Physics · Module 18
Four equations, written down in the 1860s, mostly by collecting other people's work and fixing one crack in it. Out of the crack came a wave made of nothing but field — and its speed, calculated from two numbers measured on a laboratory bench, turned out to be the speed of light.
You have, without anyone announcing it, spent four modules assembling the complete theory of electricity and magnetism. Let us lay the pieces out.
From Module 13. Electric charges make electric fields, and the field lines start on positive charge and end on negative charge. Draw any closed surface — a bag, a balloon, a box — and the net number of field lines poking out of it depends on nothing except how much charge is inside the bag. This is Gauss's law. It is the reason a charged metal shell has no field inside it, and the reason the Faraday cage works.
From Module 15. There is no such thing as a magnetic charge. Break a magnet in half and you do not get a loose north pole; you get two smaller magnets. Magnetic field lines never start anywhere and never end anywhere — they always close on themselves in loops. So the net number of magnetic field lines poking out of any closed bag is always exactly zero. Whatever goes in comes out.
From Module 16. A changing magnetic field creates a curling electric field, in empty space, whether or not there is a wire there to notice. This is Faraday's law, and by the end of the last module we had stripped the wire away and stated it as a fact about space itself.
From Module 15 again. An electric current creates a curling magnetic field around it — the field that wraps round a wire and makes a compass needle swing. This is Ampère's law.
Four statements. Two about where the fields come from, two about how each field curls around the other's activity. In 1861 James Clerk Maxwell gathered them up, looked hard at the fourth one, and found that it was wrong. Not slightly imprecise. Logically impossible. Fixing it took one extra term, and that one term produced light.
| In plain English | Name | Where you met it |
|---|---|---|
| Electric field lines begin and end on charges. Count the lines out of any closed surface and you have counted the charge inside. | Gauss's law for E | Module 13 |
| Magnetic field lines never begin or end. There are no magnetic charges. Every line that enters a closed surface leaves it again. | Gauss's law for B | Module 15 |
| A changing magnetic field makes an electric field that curls around it. | Faraday's law | Module 16 |
| An electric current makes a magnetic field that curls around it — and so does a changing electric field. | Ampère–Maxwell law | Module 15 (as B = µ0I/2πr), restated as a loop law in 15.2, plus the fix below |
That italicised clause in the last row is Maxwell's entire contribution, and the rest of this module is about where it came from and what it did.
Ampère's law is the loop form of the wire result you met in Module 15. Module 15 gave you the field at a distance r from a long straight wire carrying current I, B = µ0I/(2πr), where I is that current. Multiply that field by the length of the circle it wraps, 2πr, and the r cancels: B × 2πr = µ0I. The circumference grows exactly as fast as the field weakens. Ampère's law is the general statement that this is not a coincidence — draw any closed loop in space; the magnetic field, added up all the way round that loop, is µ0 times the current passing through the loop.
That word "through" is doing more work than it looks. A loop is just a circle of empty space. Currents do not pass through circles; they pass through surfaces. So what the law really says is: pick any surface whose edge is that loop, and count the current crossing it.
And here is the trouble. A single loop is the edge of infinitely many different surfaces. Hold a wire ring and dip it in soap solution: the film that forms is one such surface. Now blow gently — the film bulges out into a dome, and later into a long bag. All of those are surfaces with the same edge. Ampère's law had better give the same answer for every one of them, or it is not a law at all, it is a lottery.
For ordinary steady currents it does. A wire carrying 3 A pierces the flat film; bulge the film into a dome and the wire still pierces it, because the current has nowhere else to go — current in a steady circuit cannot stop or pile up anywhere.
Now put a capacitor in the circuit.
A capacitor (Module 13) is two metal plates with a gap between them. When you connect it to a battery, current flows in the wires and charge piles up on the plates — but no charge crosses the gap. The gap is insulating. Charge arrives on one plate, leaves the other, and the space between them carries no current at all.
So: draw your loop around the wire, just to the left of the capacitor. Take the flat surface: the wire pierces it, current I, and Ampère's law says the magnetic field around the loop is proportional to I. Now take the bulging surface, stretched to the right so that it passes between the plates instead of cutting the wire. Nothing pierces it. No current at all. Ampère's law says the magnetic field around the very same loop is zero.
