Static cling, compasses, generators, radio and sunlight all follow from four short statements about how fields spread out and how they swirl. Learn what each one says, find the single term Maxwell added, and watch the speed of light fall out of the arithmetic.
Common mix-up: Maxwell didn’t discover “Maxwell’s equations.” Three of the four were already known from Gauss, Faraday and Ampère, and the fourth (no lone magnetic poles) was simply how magnets behave. Maxwell’s own contribution was one extra term, the displacement current, and it is the term that makes light possible. The tidy four-line vector form came later still, from Oliver Heaviside in the 1880s.
A field is a value at every point in space: an arrow with a size and a direction. The electric field E is the force per coulomb a test charge would feel there; the magnetic field B is the matching thing for moving charges. Chapter 1 and Chapter 2 built both from scratch.
Faced with a field filling space, there are really only two useful questions to ask about it. The first: how much of it flows out through a closed surface? Picture the field lines as streams and wrap a bag around a region. If more lines leave than enter, something inside is producing them. That net outflow is called flux.
The second: how much does it swirl around a closed loop? Walk around a loop and add up how much the field pushes you along your path. If the total isn’t zero, the field circulates. That total is called circulation.
Two fields, two questions: four equations. Two are flux laws (Gauss for E, Gauss for B) and say what makes field lines start and end. Two are circulation laws (Faraday, Ampère–Maxwell) and say what makes field lines curl. That’s the whole map. Everything else in this site, from antennas to rainbows, is these four applied to different situations.
A useful everyday check: water from a garden sprinkler has flux (it comes out of the head) but no circulation. Water draining from a bathtub has circulation (it spirals) but, around the drain, little net flux through a bag that doesn’t include the plughole. Real fields can have either, both, or neither.
A theorem of Helmholtz says that a vector field that dies away at infinity is completely fixed once you know its divergence (the flux per unit volume, at every point) and its curl (the circulation per unit area, at every point). So the two questions are not just convenient; together they are enough. Maxwell’s equations give exactly that information for E and for B: the divergence of each, and the curl of each.
The integral forms used in this chapter and the differential forms at the end say the same thing. The divergence theorem turns a surface integral of flux into a volume integral of divergence, and Stokes’s theorem turns a loop integral of circulation into a surface integral of curl.
Choose a law. In the two Gauss modes, drag the charge or the magnet in and out of the dashed bag (or focus the canvas and use the arrow keys). In Faraday and Ampère modes, use the slider, and drag the dashed loop in Ampère mode to resize it.
In plain words: electric field lines start on positive charges and end on negative ones, and the number of lines leaving any closed surface tells you exactly how much charge is inside. Nothing outside the surface affects the total, however close it sits.
That last part is surprising. A charge just outside the bag certainly pushes field through the bag, but every line it sends in also comes back out somewhere else. Its net contribution is zero. The bag can be a sphere, a cube or a potato; the shape doesn’t matter either.
Gauss’s law is Coulomb’s law said in a smarter way. Put a sphere of radius r around a point charge q. By symmetry the field has the same size everywhere on it and points straight out, so the flux is just E times the sphere’s area, 4πr². Set that equal to q/ε₀ and you get E = q/(4πε₀r²): the inverse-square law, with 1/(4πε₀) = 8.988 × 10⁹ N·m²/C². A 1 nanocoulomb charge makes a field of 8.99 volts per meter at 1 meter.
A real example: the charge on the Earth. On a clear day there is an electric field of about 100 volts per meter pointing down at the ground. Wrap a Gauss bag around the planet just above the surface: flux in = 100 V/m × 5.10 × 10¹⁴ m² of surface. Multiply by ε₀ and the Earth turns out to carry about −450,000 coulombs. Spread over the surface that is only 0.9 nanocoulombs per square meter, about 5.5 billion extra electrons per square meter, which is why you don’t notice it. (The fair-weather field varies with weather and place; 100 V/m is a typical value.)
Gauss’s law also explains why a car or an airplane is a safe place in a lightning storm, and why the charge on a metal object always sits on its outer surface. Inside a conductor in equilibrium, E is zero. A Gauss bag drawn anywhere inside the metal therefore has zero flux, so it must enclose zero net charge. Any excess charge has nowhere to live except the skin.
