The field with no ends, and the force that does no work

Every electric field line drawn on this site starts on a positive charge and ends on a negative one. That is not a drawing convention — it is the counting argument that gives Gauss’s law, and the whole of electrostatics rests on lines having two ends.

Magnetic field lines have none. They close on themselves, every one of them, everywhere, always. No exception has ever been found, and the search has been thorough — a search for the one thing whose absence a null result can establish.

What the figure is computing #

The lines come from summing the contributions of the current itself, and the summation is worth stating because it is the magnetic counterpart of the inverse-square law.

A short element of current produces a field that circles it — perpendicular to the current and perpendicular to the line joining the element to the point in question, with magnitude falling as the inverse square of the distance. That is the Biot–Savart law, and everything in the figures on this page is its sum over the conductors shown.

That exponent is the same result as the falloff from a line of charge, and for the same geometric reason: contributions from an extended source add up along its length, and the sum turns one power of the distance into a constant. The mechanism is different — a circulation rather than a divergence — and the counting is identical.

The lines being circles is worth pausing on. There is no beginning; a circle has no privileged point. Any attempt to say “the field starts here and goes to there” for a straight wire has no possible answer, and it is the first hint that magnetic lines are objects of a different kind.

The law that says nothing is there #

The absence has a formal statement, and it is the shortest of Maxwell’s four equations:

$$\oint \mathbf{B} \cdot d\mathbf{A} = 0.$$

The magnetic flux through any closed surface, anywhere, of any shape, is exactly zero. Compare that with the electric version, whose right-hand side is the enclosed charge divided by a constant. The magnetic equation is the same equation with the source term removed, and the removal is the content.

For an electric field, a closed surface around a charge has every line crossing it exactly once on the way out. For a magnetic field that picture is impossible: whatever surface is drawn, every line entering it also leaves, so the net count is zero for every surface and every arrangement. That is the law with no ends in it — not a statement that magnetic charge has not been found, but a statement built into the way the field is defined from a current.

The everyday demonstration costs nothing. Cut a bar magnet in half and the result is two bar magnets, each with a north and a south end. Cut those in half and the same thing happens. There is no procedure — cutting, grinding, heating, dissolving — that produces an isolated north pole, and the reason is that a bar magnet’s poles were never objects. They are where the lines happen to leave the iron, and cutting the iron simply gives the lines two more places to do it.

The search, and what its failure is worth #

An isolated magnetic charge — a monopole — is not forbidden by anything obvious. Maxwell’s equations accommodate one comfortably; adding a magnetic charge density to the right-hand side above makes the four equations more symmetric rather than less, which is the kind of aesthetic argument that usually turns out to be right.

Dirac showed in 1931 that the existence of even one monopole anywhere in the universe would explain something otherwise unexplained: that electric charge comes in exact multiples of a fundamental unit. The argument is a consistency condition between the monopole’s field and the quantum phase of a charged particle moving around it, and it forces the product of the two charges to be quantised. Charge quantisation is an experimental fact with no other accepted explanation, so this is a genuine argument for monopoles rather than a curiosity.

They have been looked for accordingly: in cosmic rays, in accelerator collisions, in seawater, in moon rock, in iron ore, in polar ice. One candidate event was recorded, by a single detector at Stanford on the evening of 14 February 1982 — a clean signal of exactly the expected size, in an instrument built for the purpose. Nothing like it has been seen since, by that detector or by the much larger ones built afterwards, and it is generally regarded as unexplained rather than as a discovery.

So the law $\oint \mathbf{B} \cdot d\mathbf{A} = 0$ is an experimental statement of enormous precision that could be falsified tomorrow by a single event. That is an unusual status for one of the four fundamental equations of a subject, and it is worth holding rather than smoothing over: the equation is written as an identity and believed as a measurement.

The force that cannot do work #

The second peculiarity of the magnetic field follows from the shape of the force it exerts, and it is a claim that sounds flatly false.

The force on a charge $q$ moving with velocity $\mathbf{v}$ is $q\,\mathbf{v} \times \mathbf{B}$: perpendicular to the velocity, always. A force perpendicular to the motion does no work, ever — that is the definition of work — and it therefore cannot change a particle’s kinetic energy. It can only turn it — which is exactly the circular motion a mass spectrometer reads as a momentum.

A magnetic field does no work on anything.

The obvious objection is a magnet picking up a paperclip, which certainly looks like work being done. The resolution is that the work is done by other agents. The magnetic force organises what happens; the energy comes from the field’s own stored energy, and from the internal electric forces that hold the current loops of magnetised matter together. In every case where a magnetic system is analysed carefully, the energy is traced to something that is not the magnetic force, and the accounting closes.

That is a genuinely uncomfortable result and it is not a technicality. It is why a charged particle in a magnetic field moves in a circle at constant speed forever rather than being accelerated, why magnetic confinement can hold a plasma without heating it, and why the energy in a motor comes from the current source rather than from the magnets. The distinction between organising a motion and paying for it is the same one an energy landscape makes between a constraint and a force.

