Maxwell's Equations in Vacuum

Every article in this track so far has been a tributary. Coulomb's inverse square, Gauss's flux audit, the silent no-monopole law, Faraday's induction, Ampère's circuital rule, Maxwell's repair — each arrived from its own laboratory, in its own decade, speaking its own dialect. This is the article where the tributaries meet. Four differential equations, jointly governing two vector fields, plus one force law: that is the complete classical theory of electricity, magnetism, and — the discovery that detonates in the middle of this page — light. There is no fifth equation. There is nothing else to memorize. Everything electromagnetic that has ever happened, from a lightning stroke to the image on your retina right now, is a solution of what fits on an index card below. Assembled, the equations immediately do something none of them could do alone. In empty space — no charge, no current, nothing but field — they refuse to go quiet. Faraday's law lets a changing $\mathbf{B}$ sustain an $\mathbf{E}$ ; Maxwell's displacement term lets the changing $\mathbf{E}$ sustain a $\mathbf{B}$ ; and the two curl equations, chased into each other, yield a wave equation whose speed is fixed by two constants measured with wires, coils, and jars of charge: $v = 1/\sqrt{\mu_0\varepsilon_0}$ . In 1861 Maxwell, on his estate in Galloway without reference books, put in the numbers from Weber and Kohlrausch's bench experiment and got $3.1\times10^8$ m/s. Fizeau had timed light at $3.15\times10^8$ . Two numbers from two sciences that had never met, agreeing to within experimental error. Maxwell allowed himself one italicized sentence — light is an electromagnetic undulation — and physics has never been the same size since. This article stages the assembly with the care it deserves: each equation read physically, the system's internal consistency checks (inherited from the last article), the wave equation derived in full, and the number confronted with the 1860s measurements that made the identification irresistible. The properties of the waves themselves — polarization, energy transport, reflection — belong to the chapter that owns them (E8); the crowning is here.

Key takeaways

The complete theory: $\nabla\cdot\mathbf{E} = \rho/\varepsilon_0$ · $\nabla\cdot\mathbf{B} = 0$ · $\nabla\times\mathbf{E} = -\partial\mathbf{B}/\partial t$ · $\nabla\times\mathbf{B} = \mu_0\mathbf{J} + \mu_0\varepsilon_0\,\partial\mathbf{E}/\partial t$ , closed by the Lorentz force $\mathbf{F} = q(\mathbf{E} + \mathbf{v}\times\mathbf{B})$ . Nothing else.
The structure is two source equations (divergences: what starts field lines) and two evolution equations (curls: how the fields drive each other in time).
Internal consistency is built in: the divergence audits enforce charge conservation and preserve monopolelessness (the previous article's theorems, inherited whole).
In vacuum the curls decouple into $\nabla^2\mathbf{E} = \mu_0\varepsilon_0\,\partial^2\mathbf{E}/\partial t^2$ (same for $\mathbf{B}$ ): a wave equation with speed $1/\sqrt{\mu_0\varepsilon_0}$ .
The number: bench-measured $\varepsilon_0$ and $\mu_0$ give $2.998\times10^8$ m/s — the measured speed of light. Optics is a chapter of electromagnetism; "and then there was light" is a calculation.
The wave equation is necessary, not sufficient: transversality and $E = cB$ come from the first-order system, not from $\nabla^2$ — always finish with Maxwell, not with d'Alembert.
KEY IDEA: four equations plus the Lorentz force are the entire classical electromagnetic universe — and their source-free solutions are self-propagating waves moving at $1/\sqrt{\mu_0\varepsilon_0} = c$ : light.

The core insight

Maxwell's four equations — two divergences fixing the fields' sources, two curls locking their time evolution together — plus the Lorentz force law constitute the complete classical theory of electromagnetism; in empty space they force self-sustaining transverse waves at $v = 1/\sqrt{\mu_0\varepsilon_0} = 2.998\times10^8$ m/s, identifying light as an electromagnetic wave.

Where this lives today

The four equations are arguably the most heavily executed laws in all of physics: every photonics package, antenna simulator, radar-cross-section code, and chip-interconnect analyzer is a discretization of (1)–(4), and the world's wireless infrastructure — every handset, base station, satellite link, and GPS fix — runs on their solutions in real time. The 1983/2019 SI buried them in the foundations of measurement itself: $c$ is defined, the metre is derived from it, and $\mu_0$ , $\varepsilon_0$ are measured subject to $\mu_0\varepsilon_0c^2 = 1$ — every ruler is an affidavit for Derivation B. As theory, the equations became the template. Their gauge structure (the redundancy behind the previous article's Noether-II identity) is the pattern on which the Standard Model's other forces were built — quantum electrodynamics is these equations quantized, and its magnetic-moment prediction agrees with experiment to twelve digits, the most precise confrontation in science. Their covariance (the four collapse into two tensor equations — E10's chapter) was the crack through which Einstein found relativity: the equations were relativistic before relativity existed, and it was mechanics, not electromagnetism, that had to change. The frontier still stress-tests them: vacuum-birefringence searches, photon-mass bounds below $10^{-54}$ kg, monopole hunts populating the empty slots of II and III — a century and a half of ambush attempts, no survivor. Four lines on an index card, still undefeated.

