The force between pieces of a current

The field that wraps a current introduced Ampère’s law as a statement about circulation: go once round a closed path, add up the field, and the answer counts the current threaded through. The field outside the solenoid and the law that is always true and rarely useful found the limits of that law, and the field that points against the magnet it is in and a potential that does not come back to itself followed it into matter and into a scalar description. In the first of those an aside noted that the law bearing Ampère’s name is not the law Ampère wrote. What he wrote, over six years from 1820, was a force law between two short pieces of current — and the aside said, without arguing it, that such a force is not a measurable thing.

That claim deserves an argument, because it is strange. The force between two charges is measurable, and so is the force between two magnets. Why should the force between two short pieces of wire carrying current not be? The answer is that there are at least two laws for it that disagree about almost every pair of pieces, including about whether the two forces are equal and opposite, and that no experiment can ever tell them apart. Both are correct. What that means is the subject of this essay.

Two laws for one pair #

Consider two short elements of current, $I_1\,d\mathbf{l}_1$ and $I_2\,d\mathbf{l}_2$, separated by $\mathbf{r}$ pointing from the first to the second. The law used today follows from the field each element makes by the Biot–Savart rule and the force $I\,d\mathbf{l}\times\mathbf{B}$ that a field exerts on an element. Written out, the force on element 2 is

$$d\mathbf{F}_{12} = \frac{\mu_0 I_1 I_2}{4\pi}\,\frac{d\mathbf{l}_2 \times (d\mathbf{l}_1 \times \hat{\mathbf{r}})}{r^2},$$

often called Grassmann’s law after Hermann Grassmann, who wrote it down in 1845. Ampère’s own law, published in its final form in 1826, is

$$d\mathbf{F}_{12} = -\frac{\mu_0 I_1 I_2}{4\pi}\,\frac{\hat{\mathbf{r}}}{r^2}\left[2\,d\mathbf{l}_1\cdot d\mathbf{l}_2 - 3\,(d\mathbf{l}_1\cdot\hat{\mathbf{r}})(d\mathbf{l}_2\cdot\hat{\mathbf{r}})\right].$$

Ampère’s force points along the line joining the two elements, like the force between two charges, and it is symmetric: swap the elements and the force reverses exactly. Grassmann’s force is perpendicular to the element it acts on, and nothing makes it symmetric.

The figure takes the simplest disagreement. Let element 1 point straight at element 2. By the Biot–Savart rule an element makes no field along its own line, so element 1 exerts no force on element 2 at all. Element 2, turned at 45°, does make a field at element 1, and pushes it sideways. The two forces are not equal and opposite; one of them is zero. By Ampère’s law both forces are 0.71 units, along the line, one pushing each way. The two laws are not approximations of each other. They disagree about the direction of the force, its size, and whether the third law holds.

The third law, pair by pair #

Ampère built his law to satisfy Newton’s third law. It was one of his starting assumptions, taken from the mechanics of his day: the force between two particles of anything acts along the line between them, and is equal and opposite. With that assumption and four null experiments of great ingenuity — arrangements in which a movable circuit was shown to feel no force — he fixed the numbers 2 and 3 in the bracket.

The Biot–Savart law breaks the third law for almost every pair. The figure plots the amount by which the two forces of an isolated pair fail to cancel, as element 2 is turned. Ampère’s law gives zero throughout. Grassmann’s gives zero only when the elements are parallel, and a full unit when element 2 is perpendicular to the line: then element 2’s field at element 1 is at its strongest and element 1 still exerts nothing on element 2.

The pull that has to equal the pull back found that a failure of the third law between the gravitating parts of a body would let the body push itself through space, and treated that as a test worth making to a part in $10^{14}$. Here a widely taught law fails the third law between pairs by a factor of order one, and nothing pushes itself anywhere. The reason is the one the rest of this essay is about: a current element never occurs alone. Charge is conserved, a steady current cannot begin or end, and the elements come in closed circuits. The imbalance is real for every pair and it always sums to zero over the circuit.

