Working Note · Electromagnetism · Part IV
Maxwell's Equations: How Light is Born
Four short equations that explained magnets, generators, radio waves—and, unexpectedly for Maxwell himself, turned out to be the formula for light itself.
In the 1860s, James Clerk Maxwell gathered the disparate experimental laws of Coulomb, Gauss, Ampère, and Faraday into a system of four equations—and discovered a missing piece: without it, the equations violated charge conservation. By adding the “displacement current,” he not only fixed the mathematics—the equations immediately predicted waves traveling at a speed that perfectly matched the speed of light. Light turned out to be an electromagnetic wave. In this note—all four equations individually, then the assembly: how a self-propagating wave emerges from them.
Four Equations in a Nutshell
| № | Name | Meaning in One Phrase |
|---|---|---|
| I | Gauss (E) | charge—the source of an electric field spreading outward |
| II | Gauss (B) | magnetic field has no “charge” source—it is always closed |
| III | Faraday | changing magnetism creates a vortex electric field |
| IV | Ampère-Maxwell | current AND changing electric field create a vortex magnetic field |
The symmetry is almost perfect: III and IV are mirror statements (“changing B creates E” and “changing E creates B”), while I and II are “charge exists for E, not for B.” This near-symmetry and the “lack” of one term in the original Ampère's law led Maxwell to think that something was missing—and led to the discovery of the displacement current (§5).
Gauss's Law: Charge as a Source of Field
Formulation. The electric field “diverges” from a charge in proportion to its magnitude:
In the integral, more visual form: the flux of the electric field through any closed surface is proportional to the charge inside it. Visually—imagine a charge as a “faucet” from which jets of field lines stream in all directions; no matter what surface you draw around the faucet, the same amount of “water” passes through it, depending only on the strength of the faucet (the magnitude of the charge), not on the shape or size of the surface.
If you apply Gauss's law to a single point charge and compute the field on a sphere of radius r around it, you get exactly Coulomb's law (the force between two charges falls off as 1/r²)—historically earlier, discovered experimentally back in 1785, 80 years before Maxwell. Gauss's law is not a new fact, but a more general and powerful reformulation of the same thing.
Gauss's Law for B: No Monopoles
Formulation. The magnetic field has no source—it never “begins” or “ends” anywhere:
Break a magnet in half, hoping to get a separate “north” piece and a separate “south” piece—it won’t work: each fragment will again have both poles. Magnetic field lines are always closed on themselves (exiting from the north pole and inevitably returning to the south), unlike electric lines, which start on positive charge and terminate on negative charge (or go off to infinity).
Some theories beyond Maxwell's equations (in particular, Grand Unified Theory) predict the existence of magnetic monopoles—isolated “north” or “south” charges. They have been sought for decades (including the famous false signal in Blas Cabrera’s detector in 1982), but convincingly not found—within classical electrodynamics, the law ∇·B = 0 remains an exact experimental fact, not merely a convenient approximation.
Faraday's Law: Magnetism Creates Current
Formulation. A time-varying magnetic field produces a vortex electric field—and hence, a current in a closed conductor:
This is exactly how any electric generator works: mechanically rotate a magnet near a coil (or the coil near the magnet—by the law, only the relative rate of change of flux matters), the flux constantly changes, and an EMF is continuously induced in the wire. The minus sign—Lenz's rule: the induced current flows in such a direction that its own magnetic field opposes the change that produced it (otherwise you'd have a perpetual motion machine—the effect would “feed” itself ever stronger, which is forbidden by the law of conservation of energy).
Numerical example: a coil with N = 100 turns is in a magnetic field, the flux through one turn grows uniformly from 0.02 Wb to 0.05 Wb in 0.5 s. Find the induced EMF.
The magnitude of the EMF is 6 volts; the sign only indicates the direction of the current (opposing the increase in flux), not the magnitude.
Ampère-Maxwell Law: Current Creates Magnetism
Formulation. Electric current AND a time-varying electric field produce a vortex magnetic field:
The first term (μ0J)—this is the original Ampère's law (1826): current creates a circular magnetic field around itself, easily verified with a compass next to a current-carrying wire. The second term—Maxwell's own contribution, “displacement current”: even where there is no real current (for example, between the plates of a charging capacitor, in a vacuum), a rapidly changing electric field produces the same effect—creates a magnetic field.
Without the displacement current, Ampère's law in its old form mathematically contradicted charge conservation when a circuit is broken (for example, at a capacitor): applied to two different surfaces spanning the same loop, it gave different answers—absurd. Adding the term μ0ε0∂E/∂t resolved the contradiction—and, as it turned out, also made the equations symmetric and capable of describing a self-propagating wave (§6). A rare case in the history of physics where fixing a mathematical inconsistency led to a great discovery.
Walkthrough Example: The Birth of Light
Let's put together Faraday's law (§4) and the Ampère-Maxwell law (§5)—and see why the electromagnetic field can propagate on its own, without wires or magnets.
- Suppose somewhere in space the electric field E begins to change over time (e.g., due to charge oscillation in an antenna).
- By the Ampère-Maxwell law (§5, second term) this changing E right here produces a vortex magnetic field B—nearby, in a neighboring point in space, without any wire.
- But this B also has no reason to be constant—it too changes over time. And by Faraday's law (§4) the changing B produces a vortex E—now a bit further from the source.
- The result is a chain reaction: E creates B, B creates E, each step a bit further from the source. The field “runs away” from the antenna on its own, without external assistance—this is the electromagnetic wave.
