To understand how a magnet works, you need to understand that the electron doesn’t have an electric field or a magnetic field, it has an electromagnetic field. In fact it is electromagnetic field. We made it in gamma-gamma pair production, such that a 511keV electromagnetic wave is configured as a spin ½ standing wave. Hence the wave nature of matter. When you wrap a sinusoidal electro-magnetic field variation into a twisted double loop, the minimum and maximum field variation combine, along with all points in between, to leave you with an all-round standing field. But as Feynman said, there’s a circulation of energy in this so-called static condition. There is angular momentum in the field. The electron electromagnetic field looks static, but actually it’s dynamical, as suggested by the Poynting vector and evidenced by the Einstein-de Haas effect. The angular momentum is real. That’s why we have the right-hand-rule and the screw nature of electromagnetism. That’s why Hestenes used the word rotor. That’s why Radu and Volkov used the word vorton. That’s why Ehrenfest used the word spinor:
It’s a three dimensional thing, but it’s easier to grasp the 2D version. The point to appreciate is that the electromagnetic field is somewhat similar to the gravitomagnetic field. It’s a “twist” field. Note though that if you’re moving through it or it’s moving through you, you might think of it as a “turn” field. That’s what a magnetic field is. That’s why in 1871 Maxwell said this: “According to Ampère and all his followers, however, electric currents are regarded as a species of translation, and magnetic force as depending on rotation”. The Faraday effect is where a magnetic field rotates the polarization of light. We can even make a magnet by turning a piece of iron, as per the Barnett effect. We don’t call it rot for nothing. That’s why the simplest magnet is the electron itself. It behaves like a tiny bar magnet because it is in itself a rotor. However it has an electromagnetic field rather than a magnetic field, and we need to look at something that doesn’t.
The current in the wire
The current in the wire is the simplest situation in which we see a magnetic field. It’s quite easy to understand why. The wire is made of copper. We have electrons that move and we have copper ions that don’t. Whilst a copper ion is far more complicated than an electron, it has the opposite charge, so let’s simplify it to the opposite spinor. Then we can draw two lines of spinors like this:
Unattributed images are drawn by me. Note the optical illusion here, the arrow appears to slant downwards
The electrons are on the top row whilst the copper ions are underneath. There’s two sets of opposite twist fields, so they cancel. Hence you don’t see any net twist field. So there is no net charge, and no linear electric force, so we say there’s no electric field present. However the electrons are moving whilst the copper ions are not. And if you’re moving through a twist field or it’s moving through you, you might think of it as a turn field. You have relative motion with respect to the electrons only, so what you see is a net turn field. A magnetic field. We use the right hand rule to depict the concentric magnetic field lines, then we use Fleming’s right hand rule to remember how the electron moves:
The magnetic field lines point away from us where the electron is located. The electron is initially moving right to left, and there’s a rotational force orthogonal to both the magnetic field line and the electron motion. So the electron moves clockwise in a near-circular fashion around the magnetic field line. The electron goes around and around clockwise like a left-handed boomerang. A positron goes around and around anticlockwise like a right-handed boomerang.
Note though that the field around the wire isn’t uniform, because it diminishes with distance. Hence the electron path isn’t quite circular. Also note that the wire is part of a circuit, so it’s already a current loop. When you bend the wire into a more pronounced loop you’re also altering the magnetic field, revealing its dipole nature. Then when you bend the wire further into multiple loops, what you’ve got is a solenoid. Then the magnetic field is the same as that of a bar magnet:
Image courtesy of Rod Nave’s hyperphysics
The magnetic field inside the solenoid is uniform, and the field lines are parallel. When you throw an electron through a clockwise solenoid, its motion is helical and clockwise.
Image © Keith Gibbs 2013 see schoolphysics
When you throw a positron through the solenoid, its motion is helical and anticlockwise. Either way there’s a longitudinal motion down through the solenoid, but the rotational component is still present. The electron still goes around and around clockwise like a left-handed boomerang. The positron still goes around and around anticlockwise like a right-handed boomerang. Now why might that be?
