The force that does no work

Of the four ways a field can act on a charge, the magnetic one is the odd case. An electric field pushes along itself, so a charge released in one gains energy in a straightforward way. A magnetic field pushes at right angles to the motion, and at right angles to itself as well, and the consequence of the first of those is a prohibition rather than an effect: it can never change a particle’s energy at all.

The prohibition is worth stating carefully, because it is easy to slide off. Work is $\mathbf{F}\cdot\mathbf{v}$, and $q(\mathbf{v}\times\mathbf{B})\cdot\mathbf{v}$ is zero identically — not approximately, not on average, but at every instant and for every field configuration. So the speed of a charged particle in a magnetic field is a constant of the motion, whatever the field does, however it varies in space, however complicated the path becomes.

What a curved track is telling anyone who reads it #

Because the speed is fixed, the only thing a magnetic field can do is bend, and the amount of bending carries information. Balancing $qvB$ against $mv^2/r$ gives

$$r = \frac{mv}{qB} = \frac{p}{qB},$$

and the quantity that appears is the momentum. A track’s curvature in a known field is a momentum, measured directly, with no need to know what the particle is.

Why a constant speed is still an acceleration is the step the whole essay rests on: the change in a velocity vector of constant length points toward the centre, and its magnitude is $v^2/r$. So a curved track tells anyone watching that a force is acting, and tells them nothing at all about whether that force is doing work — which is the distinction a magnetic field exploits.

This is why every detector built to look at charged particles has a magnet round it. The cloud chamber, the bubble chamber, the wire chamber and the silicon tracker all measure the same thing in the same way: a sagitta, converted to a radius, converted to a momentum. The sign of the curvature gives the sign of the charge, which is how the positron was identified in 1932 — Anderson’s photograph shows a track that curves the wrong way and loses energy in a lead plate, so the direction of travel is known and the charge follows.

The period that forgot to ask the speed #

The most useful consequence of the cancellation is one step further on. The time to go once round is the circumference over the speed,

$$T = \frac{2\pi r}{v} = \frac{2\pi m}{qB},$$

and the speed has vanished. A slow particle goes round a small circle and a fast one round a large circle in exactly the same time.

That is a remarkable thing to be true, and it was worth a machine. Lawrence’s cyclotron of 1932 exploits it directly: two D-shaped electrodes with a gap between them, an alternating voltage across the gap at the frequency $qB/2\pi m$, and a magnetic field to bring the particle back to the gap. Every crossing adds energy, the radius grows, the period does not, and the same fixed-frequency oscillator keeps working from the first turn to the last. The whole apparatus fitted in a hand.

The two numbers on that figure decide the whole layout of twentieth-century accelerator physics. A proton stays classical to a good ten megavolts, which was far beyond what any early machine could reach, so the cyclotron worked and kept working. An electron leaves the classical regime at five kilovolts, which is a school demonstration, so the trick never worked for electrons at all. Machines that accelerate them either sweep the frequency, or arrange the field and the frequency together so that a bunch stays in step — and either way the simplicity that made the cyclotron possible is gone.

The energy that appears in that γ is the same one that appears everywhere else in relativity, and the failure of the cyclotron above it is one of the few places where a relativistic correction announces itself as a machine that stops working rather than as a discrepancy in the fourth decimal place.

Crossing it with an electric field #

Because the magnetic force is proportional to the speed and the electric force is not, putting the two at right angles produces a device that responds to speed alone.

The selected speed $E/B$ contains no mass and no charge, so the device sorts by velocity and is completely blind to what is going through it. Thomson used exactly this arrangement in 1897 — first balancing the two forces to get the speed, then switching off the electric field to get the deflection, and dividing one into the other to get the charge-to-mass ratio. The number he obtained was a thousand times larger than for any ion, which is how the electron was discovered: not by seeing one, but by measuring a ratio that nothing known could have.

Fixing the energy to get at the mass #

A curvature alone gives a momentum, which is a product of two things nobody has separately. The standard fix is to fix one of them first.

The square root is the whole character of the instrument.

It means a spectrometer separates light isotopes easily and heavy ones with difficulty, because the same one atomic mass unit is a smaller fraction of a larger mass and the square root halves what is left. It also means resolution improves as the square root of the radius, which is why mass spectrometers grew and why the modern ones abandoned the geometry entirely in favour of measuring a cyclotron frequency, which is linear in the mass rather than square-rooted and can be counted rather than located.

Aston’s spectrograph of 1919 used this arrangement to establish that neon is two elements’ worth of isotopes and not one, that isotopic masses are very nearly integers, and — by measuring how far from integers they are — to produce the first table of nuclear binding energies. The whole of nuclear energetics rests on a set of positions measured on a photographic plate to a fraction of a millimetre.

Why the answer stops being a mass difference at high energy is that kinetic energy departs from the Newtonian parabola. A mass spectrometer fixes the energy and reads the radius, and the relation between them assumes $\tfrac12mv^2$ — so above a few per cent of light speed the instrument is measuring momentum rather than mass, and the correction is not optional. That is why isotope separation and particle physics use the same geometry and different arithmetic.