Gauss’s law works only because the field of a point charge falls off as exactly 1/r². The area of a sphere grows as r², so field times area stays constant: the same flux at every radius. If the force went as 1/r^2.01, flux would leak away with distance, and the law would fail.
This makes Gauss’s law a precision test. If Coulomb’s law weren’t exactly inverse-square, a charged hollow metal shell would have a small field inside it. Experiments since Cavendish in the 1770s have looked for one; modern versions limit any deviation in the exponent to below about one part in 10¹⁶. The same experiments set an upper bound on the mass of the photon.
Gauss’s law holds for moving charges and changing fields too. Coulomb’s formula does not: the field of a moving charge is squashed and delayed (see Chapter 4). That is why Gauss, not Coulomb, is one of the four.
In plain words: there is no magnetic charge. Magnetic field lines never start or stop; they always close on themselves. So the magnetic flux out of any closed surface, however you draw it, is exactly zero.
This is the one law that says what isn’t there. Outside a bar magnet, lines leave the north end and curve around to the south end. That makes it look as if the north pole is a source, like a positive charge. But follow a line into the south end and it keeps going, straight through the magnet’s body, and comes out at the north end again. Wrap a bag around just the north pole and every line that leaves through the air comes back in through the iron.
A real example: cutting a magnet. Snap a fridge magnet or a compass needle in two and you get two complete magnets, each with a north and a south. Cut again and again, down to a single iron atom, and the atom is still a tiny magnet: its magnetism comes from electron spins and orbits, which are circulating charge, not magnetic charge.
The same rule shapes the Earth’s field. Field lines leave the southern hemisphere, arc through space, and dive back in near the north geographic pole, about 25 microtesla at the equator and 65 near the poles. Every line that goes out comes back.
Physicists badly want this law to be broken. In 1931 Paul Dirac showed that if even one magnetic monopole existed anywhere in the universe, electric charge would have to come in whole-number multiples of a basic unit, which it does. Grand unified theories predict monopoles too. Searches in cosmic rays, moon rock and collisions at the Large Hadron Collider (the MoEDAL experiment) have found none.
If a magnetic charge qm existed, this equation would read ∮ B · dA = μ₀ qm, and Faraday’s law would gain a “magnetic current” term. The four equations would become perfectly symmetric between E and B.
Dirac’s argument links the two kinds of charge: the product of the smallest electric charge e and the smallest magnetic charge g must be a whole multiple of h/μ₀ (in these units, e·g = n·h/μ₀). With e = 1.602 × 10⁻¹⁹ C, the smallest allowed magnetic charge would be enormous compared with e, which is one reason a monopole would be so easy to spot if one ever flew through a detector.
Because ∮ B · dA = 0 for every surface, B can always be written as the curl of another field, the vector potential A. That is the starting point for how quantum mechanics handles magnetism.
In plain words: whenever the magnetic flux through a loop changes, an electric field circulates around that loop. If the loop is a wire, that circulating field pushes charges along it, and a current flows. The faster the flux changes, the bigger the push, which is called the electromotive force or EMF.
The key word is changing. A strong steady magnet sitting inside a coil does nothing. Move the magnet, turn the coil, or switch the field up or down, and current flows. Michael Faraday found this in 1831 by switching current on and off in one coil and watching a needle twitch in a second coil wound on the same iron ring.
The minus sign in the equation is Lenz’s law: the induced current always flows so as to oppose the change that caused it. Push a magnet toward a coil and the coil’s own field pushes back. That’s why a generator gets harder to turn when you draw more current from it: energy in, as work against that opposition, equals energy out.
A real example: a generator. Spin a coil of 100 turns, 10 cm × 10 cm, at 60 revolutions per second in a field of 0.1 tesla. The flux through it swings sinusoidally, and the peak EMF is N·B·A·ω = 100 × 0.1 T × 0.01 m² × (2π × 60 per second) = 37.7 volts. Every power station, from a coal plant to a wind turbine, is a scaled-up version of that coil.
The same law runs transformers (a changing current in one coil induces voltage in a second), induction cooktops (tens of kilohertz of changing field induce currents that heat the pan itself), wireless phone chargers (around 100 to 200 kilohertz), electric-guitar pickups, card readers and the regenerative brakes on an electric car.