Why the electrons goes around and around
When you look into why a boomerang flies in a circular path, you soon find that it’s because of gyroscopic precession. Then it’s the work of but moments to find articles such as chapter 15 of Nicholas Turro’s Modern Molecular Photochemistry. In section 6 he says this: “Since the mechanics of the precessional motion of a gyroscope in the presence of gravity are of the same mathematical form as the mechanics of a magnetic moment associated with a spinning charged body in the presence of a magnetic field, we postulate that the vector due to the magnetic moment of the quantum magnet undergoes precessional motion in an applied magnetic field”. A gravitational analogy is also used in the Wikipedia gyromagnetic ratio article: “The earth’s gravitational attraction applies a force or torque to the gyroscope in the vertical direction, and the angular momentum vector along the axis of the gyroscope rotates slowly about a vertical line through the pivot”. Follow the link to Larmor precession and you can read that the spin angular momentum of an electron “precesses counter-clockwise about the direction of the magnetic field”. After that you soon find the Larmor radius. That’s also known as the gyroradius. It’s the radius of the circular motion of a charged particle in a uniform magnetic field. It seems fairly clear that the electron moves rotationally in a magnetic field because of gyromagnetic precession. Particularly since the positron goes the other way. See the MRI article by Allen D Elster: “two particles with positive and negative gyromagnetic ratios precess in opposite directions”.
Around and around and around and around
As for why precession results in rotational motion, remember that the electron is a “spinor” with a Poynting vector. When the electron is at rest think of it as light going around a circular path. When the electron is moving right to left, think of it as light going around a helical path. If you were to precess the front face of the helix, the whole electron would move helically, like it does when thrown through a solenoid. To visualize this, use a Slinky spring. Stretch it out straight to emulate an electron moving in a straight line. Then form it into a helix to emulate the helical motion of the electron through the solenoid. You may need the Lucky Penny Slinky Spiral to do this. It’s a Slinky with a spring-steel ribbon through the middle, plus end-caps. To form your Slinky into a helix, you just turn the ends. Then you have a helical helix. Not quite a double helix. If anything it’s more like a helix squared. It goes around and around, and around and around:
Lucky Penny Slinky Spiral, see Youtube
If every coil was a disc, it would be a tilted disc, precessing as you traced your way along the slinky helix. Throwing the electron along the current-in-the-wire is like throwing it through the side of a solenoid, or emitting it from the electron gun in a Teltron tube. Then it has no motion along the field lines, only around the field lines. The precession inclination angle is 90 degrees, and the motion is circular rather than helical. It’s like forming your Slinky into a circle rather than a helix:
Note though that the magnetic field within a solenoid or a Teltron tube is uniform, so the electron path is circular or helical. However the magnetic field around the current-in-the-wire is not uniform. It diminishes with distance in the wire. So the electron path is near circular. This is important. This is why a magnet is a magnet. An iron cylinder will be attracted into a solenoid, but once it’s inside the solenoid the magnetic field is uniform, so there’s no more force on it. There’s no net force on a magnet in a uniform magnetic field, because the force is equal and opposite on opposite poles.
The force on a charged particle
When the particle is going around and around in a uniform magnetic field, there’s no net force on the particle. There’s a turning force, a torque, but this is symmetrical, and the motion of the particle is uniform. See Magnetic Torque and Magnetic Force from Michael Salvati at NYU physics for more about this. You can read that “in a uniform magnetic field there is no net force on the particle, but there is a torque and the angular momentum will, in some sense, precess about the magnetic field”. The particle goes around and around, but it doesn’t speed up, and it doesn’t really go anywhere. However when the particle is in a non-uniform magnetic field there is a net force on the particle. There’s a non-rotational force on the silver atoms in the Stern-Gerlach experiment because the torque is not balanced. That’s’ why the “spin up” silver atoms go up, and the “spin down” silver atoms go down. See Chris Skilbeck’s Cronondon for another description of the Stern-Gerlach experiment. Also see Particle Spin and the Stern-Gerlach Experiment by James Cresser: “the non-uniformity of the field means that the atoms experience a sideways force”. There’s a net force on the silver atoms, and it isn’t there because of length contraction. The drift velocity of the current in the wire is circa one metre per hour. The speed of the silver atoms is circa 660 metres per second. That force is there because the silver atoms are moving through a non-uniform magnetic field, and because they’re either spin up or spin down. It’s similar for electrons moving parallel to the wire, either in the same direction as the current, or not.
