If you’ve ever read Maxwell’s On Physical Lines of Force, you may have noticed this: “a motion of translation along an axis cannot produce a rotation about that axis unless it meets with some special mechanism, like that of a screw”. Maxwell was referring to what I can only describe as the screw nature of electromagnetism. If you have a pump-action screwdriver you’ll appreciate that linear force is converted into rotational force. That’s like an electric motor: current flows through the wire, and the motor turns. When you use an ordinary screwdriver, rotational force is converted into linear force, and the screw is driven into the wood. That’s like a dynamo: turn the rotor, and current flows. Hence the right-hand rule applies not just to electromagnetism, but to screw threads too:
GNUFDL right-hand rule image by Jfmelero see Wikipedia
In the Wikipedia right-hand rule article it’s called Ampère’s right hand screw rule. Other sources refer to Maxwell’s right hand grip rule. Some refer to the right handed cork screw rule and some even show you a picture of a screw:
Image by Kiran Daware, see electrical easy
Do a google search on corkscrew rule images. They are ubiquitous. So much so that the simplest demonstration of the translation-rotation principle is Michael Faraday’s homopolar motor of 1821. It featured a battery, a wire and a bath of mercury, but what do we see at the top of the Wikipedia homopolar motor article? A contemporary homopolar motor featuring a battery, a wire, a magnet, and a screw.
The electromagnetic field is not depicted
However when you read about the electromagnetic field in contemporary texts, the screw nature of electro-magnetism tends to be absent. And somewhat surprisingly, there are very few depictions of the electromagnetic field. You can see depictions of a gravitational field, depictions of a gravitomagnetic field, depictions of an electric field, and depictions of a magnetic field. But there are virtually no depictions of the electromagnetic field. What you see instead is pictures of electric fields and magnetic fields. You can find a few pictures that show both together, like figure 27-6 in the Feynman lectures. There’s also the anapole field in the 2013 Popular Mechanics article doughnut-shaped electromagnetic fields may explain dark matter:
Image credit: Michael Smeltzer/Vanderbilt
But pictures of the electromagnetic field are as rare as hen’s teeth. There’s a gap between the talk and the walk. You can read on Wikipedia that “over time, it was realized that the electric and magnetic fields are better thought of as two parts of a greater whole – the electromagnetic field”. You can also read what Oleg Jefimenko said: “neither Maxwell’s equations nor their solutions indicate an existence of causal links between electric and magnetic fields. Therefore, we must conclude that an electromagnetic field is a dual entity”. In addition you can read where John Jackson said “one should properly speak of the electromagnetic field Fμv rather than E or B separately”. But that’s in section 11. Not section 1. And people do speak of E and B separately, all the time. There’s an awful lot of places where you can read about the electric field and the magnetic field as if they’re two separate entities. And there’s virtually no places where you can see the electromagnetic field as the single entity it’s supposed to be. Sometimes it feels like Maxwell’s unification never happened.
You don’t create a magnetic field for the electron when you move
Sometimes it feels like Hermann Minkowski’s Space and Time never happened either. Towards the end, near figure 3, Minkowski said this: “In the description of the field caused by the electron itself, then it will appear that the division of the field into electric and magnetic forces is a relative one with respect to the time-axis assumed; the two forces considered together can most vividly be described by a certain analogy to the force-screw in mechanics; the analogy is, however, imperfect”. He was talking about the electron’s field. That’s an electromagnetic field. And he referred to electric and magnetic force. For charged particles, electric force is linear, whilst magnetic force is rotational. They’re related by a screw analogy, because of the screw nature of electromagnetism: a motion of translation along an axis cannot produce a rotation about that axis unless it meets with some special mechanism, like that of a screw. Because in ∇ × E = −∂B/∂t the equals sign is an “is”. The curl of E is the time-rate of change of B. But we don’t read about this. What we read instead that you create a magnetic field when you move a charged particle such as an electron. It just isn’t true. You know this, because you know that motion is relative. We could leave the electron exactly where it is and move you relative to it. However your motion doesn’t create a magnetic field for the electron. Because all along the electron had an electromagnetic field. Not an electric field, not a magnetic field, an electromagnetic field. You merely see a different aspect of that field when you move, that’s all. Because that field has a screw nature, and because it takes two to tango.