Turning the measurement into a count #

Every instrument above reads a position: a radius, a deflection, a landing point on a plate. Positions are hard to measure well — a part in $10^4$ is respectable — and the accuracy of the answer is the accuracy of the ruler.

The modern arrangement measures a frequency instead, and frequencies are the best-measured quantities in physics by a margin of several orders of magnitude. A Penning trap holds a single ion using a strong uniform magnetic field for the radial confinement and a weak electrostatic well for the axial, and then simply listens: the ion’s cyclotron motion induces an image current in the electrodes, the current is amplified, and the frequency is counted for as long as the ion sits there, which can be months.

Since $\omega_c=qB/m$, a frequency ratio between two species in the same field is a mass ratio, and the field cancels. Masses obtained this way are known to eleven or twelve significant figures. The proton-to-electron mass ratio, 1836.152673426, is a cyclotron frequency ratio; so is the value of the atomic mass unit; so are the mass differences that decide whether a nuclear decay is energetically allowed at all, where the answer turns on the eleventh digit of two numbers that are nearly equal.

The physics has not changed at all from the figure at the top of this page. What changed is which property of the circle is read out — and the fact that the period does not depend on the speed, which was a convenience for Lawrence, is the reason the measurement is possible: the ion’s orbit is not controlled, does not need to be, and the answer does not care.

Where the expression came from #

The force law is usually written as one vector equation and was assembled from several unconnected pieces. Ørsted’s accidental observation in 1820 that a current deflects a compass gave the existence of the effect; Ampère established within weeks that two currents attract or repel; Biot and Savart wrote down the field of a wire; Faraday spent thirty years demonstrating that the effect is mediated by something occupying the space between, and that a changing field of one kind makes the other.

The force on a moving charge — as opposed to on a current-carrying wire — was not obvious, because nobody was sure that a current was moving charge. J. J. Thomson wrote a version of it in 1881 and got the coefficient wrong by a factor of two; Heaviside corrected it in 1889; Lorentz gave it its final form in the 1890s as part of a theory in which charges and the field are separate things and the field acts locally on each charge. The name attached to Lorentz because his was the version embedded in a working electron theory.

The historical order matters for one reason. The expression was arrived at last, after the field concept was already in place, and it is therefore not a summary of experiments on wires but a statement about what a field does to a single charge. That is exactly why it survived relativity intact while almost everything around it was rewritten.

The machine that looks like a counterexample #

There is one accelerator that appears to break the rule, and working out why it does not is the best test of whether the rule has been understood.

A betatron accelerates electrons in a circle using nothing but magnets. There are no radio-frequency cavities, no electrodes, no gap to cross — a doughnut-shaped vacuum chamber between the poles of an electromagnet, and the electrons come out at tens of megavolts. It looks like a magnetic field doing work.

It is not. The magnet’s field is ramped, and a changing magnetic flux through the electron’s orbit produces an induced electric field around that orbit. That electric field is what accelerates the electrons; the magnetic field at the orbit is what bends them. Two jobs, two fields, and the magnetic one still does no work.

What makes the machine elegant is that both jobs are done by the same magnet, and doing them consistently imposes a condition. The radius has to stay fixed as the momentum rises, so the field at the orbit must rise in proportion to the momentum; and the momentum gained is the integral of the induced field, which depends on the average field inside the orbit. Requiring the two to agree gives the betatron condition: the field at the orbit must be exactly half the average field enclosed by it.

That is a statement about the shape of the pole pieces — they have to be closer together in the middle than at the orbit, in a specific ratio — and a betatron works or does not work according to whether the magnet was machined to satisfy it.

The device is now a museum piece, displaced by machines that accelerate more efficiently, and it survives as the cleanest available demonstration that “a magnetic field does no work” is a statement about a force on a charge and not about what a magnet can be part of.

The precession that is measured as a difference #

The second refinement is worth having because it is the most precise thing ever done with a magnetic field, and it turns on comparing two of this essay’s frequencies.

A charged particle with spin does two things in a uniform field: it goes round at the cyclotron frequency, and its spin precesses at a frequency proportional to its magnetic moment. If the moment were exactly what the simplest theory gives — a $g$-factor of exactly 2 — those two frequencies would be identical, and the spin would keep a fixed orientation relative to the momentum for ever.

It is not exactly 2, and the difference between the two frequencies is proportional to how much it is not:

$$\omega_a = \frac{g-2}{2}\,\frac{eB}{m}.$$

So a beam of muons circulating in a storage ring, with their spins initially aligned with their momenta, slowly rotates its spins relative to its motion — and since a muon’s decay sends its electron preferentially along the spin, counting decay electrons in a fixed direction gives an oscillation at $\omega_a$ directly.

The virtue is that the small quantity is measured on its own rather than as a difference of two large ones. The anomaly is about a part in a thousand of the moment; measuring the moment to a part in a billion would be hopeless, and measuring the anomaly itself to a part in a million is not.