The same total, from different pieces #

Take a closed circuit — here a unit circle of current — and ask what force it exerts on a short element of a second circuit outside it. Each law assigns every piece of the circle its own contribution, and the two assignments differ.

The figure plots, for each piece of the circle, how much it contributes along the direction of the final force. The two curves are different shapes. Ampère’s law makes the nearest pieces push harder and the far side pull back more; the Biot–Savart law spreads the push differently. The areas under the two curves are the same, to every digit computed. By either law the circle pushes the element with 1.2025 units of force, in the same direction.

The second figure shows how the agreement arrives. Add up the contributions piece by piece, starting at the point of the circle nearest the element, and the two running sums separate at once. Half-way round, the Biot–Savart sum stands at 0.92 of the final force and Ampère’s at 1.42. A circuit that stopped half-way — if there were such a thing — would push the element with forces differing by half their size, according to which law is true. The sums then converge, and when the circle closes they meet exactly.

The meeting is not a numerical coincidence. The difference between the two laws, for a fixed element acted upon, is the rate of change along the source of a single expression, something like a potential defined along the wire. Adding up a rate of change along a path gives the difference between the expression’s values at the two ends, and on a closed circuit the two ends are the same point. The difference therefore vanishes for the force of any closed circuit on any element, whatever its shape. And since the force on any piece of the second circuit is a sum of forces on its elements, each from a complete circuit, the two laws agree about the force on every part of every circuit, too — including the part of a circuit that the circuit itself pushes on.

What was ever measured #

Every experiment on the forces between currents has measured the force between complete circuits, or the force exerted by a complete circuit on part of another.

The figure computes the attraction between two coaxial loops three ways: from the rate at which their mutual inductance changes with separation, and by summing the forces between all pairs of elements by each law. The three numbers agree to four figures at every separation. The pairwise forces entering the two sums are completely different; the totals are not. The coupling that is the same both ways found that the mutual inductance of two loops is the same whichever is taken as the source, which is Newton’s third law for complete circuits, and here it is visible in the numbers: neither element law can disagree with it once the circuits are closed.

That is why Ampère could fix his law from experiments and be right, and why Weber, Maxwell and every later worker could confirm it and be right, while the Biot–Savart form, which gives different forces between pieces, is equally right. Maxwell, in his Treatise of 1873, wrote down the most general force between elements consistent with all the closed-circuit experiments. It contains an arbitrary function, and any choice of it is as good as any other. He called Ampère’s paper “one of the most brilliant achievements in science” and pointed out, in the same discussion, that the law it established was one of an infinite family, all of which predict the same results for every experiment that can be done.

Why the modern law was chosen anyway #

If the laws agree on everything measurable, the choice between them is made on other grounds, and the grounds favour the Biot–Savart form for two reasons.

The first is that it factors into a field and a force on a current in that field. The current in one circuit makes a magnetic field everywhere; the field acts on the current in the other. Ampère’s law does not factor this way. It is an action between pairs of elements with no intermediate field, in the style of Newton’s gravity, and it was Maxwell’s programme to replace such actions with fields that carry energy and momentum through space. Once the field is taken as real, the force on an element is fixed by the field at that element and nothing else, and it has to be perpendicular to the element.

The second is that the field description extends to changing currents and to moving charges, and Ampère’s does not. A changing current radiates, the field carries momentum as well as energy, and the momentum of something that is not moving found that a static field can hold momentum that the matter’s forces must balance. In such situations the forces on the matter alone need not be equal and opposite at every instant; the field takes up the difference. The Biot–Savart law’s failure of the third law pair by pair is the static shadow of that fact. Ampère’s law, which enforces the third law between elements, has no room for momentum in the field, and it gives the wrong answer as soon as the currents change fast enough for that momentum to matter.

The same puzzle with two moving charges #

A current element is a stream of moving charges, and the puzzle does not go away when the element is replaced by a single charge. Take two charges moving at right angles, the first heading straight towards the point where the second is, the second crossing that line. The magnetic field of a moving charge vanishes along its own line of motion, exactly as the field of a current element does, so at that instant the first charge exerts no magnetic force on the second. The second charge’s field at the first is not zero, and the first feels a magnetic push. The magnetic forces between the two are not equal and opposite, and there is no circuit to rescue the sum.