The force on the electron
The magnetic field surrounding the current-in-the-wire is not uniform. The field gets weaker as you move away from the wire, so the electron motion isn’t quite circular. The Larmor radius nearer the wire is less than it is further away from the wire, and the result is drift. See Lyndsay Fletcher’s lecture for more. Throw the electron along the wire from right to left into the oncoming current, and the electron spins up towards the wire clockwise, then down again. Throw a positron along the wire from right to left into the oncoming current, and the positron spins down away from the wire clockwise, then up again:
Note though that the Stern Gerlach experiment employs silver atoms rather than electrons. It is said to be infeasible for electrons, though some authors disagree. Whether it is or it isn’t, the moot point is that an atom is a combination of opposite-charged particles, wherein the Lorentz forces nearly cancel. And when they nearly cancel, there’s a force on the atom.
The force on an atom
If we simplify the atom down to one positron and one electron, then when we throw it to the left the stronger force is downwards, and when we throw it to the right the stronger force is upwards. But if we swap the positions of the electron and the positron, when we throw our atom to the left the stronger force is upwards, and when we throw it to the right the stronger force is downwards:
Hence the atom moves up or down depending on its direction and motion and the disposition of its spinors, which we can label as spin up or spin down. Either way the rotational forces don’t balance, so there’s a net force, one way or the other. See dipoles and magnets by Lawrence Rees. The atom moves, either up or down. If you throw it along the wire to the left into the current, it moves down away from the wire. If you throw it to the right with the current, it moves up towards the wire. And since for every action there’s a reaction, there’s a force on the wire too.
The force on the wire
So, if you throw your atom to the right with the current, the atom moves towards the wire, and the wire moves towards the atom. If you scaled this up to a string of copper atoms plus a stream of electrons, you effectively have a second wire. Hence two wires with currents flowing in the same direction are attracted to one another, as per Ampère’s Law.
Image from Rod Nave’s hyperphysics
Likewise two wires with currents flowing in the opposite direction repel one another. The force is very weak, because it’s a “trace” force that’s only there because the raw electromagnetic forces don’t quite balance. It’s a pale shadow of the very powerful Coulomb force between charged particles. When you stop the electrons moving there’s an even weaker trace force called gravity, but that’s so weak you wouldn’t notice any force between the wires. You need those currents flowing to notice a force. Like I said, you need two wires with currents flowing in the same direction for attraction, and two wires with currents flowing in the opposite direction for repulsion.
The force on a magnet
It’s the same if you bend your wires into current loops. If both currents are clockwise or anticlockwise, the loops attract. Otherwise they repel. When you scale this up to a series of loops to form a solenoid, then if both have a clockwise current they attract one another. When you turn one solenoid over it now has an anticlockwise current, and the solenoids repel. Either way, it’s because on the way into the solenoid the field is not uniform, and the forces don’t balance. It’s the same for a bar magnet, only we call one end the North pole and the other end the South pole. When you scale it up to the size of the Earth, we still call one end the North pole and the other end the South pole. We have the Aurora Borealis and the Aurora Australis because those forces don’t balance. The bottom line is this: a magnet works because it consists of spinors which exert linear and rotational forces on other spinors. Whilst the linear forces balance, the rotational forces don’t, because they diminish with distance, and the result is weaker attractive and repulsive forces between the North and South poles of a pair of magnets.
