Moving charge makes magnetism, and changing magnetism pushes charge around. Those two facts, found eleven years apart, run every motor, generator, transformer and wireless charger on Earth.
Common mix-up: a compass's north end points toward the Arctic, so people assume there's a magnetic north pole up there. Opposites attract, so it's really a magnetic south pole sitting near geographic north. The name stuck to the place, not the physics.
For centuries, magnetism and electricity looked like separate mysteries: lodestones and compasses on one side, sparks and amber on the other. In the spring of 1820, Hans Christian Ørsted was lecturing in Copenhagen with a battery, a wire and a compass on the bench. When the current flowed, the needle swung. When it stopped, the needle swung back.
The strange part was the direction. The needle didn't point at the wire or away from it. It turned sideways, so that compasses placed all around the wire lined up in circles. A current wraps a magnetic field around itself. Within weeks André-Marie Ampère in Paris had shown that two wires carrying current pull on each other, with no magnets involved at all.
The direction comes from the right-hand rule: point your right thumb along the current, and your fingers curl the way the field circles. The strength falls off simply with distance, not distance squared, because a long wire is a line, not a point.
Magnetic fields are measured in tesla (T). Ten amps, like a space heater draws, makes 200 microtesla one centimeter from the wire: four times Earth's field, which is why a compass held near a heavy cable goes haywire. A meter away it's down to 2 µT. Even 1,000 amps on an overhead power line gives only about 20 µT on the ground 10 meters below.
Coil the wire into a spring shape (a solenoid) and the circles from every turn add up inside into a strong, even field along the axis. That's an electromagnet. An MRI scanner is a solenoid wound from superconducting wire, carrying hundreds of amps with no resistance at all, to reach 1.5 or 3 tesla.
Walk around any closed loop and add up the component of B along your path at each step. Ampère's law says the total equals μ₀ times the current threading the loop: ∮ B · dl = μ₀ I. For a circle of radius r around a straight wire, the field is the same all the way round, so B × 2πr = μ₀ I, which gives the formula above.
For a long solenoid with n turns per meter, the same trick gives B = μ₀ n I inside. A coil with 1,000 turns per meter carrying 1 amp makes 1.26 millitesla. Stuffing an iron core inside multiplies that by hundreds or thousands, which is why electromagnets in scrapyards have iron cores.
Ampère's law in this form only holds for steady currents. Maxwell found the missing piece, a changing electric field acting like a current, and that fix is what predicts light. See Chapter 3.
A magnetic field pushes on a charge only if the charge is moving, and the push is always at right angles both to the motion and to the field. Sit still in a magnetic field and you feel nothing. Move along the field lines and you still feel nothing. Move across them and you get shoved sideways.
Because the push is always sideways to the motion, it can bend a path but never speed it up or slow it down. In physics language, a magnetic force does no work. A charge in a uniform field just goes round in a circle, forever, at constant speed. A faster particle swings wider; a stronger field pulls the circle tighter; a heavier particle is harder to turn.
An electron moving at 1,000 km/s in Earth's 50 µT field circles with a radius of 11.4 centimeters. A proton at the same speed is 1,836 times heavier and needs 209 meters. An electron's cyclotron frequency is 28 gigahertz per tesla: 1.4 megahertz in Earth's field. Old tube televisions steered their electron beam with magnetic coils, and mass spectrometers still sort molecules by how sharply a field bends them.
The aurora is the same physics on a planetary scale. The solar wind streams charged particles past Earth at around 400 kilometers per second. The magnetic field won't let them cross its lines easily, so they spiral along them instead, and the lines funnel down toward the magnetic poles. Some 100 to 300 kilometers up, the particles crash into the thin upper air. Oxygen glows green (557.7 nm) and red (630.0 nm); nitrogen adds blues and purples.