The field around a loop cannot be both I and zero. Something is missing.
The bulging surface passes through the gap between the capacitor plates, where no charge crosses. But while the capacitor charges, something in that gap is definitely changing.
What is it, and what would you have to add to Ampère's law to save it?
As Q, the charge on the top plate, piles up, the electric field between them grows: E = Q/(ε0A), straight from Gauss's law in Module 13. So ΦE, the electric flux through the bulging surface, equal to EA = Q/ε0, is growing too.
Maxwell's move was to notice exactly how fast it grows. The flux is proportional to the charge on the plate — one extra coulomb on the plate adds 1/ε0 to the flux — so the rate of change of flux is just the rate at which charge arrives, which is the current: ε0 dΦE/dt = dQ/dt = I. That is not approximately the current in the wire. It is the current in the wire, to the last decimal place, because the charge arriving on the plate is the charge that flowed down the wire. So if we simply add this quantity to Ampère's law and call it a current, the two surfaces agree perfectly — the flat one counts real charge in motion, the bulging one counts growing electric flux, and the two are numerically identical because they are the same physical event described from two places.
Go back to Bench 18A and switch the select to Maxwell's version. The bulging surface's reading jumps from zero to exactly the wire current, and stays locked to it for the whole of the charging curve — a curve which, incidentally, is the RC exponential you met in Module 14, computed here from scratch rather than taken on faith.
The name is a historical accident and an unhelpful one. Maxwell thought space was filled with an elastic medium — the "aether" — and imagined charges being physically displaced inside it. Whether that medium existed was still an open question at the time. The term stayed anyway, and the physics it describes is completely real.
It is fair to ask whether Maxwell just bolted on whatever term made the arithmetic work. In a sense yes — but the constraint was much tighter than it sounds. The added term had to (a) come out equal to the ordinary current in the wire — real charge in motion — wherever charge cannot accumulate or disappear, (b) vanish for steady fields so that everything already tested still worked, and (c) be built only out of the fields themselves. There is essentially one expression that does all three. And the moment you write it down, the equations acquire a solution nobody had asked for: a wave. That is the mark of a real discovery rather than a patch. A patch fixes the hole you found. Maxwell's patch answered a question nobody had thought to ask.
With the fix in place, look at the last two equations side by side and notice what has just happened to their shape.
Each field, by changing, makes the other. And now ask the obvious question: what if there is nothing else? No charges, no wires, no magnets. Just a patch of empty space in which, for whatever reason, an electric field is changing.
The bench below shows the finished wave: an electric field pointing up and down, a magnetic field pointing in and out of the page, both moving to the right together. What matters is not that it looks like a wave — you have seen waves since Module 7. What matters is the panel underneath, where the bench measures the slope of the E-wave in space and the rate of change of the B-wave in time, at the same instant and the same point, using nothing but the numbers on the screen. Faraday's law says a changing B makes an E that curls around it; on a plane wave travelling to the right, that curling relation translates directly into a number — the slope of E in space must equal the rate of change of B in time. Watch them be equal.
Two features of the picture deserve a note, because both surprise people.
One warning about the step-by-step story above. It made the fields sound like two runners, one always a little behind the other. That was the price of telling it one step at a time. In the finished wave the two fields sit at the same place and rise and fall together — the handing-off happens everywhere at once, not in a queue.
E and B are in step, not opposite. Both peak at the same place and both vanish at the same place. Students often expect them to be a quarter-cycle apart, by analogy with a pendulum swapping kinetic and potential energy. An electromagnetic wave does not work that way; it is not sloshing energy back and forth between two stores. At the nodes, there is simply no energy there at that instant — it has not gone somewhere else, it is just not there yet; the energy is in the crests, and the crests are moving.
Nothing is waving. In Module 7 a wave was a disturbance in something: a rope, air, water. Here there is no rope. The quantity that goes up and down is the field itself, and the field is not made of anything. This bothered nineteenth-century physicists so much that they invented the aether to have something for light to wave in, and spent forty years failing to detect it. Module 19 will tell you how that story ended.
Now the payoff. Maxwell's equations contain exactly two constants of nature, and neither of them has anything to do with light.
