Maxwell's equations

The formula

The differential form, in vacuum with sources:

∇·E = ρ/ε₀                        (Gauss's law)
∇·B = 0                           (Gauss's law for magnetism)
∇×E = −∂B/∂t                      (Faraday's law)
∇×B = μ₀·J + μ₀·ε₀·∂E/∂t          (Ampère–Maxwell law)

One-line readings, top to bottom:

Electric field lines start and stop on charge. Charge density ρ is the source; field diverges out of positive charge, into negative.
Magnetic field lines never start or stop. No magnetic charge exists — every flux line closes on itself. Whatever flux leaves a volume comes back in.
A magnetic field changing in time wraps an electric field around itself. This is why generators generate and why a loop near a switching converter picks up noise.
A current — or an electric field changing in time — wraps a magnetic field around itself. The second term is Maxwell's addition, and it's the one that makes radio possible.

Chase the two curl equations through each other in empty space (ρ = 0, J = 0) and you get a wave equation whose speed is fixed by two measurable constants:

c = 1/√(μ₀·ε₀) ≈ 2.998·10⁸ m/s
η₀ = √(μ₀/ε₀) ≈ 376.7 Ω          (impedance of free space)

That 376.7 Ω is why far-field E/H ratio is what it is, and it's lurking inside every antenna and path-loss formula you use.

Where you meet it

EMC chamber, radiated emissions run. Your unit fails at 240 MHz and the fix conversation is pure Maxwell: Faraday's law says the loop area between signal and return is an antenna; the aperture math on your enclosure seam is Maxwell with boundary conditions. The MIL-STD-461 limit line is the spec; the physics behind every fix is these four equations.
Signal integrity review board. The moment somebody asks "is that trace a transmission line yet?" you've crossed from circuit theory into field theory. The telegrapher's equations, characteristic impedance, and every rule of thumb about λ/10 are Maxwell solved for a specific geometry.
Antenna test range. Dipole resonant length, gain, beamwidth, near-field/far-field boundary — every number on the test plan descends from solving these equations with a radiating boundary.
Test stand instrumentation. The 60 Hz hum riding on your strain-gauge channel is Faraday's law coupling into your cable loop. Twisted pair works because it shrinks the loop area that −∂B/∂t has to act on.

How it works

The engine is the two curl equations feeding each other. A changing B makes a curling E; that E is itself changing, so it makes a curling B; and the disturbance leapfrogs through space at 1/√(μ₀·ε₀) with no medium required. Kill Maxwell's displacement-current term μ₀·ε₀·∂E/∂t and the leapfrog dies — you get magnetostatics plus induction, but no propagation. That one term is the difference between a world with radio and a world without it.

Displacement current also rescues current continuity. Charge a capacitor and conduction current flows in the wires but nothing crosses the gap — Ampère's original law gives contradictory answers depending on which surface you integrate over. The changing E in the gap counts as current, the books balance, and AC "flows through" capacitors.

Circuit theory is the small-structure limit. Kirchhoff's laws are what Maxwell collapses to when every dimension of your circuit is tiny compared to a wavelength — fields settle everywhere at once. Take the divergence of the Ampère–Maxwell law and combine it with Gauss's law and out falls charge conservation, ∇·J = −∂ρ/∂t; with no charge piling up anywhere, that becomes "current in equals current out" at every node. Faraday's law, with negligible ∂B/∂t through the circuit, becomes "voltages around a loop sum to zero." The classic mistake is carrying lumped intuition past its expiration date: at 100 MHz a 30 cm ground strap is a tenth of a wavelength and has stopped being a node. It's an inductor at best and an antenna at worst. When someone says "just ground it," Maxwell gets a vote on whether that helps.

Current continuity is the most underrated consequence of the four on the bench. Charge can't accumulate in steady operation, so return current always flows — somewhere. If you didn't give it a low-inductance path directly under the signal trace, it will find its own route, because the return settles wherever total loop inductance — the flux the loop encloses — is minimized, and at high frequency that's directly beneath the signal. Fail to provide that path and the enlarged loop becomes your emissions problem, courtesy of Faraday and Ampère–Maxwell. Half of practical EMC work is respecting where the return current wants to go. As for ∇·B = 0 itself, its bench meaning is exactly what it says: no magnetic monopoles, so flux lines always close, and magnetic shielding never terminates flux — a mu-metal can only offers it a lower-reluctance detour around the thing you're protecting.

In materials, ε₀ and μ₀ get replaced by ε and μ, conduction enters through J = σ·E, and the same four equations produce skin depth, dielectric loss, waveguide cutoff, and why your coax attenuation climbs with √f. Limits of validity: the equations are classical and linear in vacuum. They hold from DC through optics, which covers essentially all engineering; you need quantum electrodynamics only when single photons matter. For anything with a connector on it, Maxwell is exact for practical purposes.

History

The pieces came first. Michael Faraday demonstrated electromagnetic induction at the Royal Institution in 1831 and reported it to the Royal Society that November ^[1]^[2]. Ørsted's 1820 discovery that current deflects a compass needle, followed by Ampère's work through the 1820s, tied magnetism to current ^[3]^[4].

James Clerk Maxwell assembled the machine. In "On Physical Lines of Force" (1861–62) he added the displacement-current term, then noticed that the ratio Weber and Kohlrausch had measured in 1856 — about 3.107·10⁸ m/s — matched Fizeau's measured speed of light, and wrote that light could scarcely be anything other than waves in the same medium responsible for electric and magnetic phenomena ^[5]^[6]. He delivered the full theory in "A Dynamical Theory of the Electromagnetic Field," presented to the Royal Society in 1864 and published in 1865 — as twenty equations in twenty variables, since vector notation didn't exist yet ^[7]^[8]. His 1873 "Treatise on Electricity and Magnetism" consolidated the work ^[6]^[9].

The four-equation form on every whiteboard today isn't Maxwell's own. Oliver Heaviside compressed the twenty component equations into four vector equations in 1884–85, with Heinrich Hertz arriving at a similar reduction independently ^[4]^[9]. Hertz then did the part engineers respect most: between 1886 and 1888 at Karlsruhe, using spark-gap transmitters and resonant loop detectors, he generated and detected the predicted waves and showed they travel at finite speed ^[10]^[11]. Maxwell died in 1879 and never saw the confirmation. Within two decades of Hertz's spark, radio was an industry.

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Sources

Written by HE in our own words from the cited sources — engineering judgment included, your stamp still required. All entries →