The field that wraps a current

There are two ways to summarise a vector field over a region. One is to add up how much of it passes through a surface, which is flux, and that is what Gauss’s law counts. The other is to add up how much of it points along a closed path, once round, which is circulation. Electric fields have a great deal to say about the first and nothing at all about the second. Magnetic fields are the other way round.

That is Ampère’s law, and the figure is the whole of it:

$$\oint \mathbf{B}\cdot\mathrm{d}\boldsymbol\ell = \mu_0 I_{\text{enc}}.$$

Nothing about the path survives into the answer — not its radius, not whether it is centred, not whether it has corners. What survives is what it encloses. That is exactly the property flux has for electric charge, and it is the reason the two laws are always taught together despite describing quite different things.

Why the shape drops out #

The mechanism is easy to see for the circular case and worth following, because it explains why the general case works too.

Round a circle centred on the wire, the field has the same magnitude everywhere and points along the path everywhere, so the integral is $B$ times the circumference: $B\cdot2\pi r$. The field falls as $1/r$ and the circumference grows as $r$, and the two cancel exactly. That cancellation is not a coincidence — it is the same $1/r$ that appears in a two-dimensional inverse law, and it is why the answer has no length scale in it.

How the falloff depends on the shape of the source is what makes the wire’s answer so clean: a point gives $1/r^2$, a line gives $1/r$, a plane gives a constant, and each comes from the same conservation applied to a different geometry. The circulation law inherits that, which is why the field of a straight wire has one power of $r$ in it and no property of the wire at all.

For a path that is not a circle, the argument is a small piece of geometry. Moving along any path, the contribution to the integral is the field’s magnitude times the component of the step along it. The field is $\mu_0I/2\pi r$ and points azimuthally, so the contribution is $(\mu_0I/2\pi)\,\mathrm{d}\theta$ — the change in angle subtended at the wire, with the radius cancelled. Going once round a path that encloses the wire accumulates $2\pi$ of angle. Going once round a path that does not accumulates zero, because the angle goes up and comes back. The numbers in the figure are that statement, evaluated numerically rather than argued.

Where the symmetry comes from, and the trap in it #

The law relates one number to one number. The field is a vector function of three coordinates. So Ampère’s law cannot possibly determine a field on its own, and the step that makes it look as though it does is always an assumption about symmetry brought in from outside.

For a straight wire the assumption is that the field has the same magnitude at every point on a circle about the axis and points along that circle. It follows from the symmetry of the source — the arrangement is unchanged by rotating about the wire, sliding along it, or reflecting — and it is a genuine argument. But it is an argument about the source, not a consequence of the law, and it fails silently whenever the source is not symmetric.

The electrostatic case where the trap was first identified is worth having beside this one. A Gaussian surface tells the truth about the total flux for any charge distribution, and gives the field only when a symmetry argument is supplied independently. Ampère’s law is in exactly the same position: the circulation is always right, and turning it into a field requires knowing in advance that the field is constant along the path.

The electrostatic answer symmetry cannot give makes the limit concrete: charge on an irregular conductor has to be solved numerically, because no argument from symmetry applies. The magnetic analogue is the same — an awkwardly shaped circuit has a perfectly definite field that Ampère’s law will not produce, and the law has not failed. It has been asked a question it does not answer.

The practical form of the caution: if a problem cannot be solved by Ampère’s law, that is not a failure of the law but a statement that the arrangement lacks the symmetry, and the alternative is to integrate the Biot–Savart law over the actual current distribution — which always works and is usually unpleasant.

Inside the metal #

The first genuinely new result the law gives is not about the outside of a wire but the inside, where the enclosed current is no longer the whole current.

The linear rise is worth contrasting with the electrostatic case, where the field inside a conductor is zero. There is no contradiction: the electrostatic result is about charges having moved until they cancel the field, and there is nothing analogous for a steady current, which is a state of continuous motion rather than of equilibrium. A current-carrying wire has a magnetic field inside it, and the field does work on the conduction electrons in the sense that it deflects them — which is the origin of the pinch effect that squeezes a high-current plasma, and of the force that tries to crush a lightning-struck copper tube.

The linear rise also has a consequence for where the current goes. A changing current induces an electric field opposing the change, and because the internal magnetic field is largest near the surface, the opposition is largest on the axis — so at high frequency the current abandons the middle of the conductor and flows in a skin. That is the same induction at work in an unfamiliar place, and it is why a radio-frequency conductor is silver-plated rather than made of silver, and why a thick busbar carries no more alternating current than a thin one.

The two lines that give the solenoid #

The famous use of the law is a coil, and it is famous because the answer is short and because three of the four legs of the chosen path do nothing at all.