In electrostatics, the electric field has no circulation, so the voltage between two points is the same whichever path you take between them. Faraday’s law breaks that. Where a magnetic field is changing, E has curl, and the work done carrying a charge from A to B depends on the route.
A famous demonstration (MIT’s Walter Lewin made it widely known) puts two different resistors in a ring around a changing magnetic field and connects one voltmeter to each side. The two meters, touching the same two points, read different values. Nothing is broken; the meters’ leads simply enclose different amounts of changing flux. Kirchhoff’s loop rule assumes no changing flux through the circuit, and fails when there is some.
The flux rule covers two physically different cases with one formula: a loop that moves through a steady field (the push comes from the magnetic force qv × B on the moving charges) and a stationary loop in a changing field (the push comes from a genuinely induced electric field). Einstein’s 1905 relativity paper opens by pointing out that these are the same thing seen from different frames.
In plain words: a magnetic field circulates around any current. The circulation around a loop equals μ₀ times the current passing through the loop. Maxwell’s addition: a changing electric field passing through the loop counts as current too.
The first half is Ampère’s law, from the 1820s, right after Hans Christian Ørsted noticed a compass needle swing when he switched on a current. For a long straight wire it gives B = μ₀I/(2πr). Real numbers: 1 amp at 5 cm makes 4.0 microtesla, about a twelfth of the Earth’s field, enough to nudge a compass. A power line carrying 1,000 amps makes about 10 microtesla 20 meters away.
The capacitor paradox. Ampère’s law lets you pick any surface whose edge is your loop. Take a loop around the wire leading into a charging capacitor. A flat disc spanning the loop is pierced by the wire: current I. Now balloon the surface out so it passes between the plates instead. No wire goes through it and no charge crosses the gap, so the current through it is zero. Same loop, two answers. Something is missing.
Maxwell’s fix: between the plates the electric field is growing as charge piles up. Give that changing field the status of a current, ε₀ times the rate of change of electric flux, and the two surfaces agree exactly. He called it displacement current. It isn’t charge moving; it is the field itself changing.
Why nobody had noticed: it takes a lot of changing field to amount to much current. For plates 10 cm square, carrying 1 amp of displacement current needs the field in the gap to rise by 1.13 × 10¹³ volts per meter every second. In any ordinary circuit the effect hides inside the capacitor and does exactly what the wire’s current would have done. It only becomes the star of the show in empty space, where there is no current at all, which is where light lives.
In differential form Ampère’s original law reads ∇ × B = μ₀J. Take the divergence of both sides. The divergence of a curl is always zero, so this demands ∇ · J = 0: current can never pile up anywhere. But a charging capacitor plate is exactly a place where current piles up. The law contradicts charge conservation.
Charge conservation says ∇ · J = −∂ρ/∂t, and Gauss’s law says ρ = ε₀ ∇ · E. Together: ∇ · (J + ε₀ ∂E/∂t) = 0. So the combination J + ε₀ ∂E/∂t is always divergence-free, and it is the right thing to put on the right-hand side. That’s the Ampère–Maxwell law, ∇ × B = μ₀J + μ₀ε₀ ∂E/∂t. Maxwell actually reached it in 1861–62 from a mechanical model of whirling cells in the ether; the conservation argument is the clean modern route.
Press Charge or Discharge. The wire carries conduction current; the gap carries displacement current, and the two readouts always match. Slide the resistor to change how fast it happens (the animation is slowed to make it visible; the readouts show real time).
Plates 10 cm × 10 cm, 1 mm apart (88.5 picofarads), 9-volt battery. Time constant = R × C.
Now read the two circulation laws in empty space, where there are no charges and no currents. Faraday: a changing B makes a curling E. Ampère–Maxwell: a changing E makes a curling B. Each change produces the other. Without Maxwell’s term, a changing E would make nothing, and the chain would break at the first link.
With it, a disturbance can carry itself. Wiggle a charge; its changing E field makes a changing B nearby, which makes a changing E a little farther out, and so on. Once launched, the ripple doesn’t need the charge anymore. It travels on its own, through empty space, forever. Chapter 4 shows that birth in detail.
