in the loop.
Maxwell published the prediction in 1865 using the best electrical constants available to him, which came from Weber and Kohlrausch's 1856 experiment and were about 3.7% too high. He still wrote that the agreement with Fizeau's optical measurement “seems to show that light and magnetism are affections of the same substance”. The percentage column is computed on this page from each figure against today's defined value.
| how it was found | year | value, m/s | off by |
|---|
Everything numeric here is computed, not quoted. The two flux sums are genuine 1,200-point quadratures over a sphere; they land on Q/ε₀ to about one part in 10¹² and on zero to a few parts in 10⁵ of the flux actually crossing the surface — those residues are the integration grid, not physics. The induced voltage is a numerical derivative of the flux curve and is checked against N·B·A·cos α·ω. The displacement current is worked out from ε₀·d(EA)/dt and compared with the conduction current independently. The wave strip is a standard staggered-grid integration of the two curl equations with ε₀ and μ₀ as its only constants, and it measures a speed within about 0.05% of c; the shortfall is numerical dispersion on a 1 mm grid, and it shrinks if you shrink the cells.
But these are the integral forms applied to deliberately easy situations, which is not the same as solving Maxwell's equations. Every field here is either static or a single plane wave in one dimension, travelling through vacuum. Real problems have three dimensions, boundaries, conductors, dielectrics with their own ε and μ, radiation from accelerating charges, and awkward geometry — and they are solved with potentials, Green's functions or a great deal of computer time, not with a sphere and a bit of arithmetic. The capacitor picture also ignores fringing at the plate edges and assumes the field inside is uniform; the coil is treated as an ideal loop with no resistance, no inductance of its own and no back-reaction on the field that drives it. And none of this is the final word: the equations as written are the classical ones, superseded at small scales by quantum electrodynamics, of which they are the large-scale average.