Far from its sources a current loop’s field has the same shape as an electric dipole’s, which is why both are called dipoles. Close in they could hardly differ more: the electric one runs from a positive charge to a negative one, and the magnetic one closes on itself through the loop. The shared name records a resemblance at a distance and hides a complete difference at the source.

The far field that agrees and the near field that does not #

Comparing the two dipoles is the sharpest way to see what “no ends” costs and buys.

The near field is where the two part company, and it is the whole content of the law. An electric dipole’s lines begin and end; a current loop’s pass through the middle and come back round. So the far fields agree and the near fields cannot, and any attempt to model a magnet as a pair of poles gets the outside right and the inside wrong — which is exactly the error a compass cannot detect and a hole drilled through a magnet can.

At large distances the two fields are indistinguishable in form: both fall as $1/r^3$, both have the same angular pattern, and the name dipole is applied to both for that reason. A compass needle far from a current loop cannot tell whether it is being deflected by a loop or by a pair of magnetic charges, and for two centuries the pair-of-charges description was the standard one.

Near the source they differ completely. The electric dipole’s lines converge on two points where the field is infinite and the divergence is non-zero. The current loop’s lines pass through the loop and out the other side, with nothing anywhere resembling a source. Ampère argued from exactly this that all magnetism is produced by circulating currents rather than by magnetic charges, and he was right — the magnetism of iron comes from electron spin and orbital motion, both of which are current loops in the relevant sense.

What the closure costs #

A field whose lines never end is not merely a curiosity of bookkeeping. It changes what can be built, and the constraints are severe enough to have shaped whole technologies.

A magnetic field cannot be shielded by surrounding something with it. A conductor excludes electric fields by rearranging its surface charge, and there is no magnetic surface charge to rearrange. Magnetic shielding therefore works by an entirely different mechanism: a high-permeability material such as mu-metal offers the lines an easier route and they crowd into it, diverting round the volume to be protected. The lines are not stopped; they are given somewhere more attractive to go, and the enclosure has to be a closed shell of material rather than a conductive skin. This is why a room shielded to microvolts electrically may be useless magnetically, and why magnetically shielded rooms are heavy.

A field cannot be confined to a finite region. Lines that must close have to leave and return, so there is no arrangement of currents that produces a field inside a volume and nothing at all outside. A solenoid’s field spreads and returns outside it, and the only way to reduce the return field is to route it deliberately — which is what an iron yoke on a magnet is for, and why an MRI magnet is surrounded by tonnes of steel or by a second, opposing coil. The stray field is not waste; it is a requirement.

And energy cannot be stored in a magnetic field as cheaply as in an electric one. Storing energy in a magnetic field means running a current, and running a current means resistive loss unless the conductor is superconducting. Capacitors store electrostatic energy with essentially no standing loss; inductors of the same energy dissipate continuously. That asymmetry between the two halves of electromagnetism is entirely traceable to the fact that one field is made by charges sitting still and the other by charges that must keep moving.

The one thing it buys is that flux is conserved. Because no lines are created or destroyed, the flux through a closed loop of wire can only change by lines moving across the wire — and that motion is what induces a voltage. Faraday’s law is a statement about a quantity that is well defined only because $\oint \mathbf{B}\cdot d\mathbf{A} = 0$ makes the flux through a loop independent of which surface is used to span it. Every generator on the grid depends on that independence, and it is a direct consequence of the absence of monopoles.

Where magnetism actually comes from #

The deepest statement about the magnetic field is that it is not an independent thing at all, and the demonstration needs only a wire and a change of frame.

Take a current-carrying wire: positive ions at rest, electrons drifting along it. The wire is electrically neutral, so a stationary charge beside it feels no electric force. A moving charge beside it feels a magnetic force.

Now describe the same situation from a frame moving along with that charge. In this frame the charge is at rest, so it cannot feel a magnetic force — there is no $\mathbf{v}$ to cross with $\mathbf{B}$. And yet something must push it, because whether a charge is deflected is a fact that all observers agree on.

What pushes it is an electric field. In the moving frame the spacings of the positive and negative charge distributions are contracted by different amounts, because the two populations were moving at different speeds to begin with, and length contraction depends on speed. The wire that was neutral in one frame carries a net charge density in the other, and the resulting electric force is exactly the magnetic force computed in the first frame.

So magnetism is what electrostatics looks like from a moving frame. The two fields are components of one object, sliced by whichever frame is doing the describing, and the reason a magnetic force is perpendicular to velocity is that it originates in a transformation that involves the velocity.

The astonishing part is the size. Length contraction at the drift speed of electrons in a copper wire — about a tenth of a millimetre per second — is a relative effect of order $10^{-25}$. That such a fantastically tiny relativistic correction produces a force strong enough to run an electric motor is possible only because the electric force it modifies is so enormously strong: the residual after nearly complete cancellation of two vast numbers. Magnetism is a rounding error on electrostatics, made visible by the scale of what it is a rounding error on.