Electric field lines don’t work for charged particles
That means the depictions of a charged particle’s electric field are misleading. A positron doesn’t really have an outward pointing electric field. An electron doesn’t really have an inward pointing electric field:
Image by Andrew Duffy, see his PY106 course
Because each has an electromagnetic field. Yes, an electric field can be visualized on paper by drawing radial lines of force, or radial field lines if you prefer. But this visualization is not a good one for a charged particle. That’s because the force is not the same as the field. It results from electromagnetic field interactions, and it takes two to tango. An electron is not surrounded by a field of force. Nor is a positron. We only see a force when we set down a second electron, or a second positron. If we then have two positrons with no initial relative motion, they move apart linearly. We might then think that we could draw outward pointing arrowheads around each positron to represent the force pushing them apart. But if we have two electrons with no initial relative motion, they also move apart linearly too. For consistency we’d have to draw them with outward pointing arrowheads too. So the arrowheads just don’t work. Moreover they come with the mental baggage of sources and sinks. There is no outflowing from a positron. There is no inflowing for an electron, just as there is no inflowing Chicken Little waterfall of space for a gravitational field. All in all, electric field lines just don’t work for charged particles.
Magnetic field lines don’t work for charged particles
Magnetic field lines don’t work for charged particles either. They work for iron filings. You can shake them over a piece of paper and they line up like little compass needles. They mark out the field, and you can see plenty of images when you Google on magnetic lines of force. Some even show compass needles pointing along the lines of force. You are seeing something real, and something with pedigree that goes back to Faraday. He described them as physical in 1852, and those iron filings prove that there’s something physical there. Hence Maxwell wrote On Physical Lines of Force in 1861. But a charged particle doesn’t move along magnetic lines of force. An electron doesn’t scoot from the North pole of a bar magnet to the South pole. Instead it moves around the magnetic lines of force. Because of this you might say that they’re magnetic field lines as opposed to lines of magnetic force. That sounds reasonable, especially since an electron is said to have a magnetic dipole field rather like a bar magnet, and rather like a solenoid:
Image from the national mag lab, see Magnets from Mini to Mighty and EMR
Only it isn’t enough. Because when you throw an electron through a solenoid, its path will trace out a clockwise helix. And when you throw a positron through a solenoid, its path will trace out an anticlockwise helix. That’s because the rotational magnetic force on a charged particle is orthogonal to the particle motion and to the magnetic field, as per the right hand rule. It also depends upon the charge. But when you throw a magnet through a solenoid, there is no helical path. Electrons and positrons move in opposite spirals in a bubble chamber, because of the magnetic field within it. But a magnet doesn’t spiral in a magnetic field. Magnetic field lines on their own are not enough. The motion of the electron or positron in a magnetic field is a product of the particle itself and the field. It takes two to tango. And it isn’t because the electron or positron has a magnetic field. The little magnet moves directly towards the big magnet, it doesn’t move around the magnetic field lines. The electron does, not because it has an electric field with field lines pointing inwards, not because it has a magnetic field with field lines going around and around, but because it has an electromagnetic field. It’s a combination of the electric field and the magnetic field, and you need to see it to understand it.
Combining electric and magnetic field lines
The electron’s electric field is said to be radial. That’s simple enough. However the electron’s magnetic field is said to be like that of a bar magnet, which isn’t. The bar magnet’s magnetic field is like that of a solenoid. The solenoid’s magnetic field is like that of a simple loop of wire. Let’s straighten out the wire to make it even simpler. Let’s consider the magnetic field around the current in the wire. That’s the simplest magnetic field I can think of, and it’s depicted as concentric circles. So, how do we combine the radial electric field lines with the concentric magnetic field lines? Let’s ask Maxwell. In 1871 he wrote a paper entitled Remarks on the Mathematical Classification of Physical Quantities. He said this: “Another distinction among physical vectors is founded on a different principle, and divides them into those which are defined with reference to translation and those which are defined with reference to rotation. The remarkable analogies between these two classes of vectors is well pointed out by Poinsôt in his treatise on the motion of a rigid body. But the most remarkable illustration of them is derived from the two different ways in which it is possible to contemplate the relation between electricity and magnetism”. There’s the translation and rotation again. Maxwell referred to Descartes, to Helmholtz and “his great paper on vortex motion”, and to Thomson and Tait. And to Ampère. 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. I am constrained to agree with this view, because the electric current is associated with electrolysis, and other undoubted instances of translation, while magnetism is associated with the rotation of the plane of polarization of light, which, as Thomson has shown, involves actual motion of rotation”. This is the paper where he came up with convergence and curl and drew this picture:
Image by James Clerk Maxwell
It’s also where he said this: “It represents the direction and magnitude of the rotation of the subject matter carried by the vector σ. I have sought for a word which shall neither, like Rotation, Whirl, or Twirl, connote motion, nor, like Twist, indicate a helical or screw structure which is not of the nature of a vector at all. However it is in the nature of a vector field. The electric field has the convergence. The magnetic field has the curl. Curl is also known as rot, which is short for rotor. You can combine them to depict the electromagnetic field. Like this:
Then you see the rotation, and the whirl, the twirl, and the screw structure. You see the screw structure of the “spinor”. The thing that’s responsible for the screw nature of electromagnetism. The thing that’s responsible for the way that charged particles move.