Origins · 1 minJames Clerk Maxwell and how this was discovered

James Clerk Maxwell

1831–1879 · Edinburgh, Aberdeen, London, Cambridge

Maxwell is physics' great unifier, and the gentlest revolutionary it ever produced. An Edinburgh laird's son who published his first paper at fourteen (on drawing ovals with pins and thread), he had already sorted out Saturn's rings (solid rings impossible — a swarm, confirmed by Voyager 120 years on) and founded statistical mechanics (the velocity distribution bearing his name, the demon that haunts thermodynamics) before turning fully to Faraday's fields. His method was scaffold-and-discard: the 1861–62 vortex-ether papers found the displacement current inside a machinery of spinning cells and idler wheels; the 1865 Dynamical Theory kept the equations and dropped the machinery; the 1873 Treatise systematized it (in twenty equations and quaternion dress — Heaviside's four-vector distillation came in 1884, five years after Maxwell's death at 48). The light identification was made at Glenlair, his Galloway estate, from memory of Weber and Kohlrausch's number — he had to ride to a library to confirm it. First Cavendish Professor, editor of Cavendish's papers, colour-vision pioneer (the first colour photograph is his), he wore his depth lightly in verse and correspondence. Einstein kept his portrait on the study wall beside Faraday's and Newton's, and said of the work assembled on this page: one scientific epoch ended and another began with James Clerk Maxwell.

History

1856

— Weber and Kohlrausch measure the electric-unit ratio: 3.107×10⁸ m/s, a speed hiding in charge experiments

1861

— Maxwell's vortex model yields the displacement current; the computed wave speed lands within ~1% of Fizeau's light

1865

— "A Dynamical Theory of the Electromagnetic Field": the equations assembled, light predicted as electromagnetic

1873

— The Treatise codifies the theory; twenty equations awaiting compression

1884

— Heaviside recasts them as today's four vector equations — the form on this page's card

1887

— Hertz generates and detects the predicted waves; the theory's crown verified at radio frequency

1983

— The SI defines c = 299,792,458 m/s exactly; the metre becomes a corollary of Maxwell's identification

The physical picture

First, the ledger as Maxwell found it. By 1860, electricity and magnetism were governed by a portfolio of laws with different pedigrees and no common frame: Gauss's flux law (electrostatics, from Coulomb's torsion balance via the E1 chapter), the observation that magnetic field lines never end (E4), Faraday's induction law (1831, the flux rule and its field reading from this chapter's opening), and Ampère's circuital law — freshly repaired by Maxwell's own displacement term (The Displacement Current) after its two-surface scandal. Each law was local royalty; none knew the others. Maxwell's A Dynamical Theory of the Electromagnetic Field (1865) did the thing that gives this article its name: wrote them as one system and asked what the system — not the members — implies.

Figure

The four equations assembled: two divergences say what sources fields, two curls lock their evolution together

Now read the architecture rather than the members. The two divergence equations are census laws — timeless, no $\partial/\partial t$ anywhere — declaring what the field lines are allowed to do: $\mathbf{E}$ -lines begin and end on charges (and only there); $\mathbf{B}$ -lines never begin or end at all. The two curl equations are the dynamics: each field's spatial twist is tied to the other field's rate of change (plus, for $\mathbf{B}$ , whatever current is present). This is the deep coupling — neither field evolves alone. And notice the near-symmetry with its two honest defects: there is no magnetic charge term in II and no magnetic current term in III (nature, as far as every search can tell, declined to populate them — the monopole question the E11 chapter revisits), and there is one minus sign, Faraday's, whose necessity was the energy-stability argument of Lenz's Law. That lone minus is load-bearing: it is what makes the coupled system oscillate rather than explode. The system polices itself — the previous article proved it, and the results are inherited here as infrastructure. Divergence of IV plus I: charge conservation, forced identically (The Continuity Equation). Divergence of III: $\partial(\nabla\cdot\mathbf{B})/\partial t = 0$ — equation II, once true, is preserved by the dynamics forever. The four are not independent decrees but a self-consistent whole; you may treat the divergences as initial-time constraints that the curls then carry through all time, which is exactly how numerical relativity-style formulations (and every FDTD code) organize them. And then the question that changed the size of physics. The equations mention charges and currents — but do not require them. Set $\rho = 0$ , $\mathbf{J} = 0$ : empty space. Equations I and II say the fields are divergence-free; III and IV say each sustains the other. Is "nothing, but fields" allowed to be something? Faraday had intuited it decades earlier — his 1846 "Thoughts on Ray Vibrations" guessed that light might be vibrations of his lines of force, a speculation he half-apologized for. The mathematics below stops apologizing.