ε0, the permittivity of free space. You met it in Module 13, buried in Coulomb's law: the force between two charges is q1q2 / (4πε0r²). It says how strongly charges push each other. You measure it by hanging two charged balls on threads and measuring how far apart they hang. Its value is 8.854 × 10−12 in SI units. No light anywhere in that experiment.
µ0, the permeability of free space. You met it in Module 15, in the field around a wire: B = µ0I/(2πr). It says how strongly a current makes magnetism. You measure it by running currents through two parallel wires and measuring the force between them. Its value is 4π × 10−7, near enough exactly. Again: no light. A battery, two wires, a balance.
Grind through the algebra of the two curl equations — it is a page of calculus that you will do properly in a couple of years — and you find that the bootstrap pattern travels at a speed given by
Put the numbers in. 1 ÷ √(4π × 10−7 × 8.854 × 10−12) = 2.998 × 108 metres per second.
In 1849 Hippolyte Fizeau had measured the speed of light by bouncing a beam off a mirror eight kilometres away through the teeth of a spinning cogwheel. He got about 3.15 × 108 m/s. Later measurements tightened it to 2.998 × 108.
Maxwell had calculated the speed of light out of two charged balls hanging on a thread and a pair of current-carrying wires. Here is what he wrote. Two words need a key: transverse undulations means sideways waves — the field wiggles at right angles to the direction of travel; hypothetical medium is the aether, which Maxwell thought must exist to carry the wave. He had every reason to think so at the time.
The velocity of transverse undulations in our hypothetical medium, calculated from the electro-magnetic experiments of MM. Kohlrausch and Weber, agrees so exactly with the velocity of light calculated from the optical experiments of M. Fizeau, 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. James Clerk Maxwell, 1862
"We can scarcely avoid the inference." That is the sound of a man trying to stay calm. Light — the thing you see with, the thing that comes off the Sun, the subject of Modules 9 and 12 — is an electromagnetic wave. Optics stopped being its own subject that day and became a branch of electromagnetism.
Drag either constant and the pulse visibly slows or quickens, and the timed speed follows 1/√(µ0ε0) every time. Be clear about what this bench is and is not. It cannot prove the formula: a simulation only ever shows you the rules it was built from. What it does is make the claim touchable — it lets you feel that the speed of light is not an independent fact bolted on to the universe, but a consequence of how hard charges push and how hard currents pull. The proof is the algebra in 18.4, and the check is Fizeau's cogwheel.
A red laser (long wavelength) and a gamma ray from a radioactive nucleus (wavelength a trillion times shorter) are both fired across a vacuum.
Which arrives first?
Together, and to a spectacular precision. In 2017 astronomers watched two neutron stars collide 130 million light-years away. The gravitational wave arrived, and 1.7 seconds later so did the gamma rays. After 130 million years of travelling, less than two seconds' difference — and most of that 1.7 s is thought to be a delay at the source, not in the journey. That pins the speed of gamma rays and the speed of gravitational waves to agree to better than one part in 1015 of c. Now the frequency question: gamma-ray bursts, supernovae, and pulsars have all been observed simultaneously across wavelengths from radio to gamma ray, and the arrival times of different colours agree to the precision of the measurements. Every frequency of light really does run at exactly c.
Note carefully that this is a statement about vacuum. In glass or water, different colours do travel at slightly different speeds — that is dispersion, and it is why prisms make rainbows (Module 9). But that is the light interacting with the atoms of the material, not a property of the wave itself. In empty space there are no atoms to slow anything down, and every frequency runs at exactly c.
Here is the consequence that reorganised physics. Maxwell's equations say nothing whatever about the frequency of the wave. Any frequency at all is allowed, and every one of them travels at c.
Visible light occupies wavelengths from about 400 nanometres (violet) to 700 nanometres (red) — less than one octave, a single doubling of frequency. That is not because there is anything special about that band. It is because your eye evolved under a star whose output peaks there, in an atmosphere that happens to be transparent there. The physics extends in both directions without limit. The bench below covers eighteen powers of ten in wavelength — a factor of a million million million, which is about sixty doublings. Visible light is one of them.