The linear electric force between charged particles
When you set down an electron and a positron such that they have no initial relative motion, they move directly towards one another. The force is linear, and we talk of Coulomb’s Law. That says the electrostatic force between two charged particles separated by a distance r is F = ke(q1q2) / r². The ke is Coulomb’s constant which is 1/4πε0, the 4π being related to a sphere and the ε0 being vacuum permittivity. The force is attractive if the charges q1 and q2 have opposite signs, and repulsive if they don’t. But why? These two charged particles aren’t throwing photons at one another. Or boomerangs. The virtual photons that are described as exchange particles are virtual. They aren’t real photons. They only exist in the mathematics of the model. Hydrogen atoms don’t twinkle, and nor does positronium, which is often described being like light hydrogen.
The rotational magnetic force between charged particles
Positronium is an “exotic atom” that doesn’t last long. It’s depicted as the electron and positron doing a little whirligig dance of death before annihilation:
CCASA image by Manticorp/Rubber Duck, see Wikipedia
The lifetime depends on the relative spin states. If the spins are antiparallel we have 11S0 singlet state para-positronium with a lifetime of 125 picoseconds, ending up as two 511keV photons. If the spins are parallel we have 13S1 triplet state ortho-positronium with a longer lifetime of about 142 nanoseconds, ending up as three gamma photons. There’s also 2S positronium with a lifetime of about a microsecond, ending up as five photons. See papers like Observation of positronium annihilation in the 2S state by D A Cook et al where they say it appears to survive collisions with the wall of the tube. All of this is salient reminder that the electron and the positron each has an electromagnetic field, not just an electric field. If you throw an electron through a solenoid it goes around and around. If you throw an electron past a positron, the particles go around and around each other as they close in. There’s obviously more than one way they can do it, and we could get into spherical harmonics or even toroidal harmonics. But to keep things simple we talk of the Biot-Savart Law, and then say the magnetic force vector between two charged particles is given as:
The km is the magnetic force constant which is μ0/4π, the 4π being related to a sphere and the μ0 being vacuum permeability. The v1 and v2 with the overhead arrows are the particle motion vectors, and the r with the hat is a unit vector in direction r. You can read about this on Frank Wolfs’ website. You can also read on Wikipedia that the Biot-Savart law is also used in aerodynamic theory to calculate the velocity induced by vortex lines. Note this: “In Maxwell’s 1861 paper ‘On Physical Lines of Force’, magnetic field strength H was directly equated with pure vorticity (spin), whereas B was a weighted vorticity that was weighted for the density of the vortex sea. Maxwell considered magnetic permeability μ to be a measure of the density of the vortex sea”. Maxwell didn’t quite get it right of course. He said this: “Let AB, Pl. V. fig. 2, represent a current of electricity in the direction from A to B. Let the large spaces above and below AB represent the vortices, and let the small circles separating the vortices represent the layers of particles placed between them, which in our hypothesis represent electricity”. His figure 2 is shown below. There’s a honeycomb of vortices, with particles between them. Perhaps there’s a later paper where he corrected this, but if Maxwell ever made a mistake, this is it:
Image by James Clerk Maxwell
There is no vortex sea. We don’t talk of spinors for nothing. A bispinor isn’t used to describe relativistic spin ½ wave functions for nothing. Because the vortices are the particles. On such the world turns.
Why charged particles move the way they do
So, why do the particles move linearly and rotationally? We refer to the Lorentz force law for this. The linear and/or rotational force on a particle of charge q is F = qE + qv Χ B, the v being velocity, and E and B being electric and magnetic fields respectively. Is this force present because charged particles are throwing photons at one another? No. Those photons are virtual, and virtual photons are not real. Is it because of some magical mysterious action at a distance? No. Even Newton knew that, over three hundred years ago. See his 1692 letter to Richard Bentley where he said this: “That gravity should be innate, inherent, and essential to matter, so that one body may act upon another at a distance, through a vacuum, without the mediation of anything else, by and through which their action and force may be conveyed from one to another, is to me so great an absurdity, that I believe no man who has in philosophical matters a competent faculty of thinking can ever fall into it”. He was talking about gravity as opposed to electromagnetism, but the same principle applies. I suspect Descartes knew that nearly four hundred years ago. I also suspect that if you’ve read Lorentz’s Nobel lecture you may have noticed that “this force is always due to the ether in the immediate vicinity of the electron”. Replace ether with space, and look to fluid dynamics which says counter-rotating vortices attract. Kelvin talked about this. If you know that, you will also know that co-rotating vortices repel. And that vortices swirl around one another too. Then you will know that the reason charged particles move the way they do is because of their spinor nature.