A current is just a lot of moving charges, so a wire carrying current in a magnetic field feels a sideways force too: F = I L × B. That's every electric motor. And two parallel wires each feel the other's field: currents in the same direction attract, opposite directions repel. Until 2019, the ampere was defined by exactly this: 1 amp in each of two long wires 1 meter apart gives a force of 2 × 10⁻⁷ newtons per meter.
If a particle's velocity has a component along the field, that part is untouched by the field. The particle circles and drifts along at once: a helix. That's the "spiral along the field lines" in the aurora.
Where field lines converge, near a pole, the field strengthens, and a spiraling particle gets pushed back the way it came, like a ball rolling up a slope. Earth's field traps particles bouncing pole to pole between two such "magnetic mirrors" for days to years. Those are the Van Allen radiation belts.
MRI uses a related frequency. Hydrogen nuclei have their own spin magnetism and wobble (precess) in a field at 42.58 megahertz per tesla, so 127.7 MHz in a 3 T scanner. That's spin precession, not cyclotron motion, but it is the same idea of a field setting a natural frequency, and it's why an MRI is, among other things, a very loud FM-band radio.
A fridge magnet has no battery and no wire, so where's the current? Inside the atoms. Every electron behaves like a tiny bar magnet because of a quantum property called spin. Orbital motion adds a little more. In most materials electrons pair up with opposite spins and cancel, so nothing shows.
In iron, cobalt and nickel, something unusual happens. A quantum effect called the exchange interaction makes it energetically cheaper for neighboring atoms' unpaired spins to point the same way. Billions of atoms line up into patches called magnetic domains, typically micrometers to a millimeter across. In an ordinary iron nail the domains point every which way and cancel. Stroke it with a magnet, or put it in a coil, and the domains aligned with the field grow at the expense of the others. Now the nail is a magnet.
Heat undoes this. Above the Curie point, thermal jiggling beats the exchange interaction and the alignment collapses: 770 °C for iron, 354 °C for nickel, 1,115 °C for cobalt. That's also why Earth's field can't come from a giant bar magnet. Its iron core is thousands of degrees hotter than iron's Curie point. Instead, churning liquid iron in the outer core carries currents that keep regenerating the field, a self-sustaining dynamo.
Modern neodymium magnets (neodymium, iron and boron) hold about 1.3 tesla of locked-in magnetization and lose it above roughly 310 °C. They're in hard-drive heads, earbuds, electric car motors and wind turbines. Everything else responds to magnets too, just very weakly. Water is slightly repelled; in a 16-tesla magnet, physicists famously levitated a live frog.
Ferromagnetism needs atoms with unpaired electrons and spacing that makes the exchange interaction favor parallel spins. Iron, cobalt and nickel have partly filled 3d shells at just the right spacing. Manganese has more unpaired spins but the wrong spacing, so its neighbors anti-align and cancel. A few rare-earth elements, such as gadolinium (Curie point about 20 °C), join the club when cold enough.
"Soft" magnetic materials, like transformer steel, let domains swing easily and switch off when the field goes away. "Hard" ones, like neodymium magnets, pin the domain walls with crystal defects so the alignment stays. Plot magnetization against applied field and you get a loop, called hysteresis. The area of that loop is energy lost as heat every cycle, which is why transformer cores are made of soft iron.
If currents make magnetism, can magnetism make current? People tried for years by putting magnets next to wires and waiting. Nothing. In 1831 Michael Faraday found the catch: the magnet has to change. Push a magnet into a coil and a meter on the coil kicks. Hold it still and the needle drops back to zero. Pull it out and the needle kicks the other way.
What matters is magnetic flux, Φ: how much field passes through the loop, roughly field strength times the area it crosses. Flux is measured in webers (one tesla over one square meter). The voltage induced around a loop, called the EMF, equals how fast the flux through it changes. Wind N turns and each one adds its own share.