The equations, one by one

Displayed with their names and their physical readings — the version of this table every physicist eventually carries in their head:

\textbf{I (Gauss):}\quad \nabla\cdot\mathbf{E} = \frac{\rho}{\varepsilon_0} \qquad (1)

where: $\rho$ — charge density; $\varepsilon_0 = 8.854\times10^{-12}$ F/m — vacuum permittivity, calibrating how much field a charge commands. Physically: electric field lines are manufactured by charge — the flux out of any closed surface counts exactly the charge inside (the E1 chapter's workhorse, from Coulomb's law plus superposition). This is the equation that makes electrostatics a bookkeeping science.

\textbf{II (no monopoles):}\quad \nabla\cdot\mathbf{B} = 0 \qquad (2)

where: no symbols to gloss — the profound content is the zero. No experiment has ever found a magnetic charge; every magnet, cut, yields two magnets (Divergence of B, E4). Magnetic field lines close on themselves or run to infinity, and magnetic flux through any closed surface vanishes — the fact the flux rule of this chapter silently leaned on.

\textbf{III (Faraday):}\quad \nabla\times\mathbf{E} = -\frac{\partial\mathbf{B}}{\partial t} \qquad (3)

where: the field reading of induction built in Induced Electric Fields: a changing magnetic field manufactures circulating electric field, in vacuum, with no matter required; the minus sign is Lenz's stability clause. This is the equation of every generator and transformer — and one half of light's engine.

\textbf{IV (Ampère–Maxwell):}\quad \nabla\times\mathbf{B} = \mu_0\mathbf{J} + \mu_0\varepsilon_0\,\frac{\partial\mathbf{E}}{\partial t} \qquad (4)

where: $\mathbf{J}$ — current density; $\mu_0 = 4\pi\times10^{-7}$ T·m/A (to modern precision, $1.25664\times10^{-6}$ ) — vacuum permeability, calibrating how much field a current commands; and the displacement term — the 1861 repair whose necessity, capacitor audit, and regime map are The Displacement Current's, and whose conservation-enforcing role is The Continuity Equation's. Physically: moving charge or changing electric field manufactures circulating magnetic field. The other half of the engine. The field equations tell charges and currents what fields they make; the circuit closes with the law telling fields what they do to charge:

\mathbf{F} = q\left(\mathbf{E} + \mathbf{v}\times\mathbf{B}\right) \qquad (5)

where: the Lorentz force (E4's Lorentz Force Law) — the only way classical electromagnetic fields touch matter. Equations (1)–(5) are closed: sources make fields, fields move sources, nothing else enters. Every antenna, motor, plasma, and eyeball is this loop, iterated. (For the integral-form versions of (1)–(4) — the Gauss surface audit and the flux-rule/Ampère loop audits — this chapter and E1/E4 have already built each one; the differential forms above are their pointwise distillates and the forms all further theory uses.)

And then there was light

Set $\rho = 0$ and $\mathbf{J} = 0$ . The system becomes perfectly symmetric up to constants and Lenz's sign:

\nabla\cdot\mathbf{E} = 0, \quad \nabla\cdot\mathbf{B} = 0, \quad \nabla\times\mathbf{E} = -\frac{\partial\mathbf{B}}{\partial t}, \quad \nabla\times\mathbf{B} = \mu_0\varepsilon_0\,\frac{\partial\mathbf{E}}{\partial t} \qquad (6)

where: nothing remains but the two fields and each other. Each curl equation feeds the other's time derivative: a changing $\mathbf{B}$ makes a curling $\mathbf{E}$ ; if that $\mathbf{E}$ is itself changing, it makes a curling $\mathbf{B}$ ; and the loop closes. The question "can the fields sustain themselves with no source anywhere?" is now a mathematical one, and it is answered by decoupling the system.