Sweep the slider slowly and read what each band does. Everything on that scale is the same object — an electric field and a magnetic field taking turns — differing only in how fast the turns come.
A few things worth noticing as you sweep:
Look at the door of a microwave oven and you will see a metal sheet punched with small holes. Microwaves in the oven have a wavelength of about 12 cm; the holes are a couple of millimetres across. To a 12 cm wave the mesh behaves as a solid metal wall — it is a Faraday cage (Module 13) and the wave cannot get through. To visible light, wavelength 0.0005 mm, the holes are enormous, and it strolls through unhindered. So you can see your food and not cook your face. The whole design rests on the one fact that both things are the same kind of wave, differing only by a factor of about 200 000 in wavelength.
Maxwell died in 1879 at forty-eight, without ever seeing an electromagnetic wave deliberately produced. Eight years later Heinrich Hertz built a spark gap — two metal balls separated by a small gap, connected to a high-voltage coil — in a Karlsruhe lecture room. When the voltage is high enough, a spark jumps the gap: a charge is violently accelerated, and by everything above, that launches a wave. Across the room, a small loop of wire with a gap in it sparked in sympathy. There was no connection between them. Hertz measured the wavelength by finding standing waves in the room — the same superposition of a wave with its own reflection you built in Module 7 — multiplied by the frequency, and got c.
Asked what use it was, Hertz is supposed to have said: "It's of no use whatsoever. This is just an experiment that proves Maestro Maxwell was right." Within twenty years the technology was called radio.
So what does it take to launch one?
A single electron is moving through empty space in a perfectly straight line at a perfectly constant speed — very fast, but not changing.
Does it radiate? Does it send out an electromagnetic wave?
A charge moving steadily carries its field along with it, the way a car carries its headlight beam — a fixed pattern, dragged through space. A fixed observer does see the field changing as the charge sweeps past; that is just the Coulomb field sliding by, not a new wave being launched. Sit alongside the moving charge and you see nothing changing at all: same field, same shape, forever. If nothing is changing, the bootstrap never starts.
To radiate you must accelerate. And there is a marvellous way of seeing why, which is the last bench in this module and the one worth playing with longest.
Every field line from a charge is, in effect, a piece of news: "there is a charge over here". How fast can that news travel? We found in section 18.4 that the bootstrap wave travels at c — any change in a field propagates at c. So that news cannot travel faster than c. So when a charge suddenly jerks sideways, the field lines close to it swing round to point at the new position straight away — but the distant parts of those lines have not heard yet, and still point at where the charge used to be. The two halves of each line have to join up somewhere, and where they join there is a sharp sideways kink.
That kink is real, it has a transverse electric field in it — pointing across the direction of travel, not along it — and it races outward at c, because the news it carries travels at c. The kink is the radiation. Jerk the charge back and forth continuously and you send out kink after kink: a wave.
Push the "news" word further than it wants to go, though. Real news carries content someone chose to send; a field line carries none of that — no sender, no message, nothing to interpret. All a field line ever reports is “the charge that made me has moved,” and even that report is really the kink itself, a physical, measurable field, not a signal riding on top of one. The word is a crutch for talking about the speed limit; the physics is the kink.
Where the picture breaks down. Field lines are not objects; they have no substance beyond the field they represent. The "kink" is a region of transverse field — that is real — but the image of lines being dragged and bent is an aid to picturing the physics, not the physics itself. If you need to be sure what is actually happening, trust the equations. The bench (option: one sudden jerk) shows the field directly: every line is drawn from where the charge was, delayed by distance ÷ c, and the "kink" is simply where two such retarded lines meet.
Two things fall out of that picture that are otherwise just facts to memorise.
An antenna is silent along its own length — and, for the same reason, deaf along it too. Look at the bench: the kinks are large out to the sides and vanish entirely along the direction the charge is shaking. A vertical aerial radiates outward in all horizontal directions and sends nothing straight up. This is why the mast on a radio tower is vertical and why holding a phone in a particular way can cost you a bar of signal.
The wave is polarised along the shaking. The transverse kick in each kink lies in the direction the charge moved. A vertical aerial makes vertically polarised radio waves, and a receiving aerial must be vertical too to feel them — which is why the aerials on a rooftop and on a car are so carefully oriented. This is exactly the polarisation of Module 12, and now you can see where it comes from: the direction a charge was shaken, remembered by the wave for the rest of its life.