The minus sign is Lenz's law, and it's the most important minus sign in engineering. The induced current always flows so as to oppose the change that caused it. Push a north pole into a coil and the coil becomes a north pole facing it, pushing back. Pull it out and the coil turns into a south pole, tugging it back. You always have to do work to induce a current. If the sign were plus, the coil would pull the magnet in faster, inducing more current, pulling harder still: free energy, which nature does not hand out.
Drop a strong magnet down a thick copper pipe and Lenz's law is unmistakable. Copper isn't magnetic, yet the magnet drifts down slowly, because currents swirling in the pipe walls (eddy currents) oppose its fall. The energy of the fall turns into a slight warming of the pipe.
Scale it up. Spin a 100-turn coil, 10 by 10 centimeters, in a 0.5-tesla field, 60 times a second. The flux through it swings back and forth, and the peak voltage is N B A ω = 100 × 0.5 × 0.01 × 377 ≈ 188 volts. That is a generator.
There are two ways to change the flux through a loop. Move the wire through a field, and the charges in it feel the ordinary q v × B force: that's "motional EMF." Or keep the wire still and change the field, and something new happens: a changing magnetic field creates an electric field that circles around it, ∮ E · dl = −dΦ/dt, even in empty space with no wire at all.
That second kind of electric field is unlike anything in Chapter 1. Static fields start and end on charges, and carrying a charge around a loop gets you nothing. This one loops back on itself, so going round it does net work. That's how a transformer pushes current around a coil it never touches.
Faraday's law treats both cases with one formula, yet they look physically different, depending on whether you say the magnet or the coil is moving. Einstein opened his 1905 paper on special relativity by pointing out exactly this awkward asymmetry. His resolution is the last section of this chapter.
Generators. Almost every watt on the grid comes from a magnet spinning inside coils, or coils spinning inside a magnet. Steam, falling water and wind just turn the shaft. A simple two-pole generator must spin at 3,600 revolutions per minute to make North America's 60 hertz; Europe's 50 hertz needs 3,000. Run the same machine backwards, feeding it current, and it's a motor. Electric cars do both: their motors become generators when you brake.
Transformers. Wind two coils on one iron core. Alternating current in the first makes a changing flux in the core, which induces a voltage in the second. Since every turn sees the same flux, the voltage scales with the number of turns.
This is why the grid runs on AC. Power lost heating a wire is I²R. Send the same power at 100 times the voltage and you need a hundredth of the current, which cuts the heating loss by a factor of 10,000. Transformers step voltage up to hundreds of thousands of volts for long-distance lines and back down for your street. They only work on changing current; feed one steady DC and, after the first instant, nothing comes out.
Induction cooktops drive a flat coil at roughly 20 to 100 kilohertz. The rapidly changing field induces eddy currents in the bottom of an iron or steel pan, and the pan's own resistance heats it. The glass top stays cool except where the hot pan sits on it. A Qi wireless charger is a transformer with the core split in two and an air gap between: a coil in the pad at 87 to 205 kilohertz, a coil in the phone. Tap-to-pay cards do the same at 13.56 megahertz, powering a battery-free chip from the reader's field.
The induced voltage is proportional to how fast the flux changes, so at higher frequency a smaller field (and a smaller, lighter coil) does the same job. That's why a phone charger's transformer, switching at tens or hundreds of kilohertz, can be thumbnail-sized, while a 60-hertz transformer of the same power needs a heavy iron core. Large grid transformers are among the most efficient machines ever built, typically above 99%.
Eddy currents are a nuisance inside a transformer, so its core is built from thin, insulated steel sheets that break up the current loops. In an induction cooktop the eddy currents are the whole point, so the pan is a solid slab. Aluminum and copper pans conduct too well and aren't magnetic, so they heat poorly unless the cooktop is specially designed for them.
Here's a puzzle. Electrons in a copper wire carrying 1 amp (1 mm² cross-section) drift along at about 0.07 millimeters per second, roughly a quarter of a meter per hour. That's a snail's pace: about 2.4 × 10⁻¹³ of the speed of light. How can anything so slow produce a force strong enough to swing a compass?