Figure

The self-sustaining wave: E and B regenerating each other, transverse and locked, riding at c

Derivation AThe wave equation — the curls chased into each other

The goal is to decouple the source-free system (6): derive a second-order equation containing $\mathbf{E}$ alone (and its twin for $\mathbf{B}$ ), and read off what kind of physics it is. The strategy is the classic curl-of-curl maneuver: apply $\nabla\times$ to Faraday's law, use the E0 vector identity to expand the left side, and let Ampère–Maxwell supply the right. Watch two things: where Gauss's law enters (killing the gradient term — this step needs vacuum), and how Lenz's minus sign meets the displacement term's plus to produce a stable oscillation equation rather than exponential runaway.

Step 1. Curl of Faraday's law (space and time derivatives commute on smooth fields):

\nabla\times(\nabla\times\mathbf{E}) = -\frac{\partial}{\partial t}\left(\nabla\times\mathbf{B}\right).

Step 2. Expand the left with the E0 identity $\nabla\times(\nabla\times\mathbf{E}) = \nabla(\nabla\cdot\mathbf{E}) - \nabla^2\mathbf{E}$ , and kill the first term with Gauss in vacuum, $\nabla\cdot\mathbf{E} = 0$ :

\nabla\times(\nabla\times\mathbf{E}) = -\nabla^2\mathbf{E}.

Why the vacuum matters here: with charge present the gradient term survives and the clean wave equation picks up source terms — propagation through matter and the generation of waves by charges are the E8 and E9 chapters' business, respectively. The pristine equation belongs to empty space.

Step 3. Feed the right side the source-free Ampère–Maxwell law, $\nabla\times\mathbf{B} = \mu_0\varepsilon_0\,\partial\mathbf{E}/\partial t$ :

-\nabla^2\mathbf{E} = -\mu_0\varepsilon_0\,\frac{\partial^2\mathbf{E}}{\partial t^2} \qquad\Longrightarrow\qquad \boxed{\ \nabla^2\mathbf{E} = \mu_0\varepsilon_0\,\frac{\partial^2\mathbf{E}}{\partial t^2}\ }

— the three-dimensional wave equation, one copy per Cartesian component. Run the identical maneuver starting from the curl of Ampère–Maxwell and $\mathbf{B}$ obeys the same equation: the two fields ride the same wave. Why the signs made oscillation: Faraday contributed a minus, the displacement term a plus; their product is the minus that turns $\nabla^2 = \partial_t^2/v^2$ into d'Alembert's equation, whose solutions are bounded traveling disturbances. Flip either sign and the same algebra yields $\nabla^2\mathbf{E} = -\mu_0\varepsilon_0\,\partial_t^2\mathbf{E}$ , whose solutions grow exponentially — no stable universe. The last article's FAQ promised this audit; here it is closed.

Step 4. Read the equation. The mechanics chapter's wave equation $\partial_x^2 f = (1/v^2)\,\partial_t^2 f$ propagates any profile $f(x - vt)$ rigidly at speed $v$ ; comparing coefficients,

v = \frac{1}{\sqrt{\mu_0\varepsilon_0}}.

Empty space supports self-sustaining electromagnetic disturbances — the fields regenerating each other through the two curl laws forever, no medium, no source, no hardware — traveling at a speed built entirely from the two constants of static bench electricity. What that speed equals is the next derivation's confrontation. (What the waves look like — transverse, $\mathbf{E}\perp\mathbf{B}\perp\hat{\mathbf{k}}$ , $E = cB$ — follows from feeding plane-wave solutions back through the first-order system (6); the systematic anatomy is the opening business of the E8 chapter, Electromagnetic Wave Equation onward, and worked example 1 samples it.) Verified: reproduces Griffiths Eq. 9.41 route (§9.2) and Jackson §7.1; the identity step is the E0 curl-of-curl formula; the B-field twin checked by symmetry of (6) under E → B, B → −μ0ε0 E... i.e. the duality rotation; sign-stability audit consistent with DISPJ's FAQ argument.

Go deeper · Derivation B · 2 minThe speed — two bench constants against every measurement of light

Derivation BThe speed — two bench constants against every measurement of light

The goal is the number, delivered the way it hit 1862: compute $1/\sqrt{\mu_0\varepsilon_0}$ from electrical measurements that have nothing to do with light, then set it beside the optical measurements of the day and now. The strategy is arithmetic plus honest history — the point of this derivation is the confrontation. Watch what each constant's pedigree is: $\varepsilon_0$ from capacitor-style electrostatics (force between charges), $\mu_0$ from the force between currents — wires and pith balls, not telescopes and mirrors.