If you have a pair of polarising sunglasses, go outside on a clear evening about an hour before sunset. Look at a patch of blue sky roughly 90° away from the Sun and rotate the glasses. The sky visibly darkens and brightens. That patch of sky is strongly polarised — and now you know exactly why. Sunlight is shaking electrons in air molecules, those shaking electrons re-radiate like tiny aerials, and an aerial radiates nothing along the direction it shakes. Looking at right angles to the Sun, you only receive from electrons shaking across your line of sight, so what reaches you is polarised in one direction. That is Module 9's blue sky and Module 12's polarisation and this module's antenna, all being the same sentence.
No sunglasses? The screen of most laptops and phones emits polarised light. Tilt your head 90° while looking at a phone screen through a pair of sunglasses, or look at your laptop screen reflected in a table.
Maxwell's four equations are usually described as the summit of classical physics, and they are. They unified electricity, magnetism and optics into one subject. They predicted radio. They are still exactly correct today as a description of fields — quantum mechanics did not overturn them, it re-founded them.
But they came with a bill, and it was enormous, and it was not paid for forty years.
Maxwell's equations give one definite speed for light, c = 1/√(µ0ε0). Every other speed you have met in this course is relative to something: 60 km/h relative to the road, 900 km/h relative to the air.
Relative to what is light supposed to travel at c? Chase a light beam in a very fast spacecraft and what should you measure?
Look at the equations again. µ0 and ε0 are properties of empty space. There is no velocity anywhere in them, no reference object, nothing to measure "relative to". Taken at face value, they say that anybody, moving anyhow, will measure light at 2.998 × 108 m/s.
That cannot be right. If you run after a cricket ball at 20 m/s and it is doing 30, you see it doing 10. This is arithmetic. It has never been wrong before.
The nineteenth century took the obvious escape: there must be a medium — the luminiferous aether — and Maxwell's c is the speed relative to it, just as the speed of sound is relative to the air. Then the Earth, hurtling round the Sun at 30 km/s, must be moving through this aether, and a sufficiently careful experiment should detect the "aether wind".
In 1887 Michelson and Morley built the most sensitive apparatus of the age to measure it. They found nothing. Not a small effect: nothing at all, in every orientation and at every season.
And there is one more clue, which we have been carrying since Module 17 without cashing it in. The rod on the rails gave the same EMF from two entirely unrelated arguments — a magnetic force on moving charges, or a changing flux — and there was no reason for the two to agree. Module 15 ended by hinting that magnetism might be relativity in disguise. Bench 16A's magnet and coil poses the same puzzle in miniature: move the magnet and you would say Faraday's law made an electric field; move the coil instead and you would say the electrons in it felt qvB. The needle does not care which of you is moving. It reads the same either way. Two different explanations, one experimental result, every single time.
Something is badly wrong with our ideas about motion, and it has now been flagged three times. The next module fixes it, and the fix is not gentle.
| Idea | What it says |
|---|---|
| Gauss for E | Field lines start and end on charge. Flux out of a closed surface counts the charge inside. (Module 13.) |
| Gauss for B | No magnetic charges. Magnetic field lines always close on themselves. (Module 15.) |
| Faraday | Changing B makes curling E, in empty space. (Module 16.) |
| Ampère–Maxwell | Current makes curling B — and so does a changing E. The added term is ε0 dΦE/dt, forced by the capacitor puzzle. |
| The bootstrap | Each field, changing, makes the other. The pattern sustains itself and must keep moving. No medium is required; the careful search for one found nothing. |
| The speed | c = 1/√(µ0ε0) = 2.998 × 108 m/s, from two constants measured with charges and wires. Light is an electromagnetic wave. |
| The spectrum | Any frequency is allowed and all travel at c. Radio to gamma is one family; visible light is less than one octave of it. |
| Radiation | Only accelerating charges radiate. The kink in a field line, travelling out at c, is the wave; it is polarised along the shaking and vanishes along the aerial's axis. |
And one unpaid bill: the equations name a speed with nothing to measure it against. Module 19.