There is a second trick in the same experiment worth recording. A storage ring needs electric fields to confine the beam vertically, and an electric field ordinarily contributes to the spin precession and spoils the measurement. Its contribution carries a factor that vanishes at one particular Lorentz factor — 29.3, which for a muon is a momentum of 3.09 GeV/c — so the ring is run at exactly that momentum and the electric field drops out of the answer. A machine designed around the zero of a coefficient.

The other way to weigh an ion #

The magnetic spectrometer is not the only instrument for the job, and its main competitor uses no magnet at all — which makes the comparison a good way to see what the magnet was buying.

Accelerate ions through a known voltage, as before, so that every one leaves with the same kinetic energy. Then instead of bending them, let them fly down a straight evacuated tube and time their arrival. Equal energy means speed goes as the inverse square root of mass, so the flight time goes as the square root of the mass — the same square root as the radius in a magnetic instrument, arriving through a time rather than a distance.

The trade is straightforward. A time-of-flight instrument has no upper limit on the mass it can handle, because there is no magnet whose field would have to grow; it accepts every ion produced rather than scanning through them one at a time; and it is cheap, since a drift tube and a fast detector cost far less than a precision magnet. What it gives up is that its resolution is set by how well the start time and the initial energy spread are controlled, which is a harder engineering problem than making a uniform field.

The refinement that made it competitive is a reflector at the far end — an electrostatic mirror that turns the ions round. An ion with slightly too much energy penetrates further into the mirror before turning, so it takes longer inside it, and the extra time compensates the shorter time it spent in the drift. The spread in arrival times is therefore reduced without reducing the spread in energies, which is the same trick as an achromatic lens.

Which is why the mass spectra of large molecules are taken this way and the mass spectra of small ones are often not. The square root that this essay’s magnetic instrument suffers from — heavy isotopes separating badly — applies equally to both, and what differs is only how the square root is read out.

What it costs #

The prohibition that makes all this work also makes magnetism useless for the one thing it is most often assumed to do.

No magnetic field has ever accelerated anything. A magnet can hold a beam, steer it, focus it and analyse it, and it cannot add energy. Every accelerator’s energy comes from electric fields in resonant cavities, and the magnets are the steering. The confusion is easy because a magnet clearly does work when it drags an iron nail across a table — but the work there is done on the material by the internal fields set up in it, and following the energy carefully shows it coming from the field’s own stored energy and not from $q\mathbf{v}\times\mathbf{B}$.

Circular motion is not free. A charge going round a circle is accelerating, and an accelerating charge radiates. Bending a beam therefore costs energy, and the cost rises brutally with speed.

The bill is radiation. A charge moving at 0.9 of light speed radiates in a cone swept forward by aberration, with its peak at 13.4° rather than at 90°, and the power scales as $\gamma^4$ — so the magnetic force does no work and the turning costs energy anyway. That is the honest qualification on this essay’s title: the field takes nothing, and the acceleration it produces gives something away.

The field is not the whole story in matter. Everything above treats the particle as moving through vacuum in a specified field. In a medium the applied field is not the field the charge experiences, magnetisation of the surroundings adds its own, and at high densities the beam’s own field matters as well.

What the picture cannot show #

The circles at the top of this page are drawn in a plane, which is the special case in which the velocity happens to be exactly perpendicular to the field. That is almost never true. A general velocity has a component along the field, and that component is untouched — no force acts along $\mathbf{B}$ — so the real path is a helix whose pitch is set by the parallel speed and whose radius is set by the perpendicular one. Nothing in a plane figure can show that, and the helix is the case that matters for plasma confinement, for the aurora, and for cosmic rays arriving through the galaxy’s magnetic field.

The figures also assume a uniform field. In a field that varies, the guiding centre of the helix drifts — perpendicular to both the field and its gradient — and a whole formalism exists to handle it. The magnetic mirror, in which a particle spiralling into a strengthening field is turned back, is a consequence of that treatment and is what confines particles in the Van Allen belts.

The domain of validity is therefore: uniform static field, no radiation, speed either well below $c$ or with γ carried explicitly, and no medium. Outside it the qualitative statement — that the force does no work — survives every one of those extensions unchanged, which is unusual and is the reason it is worth learning as a principle rather than as a formula.

The ladder from here #

Later rungs on this anchor: the helical path and the guiding-centre approximation, with the drifts that follow from a non-uniform field; the magnetic mirror and adiabatic invariance; the Hall effect, where the same force separates charges across a conductor and reveals their sign; magnetic focusing, where a quadrupole focuses in one plane and defocuses in the other and an alternating sequence does both; and the synchrotron, where the field is ramped in step with the momentum so that the radius stays fixed.

The neighbouring ladders are the magnetic field itself, which is what the force acts in, the radiating charge, which is what bending costs, and magnetism as a relativistic effect, which is what the force turns into when the observer moves along with the charge.