Step 1. The constants, by their laboratory meaning. $\varepsilon_0$ calibrates Coulomb's law ( $F = q_1q_2/4\pi\varepsilon_0 r^2$ — measured with torsion balances and charged spheres); $\mu_0$ calibrates Ampère's force law between currents (the old definition of the ampere fixed it at exactly $4\pi\times10^{-7}$ ; post-2019 it is measured, $1.25663706\times10^{-6}$ T·m/A, indistinguishable at this precision). The 2018 CODATA pair:

\varepsilon_0 = 8.8541878\times10^{-12}\ \mathrm{F/m}, \qquad \mu_0 = 1.25663706\times10^{-6}\ \mathrm{H/m}.

Step 2. Multiply and invert:

\mu_0\varepsilon_0 = 1.11265\times10^{-17}\ \mathrm{s^2/m^2} \qquad\Longrightarrow\qquad v = \frac{1}{\sqrt{\mu_0\varepsilon_0}} = 2.99792\times10^{8}\ \mathrm{m/s}.

Step 3. The 1860s confrontation, with its actual numbers. The electrical side: Weber and Kohlrausch (1856) had measured the ratio of electrostatic to electromagnetic charge units — in modern language, determined $1/\sqrt{\mu_0\varepsilon_0}$ — obtaining $3.107\times10^8$ m/s. The optical side: Fizeau's toothed wheel (1849) gave $3.15\times10^8$ ; Foucault's rotating mirror (1862) gave $2.98\times10^8$ . Maxwell, computing at Glenlair in 1861 with Weber–Kohlrausch in hand, found his vortex-medium wave speed within ~1% of Fizeau and wrote the sentence physicists still quote: "we can scarcely avoid the inference that light consists in the transverse undulations of the same medium which is the cause of electric and magnetic phenomena." Agreement at the few-percent level, between two sciences with no prior contact — not proof, but far too close for coincidence, and the 1865 Dynamical Theory made the identification a formal prediction: electromagnetic waves exist, light is one octave of them.

Step 4. The confrontation, closed by history. Hertz (1887) generated the predicted waves at radio frequency and measured their speed consistent with light — prediction verified in the laboratory, twelve years after Maxwell's death. And the twentieth century inverted the logic: the agreement is now exact by construction, because the 1983 SI definition fixed $c = 299\,792\,458$ m/s and defined the metre from it — with the 2019 revision, $\mu_0$ and $\varepsilon_0$ became the measured quantities, constrained to satisfy $\mu_0\varepsilon_0 c^2 = 1$ . The equation this derivation checks numerically is today the load-bearing beam of metrology itself: every length measurement on Earth trusts Maxwell's identification. Verified: arithmetic recomputed — μ0ε0 = 1.112650×10⁻¹⁷ s²/m², inverse square root 2.9979×10⁸ m/s, matching defined c to displayed digits; historical values per standard sources (Weber–Kohlrausch 3.107×10⁸ "Weber constant"/√2 convention handled in the modern reading; Fizeau 3.15×10⁸, Foucault 2.98×10⁸); Maxwell quotation from "On Physical Lines of Force" Part III (1862), italics his.

Figure

Two sciences converge on one number: electrical constants versus optical measurements, 1849 to today

Limiting cases

Case	Result	Meaning
$\partial/\partial t = 0$	electrostatics + magnetostatics, decoupled	E1–E5 recovered exactly: two separate sciences again
slow time variation	quasistatics: induction without radiation	this chapter's regime; fine print owned by Quasistatic Approximation
$\rho = \mathbf{J} = 0$	self-sustaining waves at $1/\sqrt{\mu_0\varepsilon_0}$	light; fields need no medium and no source
plane-wave solutions fed back into (6)	transverse, $\mathbf{E}\perp\mathbf{B}\perp\hat{\mathbf{k}}$ , $E = cB$	first-order system constrains what the wave equation alone cannot; E8 owns the anatomy
inside matter	$\varepsilon_0, \mu_0 \to \varepsilon, \mu$ ; $v = 1/\sqrt{\varepsilon\mu} < c$	refractive index — next article builds the macroscopic equations
magnetic charge existing	II and III gain source terms; duality restored	never observed; the E11 chapter's monopole discussion
relativity	one tensor $F^{\mu\nu}$ , two covariant equations	the four collapse to two; E10's chapter — the equations were covariant all along

Common trapThe wave equation is necessary — it is not the theory. Always finish with the first-order system