Picture a positive charge moving alongside a current-carrying wire. In the lab, the wire is neutral: as many positive ions as drifting electrons. The charge moves, so it feels a magnetic force. Now ride along with the charge. It's at rest, so there can be no magnetic force on it. But it still has to be pushed, because whether it hits the wire can't depend on who's watching. Special relativity supplies the push. In this frame the ions and electrons move at different speeds, and moving things are shortened along their motion, by different amounts. The wire's charges no longer balance. It looks slightly charged, and the "magnetic" force is now plainly electric.
The imbalance is tiny, a fraction of order (v/c)². But remember from Chapter 1 how gigantic electric forces are when they don't cancel. A tiny fraction of a gigantic force is exactly the modest force you measure between two wires. Electricity and magnetism are not two forces. They are one electromagnetic field, and how much of it you call "electric" and how much "magnetic" depends on how you're moving. Maxwell's equations are the complete rulebook, and they predict, from these two effects alone, that ripples in the field travel at the speed of light.
Take 1 meter of copper wire with a 1 mm² cross-section. It holds about 8.5 × 10²² free electrons, a charge of 13,600 coulombs. If the electrons in two such wires 1 meter apart weren't cancelled by the ions, the electric repulsion would be about 3.3 × 10¹⁸ newtons per meter.
Now let each carry 1 amp. The drift speed is v = 7.3 × 10⁻⁵ m/s, so (v/c)² = 6 × 10⁻²⁶. Multiply: 3.3 × 10¹⁸ × 6 × 10⁻²⁶ ≈ 2 × 10⁻⁷ newtons per meter. That is exactly μ₀ I₁ I₂ / 2πd for 1 amp at 1 meter, the old definition of the ampere. It isn't a coincidence: the ratio of the magnetic to the electric force is μ₀ε₀v², and μ₀ε₀ is exactly 1/c².
This shortcut just compares the two force formulas. The full relativistic argument tracks the length contraction of ions and electrons separately, and lands on the same answer.
Drag the magnet through the coil (or use the position slider), or turn on auto-oscillate. Watch the voltage spike as the magnet enters, flip sign as it passes the middle, and vanish whenever it stops. The view is 40 cm wide; the coil is 2 cm in radius.
Magnet strength is its magnetic moment in A·m²; 1 A·m² is about a 1-cubic-centimeter neodymium magnet. The flux formula treats the magnet as a point dipole and the coil as one short loop, a fair simplification once the magnet is a few centimeters away.
Two long wires, seen end-on: a dot means current toward you, a cross means away. Set the currents and spacing, and drag the probe to measure the field. Same-direction currents attract; opposite currents repel. The view is 60 cm wide.
Field lines here are drawn so their spacing reflects strength: crowded lines, strong field. Set one current to zero to see a single wire's perfect circles.
Pick a particle and set its speed and the field (pointing into the screen). The circle is always drawn the same size; watch the scale bar instead. The motion is slowed down enormously so you can see it.
Uses the relativistic forms r = γmv/qB and T = 2πγm/qB, so it stays right all the way up toward light speed.
| Device | Frequency | What changes the flux | What gets induced |
|---|---|---|---|
| Power-plant generator | 60 Hz (US), 50 Hz (Europe) | A spinning electromagnet inside fixed coils | The grid's alternating voltage |
| Street transformer | 60 Hz | Alternating current in the primary coil | A lower voltage in the secondary, e.g. 7,200 V to 240 V |
| Induction cooktop | ~20–100 kHz | Alternating current in a flat coil | Eddy currents that heat an iron pan from inside |
| Qi wireless charger | 87–205 kHz | AC in a coil under the pad | Current in a matching coil in the phone |
| Tap-to-pay card | 13.56 MHz | The card reader's coil | Enough power to run the card's chip, no battery |
| Guitar pickup | ~80 Hz to a few kHz | A steel string vibrating over a magnet | The signal that goes to the amp |