The monopoles that were found, and what they are not #

There is a class of material in which objects behaving in every respect like magnetic charges appear, move, attract and repel — and the law at the head of this page is not violated by a single part in anything. Holding both facts at once is instructive.

The materials are called spin ices. Their magnetic atoms sit at the corners of tetrahedra sharing corners, and each atom’s moment is forced by the crystal to point either into or out of the tetrahedron it belongs to. The lowest-energy arrangement has two pointing in and two out of every tetrahedron — a rule with an enormous number of solutions, which is why the material has a residual entropy and why it is named after ice.

Now flip one moment. The two tetrahedra it belongs to become three-in-one-out and one-in-three-out, which is a local excess and a local deficit of magnetic flux: a pair of magnetic charges. Flip a chain of further moments and the pair separates, leaving a trail behind it that costs no energy, because every tetrahedron along the trail still satisfies the rule. So the two ends move independently, and what they feel as they do is a force falling as the inverse square of their separation — a magnetic Coulomb law, with a magnetic charge of a definite size.

They have been measured. Neutron scattering sees the characteristic pattern of a gas of such charges; relaxation experiments see them move; and the magnitude of the charge has been extracted and agrees with what the lattice spacing and the atomic moment predict.

What has not happened is a violation of anything. The true magnetic field of the sample still has zero divergence everywhere; the charges are always made in pairs and cannot be taken out of the crystal; and the flux through a surface enclosing the whole thing is exactly nothing. What obeys the monopole law is an emergent field belonging to the arrangement of the moments, and the objects are excitations of a lattice rather than particles. The distinction is the whole content — a system can produce, at low energies, behaviour whose equations are not the equations it is made of.

The claim about work, stated more carefully #

“A magnetic field does no work” is a good slogan and it is exactly true of one thing and not obviously true of another, and the difference is worth having.

The Lorentz force $q\,\mathbf{v}\times\mathbf{B}$ is perpendicular to the velocity by construction, so its power is identically zero. That is airtight, it applies to every free charge, and it is why a particle in a magnetic field circles at constant speed for ever.

An intrinsic magnetic moment is a different object. An electron’s moment does not come from anything circulating: it is a property of the particle, on the same footing as its charge, and there is no current whose $\mathbf{v}\times\mathbf{B}$ can be computed. The force on such a moment in a non-uniform field is $\nabla(\mathbf{m}\cdot\mathbf{B})$, it is not perpendicular to anything in particular, and it straightforwardly accelerates the particle — which is what a Stern–Gerlach apparatus does, and the kinetic energy the atoms gain has to come from somewhere.

Where it comes from is the field’s own stored energy, and the accounting closes. But the route is not the one the slogan suggests, and the honest statement is narrower than the one usually given: the force on a moving charge does no work; the force on an intrinsic dipole does. A great deal of published argument about whether a magnet lifting a paperclip violates the theorem dissolves once the two cases are kept apart, since iron’s magnetism is overwhelmingly the intrinsic kind.

What survives of the slogan is the part that matters for the rest of this collection. Nothing magnetic ever creates energy, the bookkeeping always closes, and the field’s role in a motor or a magnet is to route energy rather than to supply it.

Where the model stops #

Statics. Everything above is a steady current making a steady field. When either changes, the two fields become coupled — a changing magnetic field makes an electric one whose lines also close, and the potential picture stops being available.

No magnetic scalar potential, in general. Because the field circulates, there is no single-valued scalar whose gradient it is. A magnetic scalar potential can be defined in current-free regions and it is multivalued: going once around a wire changes it by a fixed amount. The honest replacement is a vector potential, which carries its own arbitrariness in a form that turned out to be one of the organising principles of modern physics.

Matter is treated as absent. Inside magnetic materials the field is the sum of the applied field and that of the material’s own aligned currents, and the relationship between the two is nonlinear, hysteretic and history-dependent. Iron does not have a magnetic permeability so much as a magnetic biography.

And the drawings are slices. These figures are two-dimensional sections through a three-dimensional field, so line density on the page is not line density in space, and the closure of a line in the plane is a statement about the plane. The same dimensional shortfall applies as in the electric case, with the extra complication that a magnetic line need not close in any single plane at all — in a general field it can wander forever on a torus without ever rejoining itself, which is a phenomenon with no electrostatic counterpart and a serious problem for magnetic confinement.

The ladder from here #

Later rungs: the Biot–Savart law derived, and the field of a straight wire, loop and solenoid computed from it. Ampère’s law, which is the circulation counterpart of Gauss’s law and does for currents what surface counting does for charges. The Lorentz force in full, and the motion of a charge in crossed fields. Magnetic dipole moment and torque, which is how a compass works and how a motor works. Magnetisation, and the difference between the three field quantities that all get called magnetic. Diamagnetism, paramagnetism and ferromagnetism, and why only the last is strong. Induction, where a changing field of one kind produces the other. Magnetic energy and the vector potential. And the monopole treated seriously, with the Dirac quantisation condition worked through.