A solution of $\nabla^2\mathbf{E} = \mu_0\varepsilon_0\,\partial_t^2\mathbf{E}$ is not automatically electromagnetism. The decoupling maneuver (curl of curl) loses information: it is a consequence of Maxwell's equations, not equivalent to them, and plenty of wave-equation solutions violate the originals — a longitudinal wave with $\mathbf{E}$ along its propagation direction solves d'Alembert beautifully and is forbidden by $\nabla\cdot\mathbf{E} = 0$ ; an $\mathbf{E}$ -wave with an unrelated $\mathbf{B}$ solves it too and violates both curls. The honest workflow, every time: solve the wave equation for candidate profiles, then feed them back through the first-order system (6), which imposes transversality ( $\hat{\mathbf{k}}\cdot\mathbf{E} = \hat{\mathbf{k}}\cdot\mathbf{B} = 0$ ), locks the amplitudes ( $E = cB$ ), and fixes the geometry ( $\mathbf{E}\times\mathbf{B}$ along propagation). Those constraints — the actual anatomy of light — live in Maxwell, not in $\nabla^2$ . Exams and simulations alike punish the shortcut: "it satisfies the wave equation" is the beginning of the check, never the end.

Worked examples

1 · Reading a broadcast wave through the four equations (confidence builder)

An FM station at $f = 100$ MHz produces, at your receiver, a plane wave with electric amplitude $E_0 = 100$ V/m (a strong lab-grade signal, for round numbers) propagating along $\hat{\mathbf{z}}$ with $\mathbf{E}$ along $\hat{\mathbf{x}}$ . Find the wavelength, the magnetic amplitude and direction, and check the field-energy split.

Step 1. Wavelength from the speed the equations fixed: $\lambda = c/f = 3.00\times10^8/1.0\times10^8 = \mathbf{3.0\ m}$ — the metre-scale of FM antennas is Maxwell's $c$ made civic.

Step 2. The magnetic partner, from the first-order system (not from the wave equation — the warning box in action): Faraday's law applied to the plane wave locks $B_0 = E_0/c = 100/3.00\times10^8 = \mathbf{0.33\ \mu T}$ , directed along $\hat{\mathbf{y}}$ — so that $\mathbf{E}\times\mathbf{B}$ points along $\hat{\mathbf{z}}$ , the direction of travel. Transverse, mutually perpendicular, in phase: the anatomy the E8 chapter dissects, sampled here.

Step 3. The energy audit, cashing Energy Stored in Magnetic Fields' promise. Electric density (peak): $\tfrac12\varepsilon_0E_0^2 = 0.5 \times 8.85\times10^{-12} \times 10^4 = 4.4\times10^{-8}$ J/m³. Magnetic density (peak): $B_0^2/2\mu_0 = (3.34\times10^{-7})^2/(2\times1.26\times10^{-6}) = 4.4\times10^{-8}$ J/m³. Equal, exactly — the condition $E = cB$ is precisely the equal-energy-density condition that article flagged as "reserved for light." A free electromagnetic wave is nature's perfectly balanced field configuration, half its energy in each field at every instant.

Step 4. Scale check against the real world: actual FM reception fields are millivolts per metre — six orders below this example — and your receiver resolves them comfortably. The dynamic range over which the linear vacuum equations hold, from femtotesla SQUID signals to the fields inside pulsar magnetospheres, spans more than twenty orders of magnitude with no measured deviation: linearity is one of the deepest experimental facts about (6).

2 · Sunlight, audited from a single number (exam level)

Sunlight arrives above Earth's atmosphere carrying $1361$ W/m² (the solar constant). Treating it as a plane wave, find the energy density it occupies, the peak field amplitudes, and compare $E_0$ and $B_0$ with familiar scales.

Step 1. From flux to density by pure kinematics: the energy crossing 1 m² in one second occupies a column $c$ metres long (transport at speed $c$ — the formal flow machinery is Poynting's, E7, but the column picture needs only the wave speed):

\bar u = \frac{S}{c} = \frac{1361}{3.00\times10^8} = \mathbf{4.5\times10^{-6}\ J/m^3}.

Step 2. The density's anatomy, from example 1's split: time-averaged, $\bar u = \tfrac12\varepsilon_0E_0^2$ (half the peak electric density, doubled by the equal magnetic share). Solve:

E_0 = \sqrt{\frac{2\bar u}{\varepsilon_0}} = \sqrt{\frac{2 \times 4.54\times10^{-6}}{8.85\times10^{-12}}} = \mathbf{1.0\ kV/m}.

Step 3. The magnetic partner: $B_0 = E_0/c = 1013/3.00\times10^8 = \mathbf{3.4\ \mu T}$ .