Substituting both equations into each other (mathematically—taking the curl of one and plugging into the other) yields the usual wave equation, and the wave speed is expressed solely through two constants—the electric and magnetic constants:
These two constants (ε0 and μ0) were known from purely electrical and purely magnetic lab experiments—measured with capacitors and coils, with no mention of optics. When Maxwell in 1862 plugged their numbers into formula (2) and got a value that matched the already measured speed of light, he wrote: “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.” Optics turned out to be a branch of electromagnetism—no one expected that.
Visible light, radio waves, X-rays, microwaves—all are the same electromagnetic wave, differing only in wavelength (the frequency of E and B oscillations). The difference between FM radio and an X-ray image is not in the physics of the phenomenon, but only in numbers.
Where This Leads: Relativity of Fields
Maxwell's equations hold another surprise, not immediately noticed: they do not change form under Lorentz transformations (that is, they are already “relativistically correct”), unlike Newtonian mechanics, which had to be modified. Moreover—what one observer calls a “pure magnetic field,” another observer moving relative to the first will see as a mixture of electric AND magnetic fields.
Classic example: a stationary charge next to a current-carrying wire experiences no electric force (the wire is electrically neutral), only magnetic (if it moves). But for an observer moving along with the charge, the situation looks different: due to Lorentz length contraction, the densities of positive and negative charges in the wire no longer exactly cancel—the wire appears slightly charged, and a real electric force appears. The same physical effect—depending on the reference frame, it can be called “magnetic” or “electric.” This connection is one of the historical paths by which Einstein arrived at special relativity (1905), precisely starting from the puzzle of Maxwell's equations, not from mechanics.
On the site, this is a separate law if you want to dive deeper:
Maxwell's Equations · Faraday's Law of Induction · Ampère's Law
Home Experiment
Wind 30–40 turns of insulated copper wire around a large iron nail, connect the ends to a AA battery (1.5 V) for a few seconds (no longer—the wire and battery heat up).
What to Observe the nail will start attracting paper clips and other small metal objects—a direct demonstration of Ampère's law (§5, first term): the current in the wire creates a magnetic field, and winding it into a coil concentrates and amplifies this field inside and around the nail (which itself becomes magnetized, enhancing the effect).
Comb your dry hair with a plastic comb (or rub a balloon against a wool sweater) and bring it close to a thin stream of water from a tap.
What to Observe the water stream will noticeably bend toward the comb. Friction “knocked” electrons from one material to the other, creating a charge on the comb—and according to Gauss's law (§2) there is an electric field around the charge, which induces a weak opposite charge on the neutral but polarizable water molecules and attracts them (the effect is called dielectric polarization).
Check Your Understanding
First think for yourself, then open the solution. Every solution follows the scheme: Given → Law → Solution → Answer.
1. An FM radio station broadcasts at a frequency of 100 MHz. What is the wavelength of this radio signal?
Given f = 100 MHz = 10&sup8 Hz, c ≈ 3×10&sup8 m/s (§6).
Law the relation between wavelength, frequency, and speed for any wave: λ = c/f.
Solution λ = 3×10&sup8/10&sup8 = 3 m.
Answer 3 meters—roughly the length of a small antenna on a radio tower (a quarter-wave, 0.75 m, is a standard practical length).
2. A coil of 50 turns is in a field where the magnetic flux drops from 0.03 Wb to 0.01 Wb in 0.4 s. Find the magnitude of the induced EMF.
Given N = 50, ΔΦ = 0.01−0.03 = −0.02 Wb, Δt = 0.4 s.
Law Faraday's law (§4): ε = −NΔΦ/Δt.
Solution ε = −50·(−0.02)/0.4 = 2.5 V.
Answer 2.5 V.
3. X-ray radiation has a wavelength of about 1 nanometer (10⁻⁹ m). Is it the same physical nature as visible light, or different phenomena? Justify, referring to a specific conclusion from Maxwell's equations.
Law the derivation of the wave equation from Maxwell's equations (§6)—the unified nature of electromagnetic waves.
Solution Maxwell's equations contain no parameter that distinguishes “light” from “X-rays”—from them follows a single wave equation with a single characteristic speed c. Different names (radio, light, X-rays) are just different ranges of the same quantity: wavelength (or frequency).
Answer same nature—electromagnetic wave, the only difference being wavelength (for X-rays it is about 500,000 times shorter than for green light).
4. When you try to break a magnet into a “separate north pole” and “separate south pole,” why do you always end up with two new complete magnets? Which Maxwell equation forbids this from being violated?
Law Gauss's law for magnetic field (§3): ∇·B = 0.
Solution the zero on the right-hand side (unlike Gauss's law for E, §2, where charge density appears on the right) means: the magnetic field has no analogue of a source charge—field lines must be closed and cannot “terminate” on an isolated pole, no matter how many times you divide the magnet.
Answer Gauss's law for B—the absence of magnetic monopoles is embedded directly in the form of the equation.
5. A charging capacitor in a vacuum creates a magnetic field around itself, although there is no real current between its plates (charge simply accumulates on the plates). How does Maxwell's equations explain this?
Law Ampère-Maxwell law (§5), second term—displacement current.
Solution between the capacitor plates, the electric field rapidly increases as charge accumulates—this is ∂E/∂t, the “displacement current,” which according to the equation (§5) acts exactly like a real conduction current and creates a vortex magnetic field around the region between the plates.
Answer displacement current—this is precisely the term Maxwell added to the old Ampère's law to explain such cases.
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