Secrets of the ancients
I think that what’s really interesting about all this is that it goes back two hundred years. See the Princeton article on Oersted’s Theory. It says this: “Oersted proposed that a vortex-like, circular distortion existed around the wire”. It says Oersted said “motion in circles, joined with a progressive motion, according to the length of the conductor, ought to form a conchoidal or spiral line”. You can read Oersted’s short paper on a French website called Ampère and the history of electricity. Rare were those physicists who accepted Oersted’s vortex explanation, but Ampère was one of them. In 1820 Ampère invented the word solenoid, and noticed that a solenoid had a magnetic field like a bar magnet. He said “if the spirals acted like magnets, it was because magnets owe their magnetism to electric currents in planes perpendicular to their axes”. Search on vortex and you can find Ampère’s researches in the science of electrodynamics: That’s where Ampère said this: “Is it not evident, then, to all how memorable would that discovery be that would rigorously establish the fact that to magnetize a needle is to excite, to put in motion around each molecule of the steel, a small, circular, electrical vortex?” Ampère said they were a version of Descartes’ planetary vortices, on a smaller scale. He also said this: “There are on the surface and in the interior of a magnet as many electric currents in planes perpendicular to the axis of this magnet that one can imagine a series of lines forming closed curves without cutting across each other”. Here’s the picture, it’s figure 15 on page 32 of Exposé des nouvelles découvertes sur l’électricité et le magnétisme:
Image by Ampère and Babinet
This suggests macroscopic electric currents, but he referred to “un infinité de courants electrique situés tout autour dans le plans perpendiculaires à l’axe”. An infinity of electric currents situated all around in the planes perpendicular to the axis. And note this from Ampère and the history of electricity: “starting on 15 January 1821, moved by a suggestion by Fresnel, he presented another hypothesis, that of currents around elemental particles [“courants particulaires”]. Each particle of the magnet would be surrounded by a circular current around an axis parallel to that of the magnet”. In a later paper in 1826 Ampère referred again to particulate currents, and to an electrodynamic helix. Impressive stuff, and as the Princeton article says, “Ampère’s model is startlingly close to the truth. There are no circular molecular currents after all, but the spinning electrons act just like loops of current”. What a pity Feynman didn’t pursue the circulation of energy associated with the Poynting vector. If he had, he would have been able to explain how a magnet works. He wouldn’t have had to say “I can’t explain attraction in terms of anything that’s familiar to you”. In his best Brooklyn accent, he could have explained a magnet in about two minutes forty.
How a magnet works
Interviewer: What I want to know, is what’s going on between these two bits of metal?
Feynman: They repel each other. As to why, well now, the thing you have to know about, is the wave nature of matter. And that the electron is a “spinor”. It’s a wave going around and around [Feynman rolls his finger]. We know this because of something called the Einstein-de Haas effect. Think of the electron as something like a cyclone. Other particles are like anti-cyclones, so particles attract each other and go around each other like counter-rotating vortices. [Rolls two index fingers around one another]. Or they repel each other like co-rotating vortices. [Rolls fingers backwards]. In an ordinary material all the forces cancel each other out, but in the current-in-the wire, which is the simplest magnet, they don’t. That’s because the electrons are moving [stabs index finger] whilst the copper ions are not. Because of this, another nearby electron moving in the same direction as the current is subject to a net rotational force [rolls finger]. It goes around and around just like a boomerang. But this rotational motion isn’t uniform because the field diminishes with distance, so the electron drifts towards the wire. There’s a net force towards the wire, and since for every action there’s a reaction, there’s also a force on the wire towards the electron. The opposite is true if the electron is moving in the opposite direction to the current. Then when you have a whole lot of electrons in a copper wire doing this sort of thing, the same rules apply. So two wires with currents going the same way attract, and two wires with currents going the opposite way repel. Then when you bend your wires into multiple loops, each is a solenoid. When your electrons are going the same way round both solenoids, the solenoids attract. But flip one solenoid over and the electrons are now going the opposite way, so now the solenoids repel. A bar magnet is the same. André-Marie Ampère worked that out in 1820. Smart guy. A bar magnet has the same sort of magnetic field as a solenoid because electrons are spinors going around and around, all the same way. [Rolls his finger]. Look at this picture from a book he wrote. [Holds up book]. See how they’re all going around the same way? When you break a magnet in half, each half still has a North pole and a South pole, because the North pole is just looking at all this rotation from above, and the South pole is just looking at it from below. That’s why there are no magnetic monopoles. You want to know about them too?
Interviewer: Yes please.