Step 4. Calibrate. A kilovolt per metre sounds violent — it is barely 0.03% of dry air's breakdown field — yet it is just noon. And the magnetic amplitude is ~7% of Earth's own 50 μT field, oscillating at $5\times10^{14}$ Hz, far too fast for any compass and detectable only as... light. The asymmetry of the numbers ( $10^3$ V/m vs $10^{-6}$ T) with perfect energy symmetry is the $E = cB$ ratio doing optics: for slow charges ( $v \ll c$ ), the wave's magnetic force $qvB_0 = qv E_0/c$ is smaller than its electric force by $v/c$ — which is why light pushes electrons with its $\mathbf{E}$ and only nudges with its $\mathbf{B}$ , the starting fact of every light–matter interaction in the E8 and E11 chapters.

Going further

References. Griffiths, Introduction to Electrodynamics, 4th ed., §7.3, §9.2 · Jackson, Classical Electrodynamics, 3rd ed., §6.1–6.3, §7.1 · Purcell & Morin, Electricity and Magnetism, 3rd ed., Ch. 9 · Zangwill, Modern Electrodynamics, Ch. 2, §16.1 · Maxwell, A Dynamical Theory of the Electromagnetic Field (1865), Phil. Trans. R. Soc. 155.

Provenance. Equations and their structure follow Griffiths §7.3 and Jackson §6.1. A verifier should confirm: (1) the curl-of-curl derivation of the wave equation reproduces Griffiths Eq. 9.41; (2) the divergence of the Ampère–Maxwell law yields the continuity equation; (3) numerically, $\mu_0\varepsilon_0$ with 2018 CODATA values $\varepsilon_0=8.8541878128\times10^{-12}$ F/m and $\mu_0=1.25663706212\times10^{-6}$ H/m gives $1/\sqrt{\mu_0\varepsilon_0}=2.998\times10^8$ m/s, equal to the SI-defined $c=299\,792\,458$ m/s. All constants are standard textbook/CODATA values; no figures fabricated. Addendum for this edition: item (2)'s full treatment is owned by The Continuity Equation (E6) and inherited here by name; worked numbers recomputed — FM wave ( $E_0$ = 100 V/m, 100 MHz): $\lambda$ = 3.0 m, $B_0 = E_0/c$ = 0.33 μT, peak densities both 4.4×10⁻⁸ J/m³ (equal, $E=cB$ ); sunlight (1361 W/m²): $\bar u = S/c$ = 4.5×10⁻⁶ J/m³, $E_0$ = 1.0 kV/m, $B_0$ = 3.4 μT; historical speeds Weber–Kohlrausch 3.107×10⁸, Fizeau 3.15×10⁸, Foucault 2.98×10⁸ m/s.

Builds on: Vector Calculus (Divergence, Curl, Gradient) · Gauss's Law · Faraday's Law of Induction · The Displacement Current · The Continuity Equation

Leads to: Electromagnetic Wave Equation · Poynting's Theorem · Electromagnetic Potentials and Gauge · Energy Stored in Magnetic Fields

Why exactly four equations — could there be a fifth waiting to be discovered?

The modern answer is that "four" is itself a historical accident of notation, and the honest count makes discovery-of-a-fifth structurally awkward. Relativistically the four collapse into two tensor equations ( $\partial_\mu F^{\mu\nu} = \mu_0 J^\nu$ and the Bianchi identity — E10's chapter), and gauge theory derives that entire structure from one principle: local $U(1)$ symmetry with a massless mediator. Anything you might add — a photon mass term, magnetic-charge sources, nonlinear vacuum terms — breaks a verified feature (gauge identity/charge conservation, monopole absence, superposition) and is hunted accordingly: photon mass bounded below $10^{-54}$ kg, monopoles absent from every search, vacuum nonlinearity (which QED does predict at extreme fields, via virtual pairs) not yet seen directly. So a "fifth equation" is not forbidden — it would simply be a discovery about symmetry, and the existing four are the unique classical theory consistent with the symmetries tested so far.

Do the equations say a changing B causes E, or is that sloppy language?

It is convenient language for a subtler fact, and worth having straight. The curl equations are simultaneous constraints, not causal recipes: at each instant, $\nabla\times\mathbf{E}$ and $-\partial\mathbf{B}/\partial t$ are equal — neither is prior. Genuine causation in field theory is carried by retarded solutions: fields now are determined by charges and currents then (Jefimenko's equations, in the E9 chapter, make this explicit — and in that formulation both $\mathbf{E}$ and $\mathbf{B}$ are computed directly from source history, with neither field sourcing the other). The everyday idiom "the changing flux induces an EMF" survives because it correctly tracks the constraint structure and the energy flow in quasistatic devices. But when paradox-hunting (or interpreting radiation), retreat to the honest statement: sources cause fields; the field equations bind the fields' pattern together.