The two facts that make three legs idle are both assumptions and both deserve to be stated as such. That the field outside a long solenoid is zero follows from the same circulation argument applied to a loop entirely outside, plus the observation that the field must fall off somewhere; it is not exactly zero for a real coil, and the residual outside field is what makes a solenoid’s stray field a nuisance in a laboratory. That the field is purely axial inside follows from symmetry and is very good in the middle of a long coil and poor near the ends.

The result’s most useful feature is what is missing. There is no radius, so the field is uniform across the bore; there is no length, so it does not matter where along the coil the measurement is made — as long as it is not near an end. A real solenoid of length ten diameters has a central field within a per cent of $\mu_0nI$, and a field at the mouth of exactly half that, which is a large and often unwelcome fact for anyone putting a sample in a magnet.

The case it cannot do #

The best way to see what the law is really claiming is to point it at something it fails on, and the nearest such thing is the simplest object in magnetism after a straight wire.

The loop has to be done by summing Biot–Savart contributions, which for the axis is a short calculation giving $B=\mu_0IR^2/2(R^2+z^2)^{3/2}$ and for anywhere else is an elliptic integral. Two loops separated by their own radius — a Helmholtz pair — give a field uniform to a part in a thousand over a useful volume, and that result is arrived at by expanding the axial expression and cancelling the second derivative. None of it is available from a circulation argument.

The counting picture underneath the electrostatic half of the pair is flux conserved as it spreads over growing shells. That is what gives Gauss’s law its power and it is exactly what Ampère’s law lacks a version of, because a magnetic field has no sources for anything to spread from. The two laws look like a matched pair and are not: one counts what is inside, the other counts what goes round.

That is the honest position: Ampère’s law is exact and almost never sufficient. The arrangements it solves are the straight wire, the solenoid, the toroid, the coaxial cable and the infinite sheet, and the list is short because the list of highly symmetric current distributions is short. Its real work is not as a calculating tool at all but as one of the four statements that, taken together, are electromagnetism.

The force the field exerts on its own wire #

The field inside the metal was described above as rising linearly to the surface, and the consequence was mentioned in a clause: a current-carrying conductor squeezes itself. That deserves more than a clause, because it is the most consequential thing on this page and it is entirely a result of the profile.

Each element of current sits in the field produced by all the others, and the force on it points inward. The tidiest way to keep the account is as a pressure. A magnetic field of magnitude $B$ carries a transverse pressure $B^2/2\mu_0$, so a wire whose surface field is $B$ is being squeezed from outside by that much and by nothing from within, since the field on the axis is zero.

The numbers decide whether it matters. The 6 mm busbar above, at 400 A, has a surface field of 13.3 mT and a magnetic pressure of 70 pascals — a thousandth of an atmosphere, and nothing at all. A lightning strike is a different arithmetic: 30 kA through a channel five millimetres across gives a surface field of 1.2 tesla and a pressure of nearly six atmospheres, which is enough to collapse a thin metal tube and is why a struck downpipe is sometimes found crushed rather than melted.

Push the current higher and the pressure wins over the material. At one tesla the magnetic pressure is four atmospheres; at a hundred tesla it is four gigapascals, which exceeds the tensile strength of anything. That is not a limitation of engineering but a statement about what a field costs to contain, and it is why every magnet above about a hundred tesla is destroyed by the pulse that makes it.

The same pressure was, for a while, a plan. If a current squeezes its own channel, then a strong enough current through a plasma should compress and heat it to fusion temperatures with no walls involved, and through the early 1950s the pinch was the leading approach to controlled fusion in Britain, the United States and the Soviet Union. It fails for a reason visible in the geometry. A column that is slightly narrower in one place has a larger field there, so it is squeezed harder there, so it narrows further — the sausage instability, growing in microseconds. A column with a slight bend has a crowded field on the inside of the bend, so it is pushed outward, so the bend grows — the kink. Both were derived and both were observed, and ZETA’s announcement of thermonuclear neutrons in 1957 was retracted the following year when the neutrons turned out to come from ions accelerated by the instabilities rather than from a hot plasma.

The magnetic-confinement machines that followed all spend most of their design effort on stabilising against exactly these two modes, and the field profile on this page is where both of them come from.

The null experiments Ampère actually did #

The law bearing Ampère’s name is not the law he found, and how he found his is worth a paragraph because the method is better than the result.

Measuring a force between two coils absolutely is hard: it requires a known current, a known geometry and a calibrated balance, and every one of those carries an error. Ampère avoided all of it by finding arrangements in which the force is exactly zero, and by measuring only the failure of a delicately suspended conductor to move.

He established four such nulls. A wire doubled back on itself, carrying current out and back along the same path, exerts no force at all — so the effects of opposed currents cancel exactly. A wire bent into tight zigzags acts on a distant circuit exactly as a straight wire along the same line does, so what matters about a current element is its displacement and not the path it wanders along. A closed circuit exerts no force on an element of current constrained to move along a circular arc centred within it. And the force between two circuits is unchanged when every linear dimension in the arrangement, and every distance between them, is multiplied by the same factor.