Cyclones and anti-cyclones
As per the Columbia heady collisions article, particles can be likened to tornadoes and hurricanes. You can liken the electron to a southern-hemisphere cyclone rotating clockwise. You can liken the positron to a northern-hemisphere cyclone rotating anticlockwise. In this analogy the positron isn’t like a meteorological anticyclone which rotates clockwise in the northern hemisphere, because that features central high pressure, not low pressure. But I’ll take the liberty of shortening anticlockwise cyclone to anti-cyclone. If you threw the cyclone past the anti-cyclone how would they move? A picture is worth a thousand words. An animation is worth ten thousand words, but for now a picture will have to do. Those two particles move towards each other, and around and around too:
Why? Not because they’re slinging photons back and force. Not because of some magical mysterious action-at-distance. But because each is a dynamical spinor in frame-dragged space. A cyclone has intrinsic spin. So does an anti-cyclone. That’s what makes it what it is. Cancel one spin with the opposite spin, and all you’ve got is wind. An electron has intrinsic spin too. So does an anti-electron. That’s what makes it what it is. Cancel one spin with the opposite spin, and all you’ve got is light. It all makes sense.
Twist and turn
Something else that makes sense is that a charged particle doesn’t have an electric field, or a magnetic field. It has an electromagnetic field. Because it’s an electromagnetic wave going around and around, and there’s only one wave there. The result of this is a standing-wave standing-field dynamical chiral spinor called an electron or a positron. The end product is not altogether unlike the “twisted space” gravitomagnetic field, but with no solid central body. It’s a wave going round and round, not a planet, not a football, and not a point particle. There is no central billiard ball, that thing in the middle is like the eye of the storm:
The state of space where a gravitational field is, is inhomogeneous. The state of space where a photon is, is curved. It’s curved one way, then the other. The state of space where an electron is, is curved, all the same way, all the way round. The state of space where a positron is, is curved, all the same way, all the way round, but the other way round. Only it isn’t just curved in two dimensions, it’s curved in three dimensions. It’s curled, it’s twisted. In a way the electron is a twistor, and its electromagnetic field is twistor space. It’s not like Penrose suggested, but it’s not far off. You could say the electron electromagnetic field is a twist field. Look at the picture above. Hold your arms out wide and imagine you’re an aeroplane. Imagine you could fly forward into this twisted space. Imagine what would happen: your wings would tilt, they would turn clockwise with the twist. The twist results in a turn. Only if you weren’t moving, if the electron was moving relative to you, you might think you were in a turn field, not a twist field. That’s what the screw nature of electromagnetism is all about. Twist and turn.
The electromagnetic field is a twist field
That’s why the electromagnetic field is a dual entity and the greater whole. Because it’s a twist field, and when you move through it, you turn. That’s why a charged particle has an electromagnetic field rather than an electric field or a magnetic field. Because it’s an electromagnetic wave going around and around, and there’s only one wave there. You cannot remove the electric aspect of the electromagnetic wave, because it’s merely the spatial derivative of potential. Nor can you remove the magnetic aspect of the electromagnetic wave, because it’s merely the time derivative of potential. Nor can you do either when the wave is going round and round in the guise of a spin ½ standing wave. In this respect the phrase electric charge is a misnomer. The charged particle has an electromagnetic field, so we ought to call it electromagnetic charge. Yes, we can contrive an ensemble of charged particles so that some aspect of their electromagnetic fields are counter-balanced and so masked. Then when we observe other charged particles, we sometimes see the linear force only, and we sometimes only the rotational force only. When we only see the linear force we talk of an electric field, when we only see the rotational force we talk of a magnetic field. To understand how this works, we have to understand how a magnet works. For some strange reason Feynman couldn’t explain it. Richard Feynman, the great explainer, couldn’t explain how a magnet works. He didn’t understand it. But amazingly, in 1820, André-Marie Ampère did.