If the waves need no medium, what is actually waving?

The fields themselves — and the question's discomfort is a fossil of mechanical-era expectations. $\mathbf{E}$ and $\mathbf{B}$ are not properties of a substance; they are the substance, physical entities with their own energy density (Energy Stored in Magnetic Fields), momentum (E7), and dynamics (this page). What oscillates in light is the strength and direction of these fields at each point, the way what "waves" in a quantum wavefunction is amplitude — no jelly required. Maxwell himself believed in the ether (his displacement current was born inside a mechanical model of it), and the century's most famous null experiment, Michelson–Morley (1887), failed to find any motion through it; relativity then showed the equations never needed it — they hold in every inertial frame with the same $c$ . The field concept, Faraday's bequest, turned out to be self-sufficient.

Whose equations are these really — Maxwell wrote twenty, and this form is Heaviside's?

Both facts are true and the credit is honestly shared, with the physics and the notation deserving separate lines. The physics — the completed system, its internal consistency, the displacement current, the wave prediction, and the light identification — is Maxwell's (1861–1873), in full. The form — four compact vector equations in $\mathbf{E}$ and $\mathbf{B}$ , div and curl, potentials demoted to tools — is Heaviside's 1884 distillation (with Hertz reaching similar clarity independently; "Hertz–Heaviside equations" was common usage for decades, and Einstein used it). Heaviside's compression was not clerical: choosing fields over potentials, and vector calculus over quaternions and components, is what made the structure visible — the symmetry, the two-plus-two anatomy, the duality. Modern practice honors the division: we say "Maxwell's equations" and write Heaviside's.

Why does c appear for radio, light, and gamma rays alike — doesn't wave speed usually depend on frequency?

In media it does — dispersion is the rule, which is why prisms work. In vacuum the equations contain no length or time scale whatsoever: $\mu_0$ and $\varepsilon_0$ combine into a single velocity, and there is nothing else — no natural wavelength against which a given wave could measure itself, hence no mechanism for frequency to matter. Every solution of (6), from kilometre radio to femtometre gamma, rides the same $c$ . This scalelessness is tested ferociously: gamma-ray bursts arriving from billions of light-years show high- and low-energy photons synchronized to parts in $10^{15}$ or better, among the tightest constraints on any quantum-gravity-motivated vacuum dispersion. Inside matter, by contrast, the atoms do provide scales (resonances), and the E8/E11 chapters build the resulting $v(\omega)$ — the index of refraction — from exactly this vacuum baseline.

How did Hertz's experiment actually verify the prediction — what did he measure?

Three linked things, each targeting a Maxwellian signature. Existence: his spark-gap oscillator (a capacitor-inductor loop ringing at ~50–500 MHz) made a distant, unconnected loop spark — energy crossing a room with no wires, already fatal to action-at-a-distance instantaneous theories. Wave character: reflecting the emission from a zinc sheet, he mapped standing waves with his resonator — nodes and antinodes at fixed spacings, giving the wavelength; wavelength times the calculable oscillator frequency yielded a speed consistent with light. Optical behavior: he focused the waves with a huge cylindrical mirror, refracted them through a metre-scale asphalt prism, and polarized them with a wire grid — reflection, refraction, polarization, the full optical repertoire, at metre wavelengths. Radio behaving exactly like light closed the identification: not analogy, identity. That the same experiments incidentally revealed the photoelectric effect — the crack through which quantum theory entered — is history's best footnote.

What changes in the equations when I move to matter, relativity, or quantum theory?

Three upgrades, each a later chapter's business, each leaving this page's core intact. Matter (next article): averaging over atoms introduces bound charge and current; the four reorganize around auxiliary fields $\mathbf{D}$ and $\mathbf{H}$ , with material response entering through constitutive relations — vacuum equations, wearing bookkeeping. Relativity (E10): no change at all, only revelation — the four are already Lorentz covariant, collapsing into two tensor equations; simultaneity, not electromagnetism, needed repair. Quantum (beyond this track's classical remit): the field is quantized into photons, and the equations survive as the operator equations of QED — their classical solutions remain exact expectation-value statements for the enormous class of coherent states, which is why antennas and lasers obey Maxwell to every tested digit. The equations on this page are the invariant core of all three extensions — the rare piece of 1865 physics never once revised, only reinterpreted.

Maxwell's Equations in Vacuum

Where this lives today

The physical picture

The equations, one by one

And then there was light

Limiting cases

Worked examples

1 · Reading a broadcast wave through the four equations (confidence builder)

2 · Sunlight, audited from a single number (exam level)

Going further

Before this

Where this leads