What rescues it is the field. The electric and magnetic fields of the two moving charges overlap, and the overlap carries momentum, $\varepsilon_0\mathbf{E}\times\mathbf{B}$ per unit volume, that changes as the charges move. The rate at which the field’s momentum changes is exactly the amount by which the forces on the two charges fail to balance. Momentum is conserved, but only when the field is counted as a thing that can hold it, which is the step Maxwell’s programme took and Ampère’s law, written in 1826, could not.

The size of the imbalance tells where it matters. Magnetism is electricity seen sideways found that the magnetic force between moving charges is the electric force multiplied by $v^2/c^2$, up to geometry. The failure of the third law between two moving charges is therefore of order $v^2/c^2$ of the electrical forces between them. For conduction electrons drifting at a tenth of a millimetre per second that is some $10^{-25}$, and it is only visible at all because in a wire the electrical forces cancel almost perfectly between the electrons and the lattice and leave the magnetic force standing alone. In a current loop it is visible as a force between pieces; the loop that behaves like a needle summed the same pieces into a single magnetic moment, where the pairwise imbalances had already vanished.

The hairpin and the exploding wire #

That has not stopped people from looking for the difference. In 1822 Ampère floated a copper wire bent into a U — his hairpin — on two troughs of mercury, passed a current through it, and saw it move away along its legs. He took this as evidence that current elements in line repel each other, which his law predicts and the Biot–Savart law does not. The hairpin, however, is part of a closed circuit completed through the mercury, and its motion is predicted identically by both laws: the push comes from the field of the rest of the circuit acting on the hairpin’s crosspiece, and the argument above guarantees the totals agree.

In the twentieth century the claim returned in another form. Peter Graneau and others reported in the 1980s that wires carrying very large currents broke into many short pieces, as if torn apart by a tension along their length, and argued that only Ampère’s longitudinal forces could do it. The difficulty is that a force between elements lying in the same wire is where the two laws are most singular — the elements are touching — and the answer depends on how the current is spread through the wire’s thickness. For a conductor of finite thickness, calculations show that both laws give the same forces on every part, and the breakup of exploding wires is attributed to thermal and elastic effects: a wire heated by a pulse of current in microseconds expands faster than it can relieve the stress, and it shatters. The claim did not survive.

What cannot be drawn #

The figures treat current elements as idealised points with a direction, and in that idealisation the laws differ between pairs. A real current is carried by charges moving through a conductor, and the forces on it are forces on those charges and on the lattice that holds them. At that level the question of which element pushes which is a question about how the momentum delivered by the field is shared among the charges, the lattice and the field itself, and the force read off a surface that touches nothing showed that only the total over a closed surface is fixed.

The figures also assume steady currents. When currents change, neither element law is the whole story: the fields are retarded, the circuit radiates, and the force between two circuits at a distance is not what either law says until the fields have had time to arrive. The pairwise picture is a way of dividing up a total that is correct for steady currents in closed circuits and for nothing else.

Still open: whether a force between parts can be defined at all #

The question whether local forces inside a current-carrying conductor are well defined — how the total electromagnetic force on a conductor is distributed through it, and whether that distribution is observable through the stresses it produces — has not entirely disappeared. The stress in a conductor carrying a large current can be measured, in principle, through the strain it produces, and the question is then what the electromagnetic stress tensor says it should be in a medium, a question entangled with the century-old debate about the momentum of fields in matter. Most physicists regard it as settled in favour of the field description; a small literature continues to argue for Ampère’s force on the grounds of the exploding-wire experiments, and it has not produced a measurement that the field description cannot explain.

The habit worth carrying away is to ask whether a quantity exists on its own before asking what its value is. Two force laws between pieces of current disagree about nearly every pair, one obeying Newton’s third law and one not, and agree exactly about every closed circuit, because their difference is a quantity that changes from one end of the source to the other and a circuit has no ends. A steady current cannot be cut into pieces, and a force between the pieces is a way of dividing a total that nature only ever delivers whole.