How fast does it go? Imagine a flat wall of field sweeping along at speed v, with E pointing up and B pointing sideways. Draw a small loop straddling the wall’s front edge. As the wall advances, magnetic flux through the loop grows at a rate set by B and v, and Faraday’s law says that requires E = vB. Draw another loop, and the Ampère–Maxwell law says the growing electric flux requires B = μ₀ε₀vE. Put one inside the other: E = v × (μ₀ε₀vE), so v² = 1/(μ₀ε₀). The speed is not a free choice. The equations fix it.
The arithmetic. μ₀ε₀ = 1.25664 × 10⁻⁶ × 8.854 × 10⁻¹² = 1.1126 × 10⁻¹⁷ s²/m². Its square root is 3.3356 × 10⁻⁹ s/m. One over that is 2.998 × 10⁸ m/s: 299,800 kilometers per second, the speed of light to four figures. (The last digits drift a little only because the constants here are rounded.)
Neither constant has anything to do with light. ε₀ comes from measuring the force between charges or the charge a capacitor holds; μ₀ from the force between two wires carrying current. In 1862, Maxwell plugged in electrical measurements made by Wilhelm Weber and Rudolf Kohlrausch in 1856 and got about 310,700 km/s. Hippolyte Fizeau had measured light in 1849 with a spinning toothed wheel: about 314,900 km/s. A 1.3% match. Maxwell wrote, in italics, that “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.” His 1865 paper derived the wave equation properly, and Heinrich Hertz made and detected the waves in a lab in 1887–88.
Shrink each surface and loop down to a point and the integral laws become statements about divergence (∇·) and curl (∇×) at every point:
∇ · E = ρ / ε₀
∇ · B = 0
∇ × E = − ∂B/∂t
∇ × B = μ₀ J + μ₀ε₀ ∂E/∂t
In vacuum set ρ = 0 and J = 0. Take the curl of Faraday’s law. The left side is ∇ × (∇ × E) = ∇(∇ · E) − ∇²E = −∇²E, because ∇ · E = 0 with no charge. The right side is −∂/∂t (∇ × B) = −μ₀ε₀ ∂²E/∂t². So:
∇²E = μ₀ε₀ ∂²E/∂t²
That is the standard wave equation, ∇²f = (1/v²) ∂²f/∂t², with v² = 1/(μ₀ε₀). B obeys the identical equation. A solution is E = E₀ sin(kx − ωt) pointing along y, with B = (E₀/c) sin(kx − ωt) along z: both perpendicular to the travel direction (transverse), in step, and with E = cB.
In materials, ε₀ and μ₀ become ε = εrε₀ and μ = μrμ₀, and waves slow to v = c/n with refractive index n = √(εrμr). For glass at visible frequencies εr ≈ 2.25, so n ≈ 1.5. Water is subtler: εr ≈ 80 for slow fields but only about 1.77 at light frequencies (n ≈ 1.33), because its molecules can’t reorient fast enough to follow light.
A modern footnote. Since 1983 the meter has been defined so that c is exactly 299,792,458 m/s, and since the 2019 SI redefinition μ₀ is a measured quantity (1.25663706… × 10⁻⁶ N/A²), with ε₀ = 1/(μ₀c²). The equation now runs backwards in practice: c is fixed and the constants are measured. The physics is unchanged.
Turn ε₀ and μ₀ up or down (multiples of their real values) and watch a pulse race from the Earth to the Moon against one moving at our real c. Try the presets; “4× ε₀, ¼ μ₀” is a trick question.
| Law | Integral form | Says | You see it in | Credit |
|---|---|---|---|---|
| Gauss | ∮E·dA = Q/ε₀ | Charge is the source of E; flux out = charge in | Static cling, lightning-safe cars, charge on the outside of metal | Gauss, 1830s (Coulomb 1785) |
| Gauss (B) | ∮B·dA = 0 | No magnetic charge; B lines always close | Cut magnets stay two-poled; Earth’s field loops | A fact of magnets; no monopole yet found |
| Faraday | ∮E·dl = −dΦB/dt | Changing B makes a curling E | Generators, transformers, induction hobs, wireless charging | Faraday, 1831 |
| Ampère–Maxwell | ∮B·dl = μ₀I + μ₀ε₀ dΦE/dt | Current, and changing E, make a curling B | Electromagnets, motors, and (via the second term) every radio wave and light beam | Ampère 1826; Maxwell’s term 1861–65 |