Each of those is a constraint on the possible form of the force law between two current elements, and together they leave essentially one candidate, which Ampère then wrote down. No absolute measurement appears anywhere in the derivation. What is measured is the absence of a deflection, which requires only that the instrument be sensitive, not that it be calibrated.

That is a method worth recognising when it recurs, and it recurs constantly — in the Eötvös experiment, in the bridge circuits of the nineteenth century, in every lock-in measurement made today. A null is easier to measure than a value, because a null has no scale to get wrong.

What an ampere used to be #

The constant $\mu_0$ in the law was exactly $4\pi\times10^{-7}$ for a hundred and thirty years, and that exactness was not a measurement — it was a definition, and this law is where it lived.

Two parallel wires a metre apart, each carrying a current $I$, attract with a force per unit length of $\mu_0I^2/2\pi d$. Fixing $\mu_0$ fixes the force, and the ampere was defined as the current which produces exactly $2\times10^{-7}$ newtons per metre between two infinitely long parallel wires a metre apart in vacuum. The definition is a straightforward reading of the law on this page, and it was realised in practice by a current balance: a coil hung from one arm of a scale, weighed against a mass.

The 2019 redefinition of the units reversed the arrangement. The elementary charge is now exact, the ampere is a count of charges per second, and $\mu_0$ became a measured quantity — known to about a part in $10^{10}$, and no longer $4\pi\times10^{-7}$ by decree. Nothing physical changed; what changed is which constant is allowed to have an uncertainty. The law is unaffected either way, which is the point of stating it in a form with a constant in it.

The name, and the correction #

Ampère’s own work of 1820–27 was not this law. He measured forces between current-carrying wires, established the force law between current elements, and built the mathematical theory of what he called electrodynamics, in seven weeks of intense work after hearing of Ørsted’s discovery. What is now called Ampère’s law was extracted from that body of work later, in the field language Faraday and Maxwell supplied.

And in the form written above it is wrong — not approximate, but inconsistent — for any situation in which the current is not steady. The reason is visible in the law’s own terms: the circulation is supposed to equal the current through any surface bounded by the loop, and for a circuit charging a capacitor, two perfectly good surfaces bounded by one loop have different currents through them. Maxwell’s repair, the displacement-current term, removes the inconsistency and turns the four equations into a set with wave solutions. So the law on this page is the steady-current special case of something larger, and its restriction is not a technicality — the missing term is the one that produces light.

What it costs, and where the model stops #

The permeability is not a constant in matter. Everything above is written for vacuum with $\mu_0$. Put iron inside the solenoid and the field is multiplied by a factor of several thousand, which is not constant, depends on the history of the sample, and saturates. The correct statement in matter uses a different field, $\mathbf{H}$, whose circulation counts only the free current and leaves the magnetisation currents to a separate term.

Nothing here is about forces. The law gives a field, and getting from a field to a force needs the separate statement about what a field does to a moving charge. Ampère himself worked entirely with forces and never used a field at all, which is a reminder that the field language is a choice — an extremely good one, but not the only description of the same experiments.

Steady means steady. The law as stated fails at any frequency high enough that the displacement term matters, which for a laboratory circuit means anything above a few megahertz once the geometry gets to be a reasonable fraction of a wavelength.

What the picture cannot show #

The four loops at the top of this page are drawn in a plane, and the paths in Ampère’s law need not be planar at all — any closed curve in space will do, however knotted, and the answer still counts what is threaded through it. That is a topological statement, and the drawing cannot make it because a page has only two dimensions to spend.

Nor can the figure show the sign convention, which does real work. Which way round the loop is traversed decides the sign of the circulation, and which side of the surface counts as positive is fixed to it by a right-hand rule. Reverse one without the other and the law appears to fail. In the four panels the sense is the same throughout and the arrows are omitted, which keeps the picture clean and hides the one thing most likely to be got wrong in practice.

The ladder from here #

Later rungs on this anchor: the field of a toroid, where the same rectangle argument gives a field that falls as $1/r$ inside the winding and is exactly zero outside; the coaxial cable, where the outer conductor’s return current cancels everything beyond it; $\mathbf{H}$ and $\mathbf{B}$ in magnetic materials, and what the circulation counts when magnetisation currents are present; the vector potential, which turns the circulation into a flux by Stokes’s theorem and is the form the law takes in any serious calculation; and the displacement current, which is the reason the law as written here is a special case.

The neighbouring ladders are Gauss’s law, which is the same accounting applied to flux and charge, and the magnetic field, whose closed lines are the geometric statement of everything above.