The same force whichever way the surface faces

The two rungs below this one treat a magnetic field as something a fluid carries. Flux freezing says the field goes where the material goes, and the prohibition inside it says the material cannot change which field line it is attached to. Neither says anything about the field pushing back.

It does push back, and the way it does is stranger than the usual description admits. The usual description is a pair: a pressure $B^2/2\mu_0$ across the field lines, and a tension $B^2/\mu_0$ along them. Both numbers are correct. Both names are useful. And presenting them as two effects conceals the fact that they are one quantity looked at from two angles — with the same magnitude at every angle, including the angles neither name covers.

Where the arrows come from #

The bookkeeping is a tensor, and it is the one an electromagnetic force can be read off a surface with rather than by adding up forces on charges. For a field with no electric part worth keeping — which is every slowly moving conductor, because a frame moving at a millionth of light speed carries an electric field a millionth as large — it is

$$T_{ij} = \frac{1}{\mu_0}\left(B_i B_j - \tfrac{1}{2}\delta_{ij} B^2\right).$$

The force per unit area transmitted across a surface whose normal is $\hat{n}$ is $T \cdot \hat{n}$. Put the field along $x$ and the normal at angle $\theta$ to it, and the arithmetic takes one line:

$$\mathbf{t} = \frac{B^2}{2\mu_0}\,(\cos\theta,\ -\sin\theta).$$

Two things about that expression are worth more than the numbers usually quoted from it. Its magnitude is $B^2/2\mu_0$ with no dependence on $\theta$ whatsoever. And its direction is $-\theta$ where the surface’s own direction is $+\theta$: a mirror image in the field, not a rotation.

So the field does not exert a pressure on some surfaces and a tension on others in the way a fluid exerts a pressure on all of them. It exerts one force of one size on every surface there is, and hands out the names according to where that force happens to point. At $\theta = 90°$ — a surface cut across the field, its normal perpendicular to the lines — the force points back along the inward normal, and gets called a pressure. At $\theta = 0$ — a surface cut along the field — it points out along the outward normal, and gets called a tension. In between it does neither.

The quoted tension of $B^2/\mu_0$, twice the magnitude of anything on the picture, is not a traction at all. It is what a slab feels: the outward pull on the face at one end plus the outward pull on the face at the other, which is two lots of $B^2/2\mu_0$. A number arrived at by adding two forces on two different surfaces has been given a name that makes it sound like a property of one, and that is where the impression of two separate effects comes from.

The same statement, as a force on a current #

Nothing in the tensor is new physics, and it is worth seeing what it is a repackaging of, because the packaging is the whole of its usefulness.

The force per unit volume on a conducting fluid carrying a current density $\mathbf{j}$ in a field $\mathbf{B}$ is $\mathbf{j} \times \mathbf{B}$, and that is where every magnetic effect in a plasma comes from. Ampère’s law reads the current off the field’s circulation, so the current can be eliminated in favour of the field alone, and the resulting expression separates into two terms:

$$\mathbf{j} \times \mathbf{B} = -\nabla\!\left(\frac{B^2}{2\mu_0}\right) + \frac{1}{\mu_0}(\mathbf{B}\cdot\nabla)\mathbf{B}.$$

The first term is the gradient of a scalar with the dimensions of a pressure, and it is where the name comes from: a region of strong field pushes toward a region of weak field exactly as a region of high gas pressure does. The second term is what a field does when its own direction changes along itself, which is curvature, and it is where the tension comes from.

That split is a choice rather than a fact. The two terms are not independent — a purely azimuthal field around a wire has a pressure gradient and a curvature that partly cancel, and separating them means separating something that arrives as one vector. The tensor is the form in which no such choice has been made, which is why the traction picture shows one arrow where the decomposition shows two.

The practical difference between the two descriptions is which questions they answer easily. The decomposition answers what accelerates this fluid element, and is what a simulation integrates. The tensor answers what force crosses this surface, and is what a person holding a magnet feels.

Four hundred kilopascals, on a fridge #

The magnetic pressure is not an astrophysical quantity that happens to be written in SI units. It is the number that decides how hard an electromagnet holds.

An iron pole face carrying a field $B$ perpendicular to it, in contact with a piece of iron, transmits a traction of $B^2/2\mu_0$ across the contact — the $\theta = 90°$ case on the first figure, the one called a pressure, and here appearing as a pull because the field is continuous across the join and the surface being held is the far side of it. At one tesla that is 398 kilopascals, which is four atmospheres, which is forty tonnes per square metre. A lifting magnet with a pole face the size of a dinner plate holds about three tonnes, and the specification sheet is that arithmetic and nothing else.

The same expression sets a ceiling on every magnet ever built. Iron saturates near 2 tesla, so an iron-cored electromagnet cannot pull harder than 1.6 megapascals however much current is put through it, and every design above that figure is a superconducting one with no iron in it. A 20-tesla superconducting coil is holding itself apart against 160 megapascals — the yield strength of structural steel — which is why the mechanical support of a large magnet costs more than the conductor and why the failure mode of one is explosive rather than electrical.

And it explains a laboratory effect that looks like a separate phenomenon. A current flowing down a column of plasma makes an azimuthal field around itself; that field’s tension pulls the column inward; and the column squeezes until its gas pressure balances the pinch. The equilibrium condition is $\beta = 1$ at the boundary, which is to say the pinch stops exactly where the two stresses of this essay are equal. Every confinement device is a variation on arranging that balance somewhere useful, and every instability they suffer from is that balance being achievable but not stable.

Whether any of it matters #

A stress of forty pascals sounds negligible and a stress of thirty-six kilopascals sounds decisive, and neither impression survives contact with what the field is competing against. The comparison that decides is the ratio of the gas pressure to the magnetic pressure, and it is called the plasma $\beta$:

$$\beta = \frac{n k T}{B^2/2\mu_0}.$$

Below one, the field’s stress exceeds the gas’s and the gas is obliged to move where the field permits. Above one, the gas wins and drags the field wherever it goes — still frozen in, still conserving flux, and with no say in the matter.

The spread is the reason magnetohydrodynamics reads as two subjects. In a corona, where $\beta$ is a thousandth, the field’s geometry is the geometry of everything, and the gas is a tracer that makes the lines visible. In the interior of the Sun, where $\beta$ is ten thousand, convection does what convection does and the field is carried about like a dye. The transition between those two regimes happens inside a single star, over a few hundred kilometres of the photosphere, which is why the surface of the Sun is where all the interesting magnetic structure is.

It also explains why fusion machines are quoted by their $\beta$. A tokamak’s whole engineering problem is to hold a hot gas away from a wall using a field, and $\beta$ is the efficiency of that arrangement — how much plasma pressure is bought per unit of magnetic pressure paid for. The figure puts a full-field tokamak near three per cent, and three per cent is not a failure of design: it is close to what the stability limits allow.

A bend is a force #

Once the field has a tension, a bent field line is a stretched string, and a stretched string that is bent pulls toward the inside of the bend. The force per unit volume is the tension divided by the radius of curvature, which is $B^2/\mu_0 R$, exactly as a violin string of tension $T$ bent to radius $R$ carries $T/R$.

The quantity that came out of that division deserves attention, because it was not put in. Force per unit volume divided by mass per unit volume is an acceleration, and the acceleration turns out to be $v_A^2/R$ where

$$v_A = \frac{B}{\sqrt{\mu_0\rho}}.$$

Nothing in the setup mentioned a speed. A tension and an inertia were divided and a squared velocity came out, in the same way that dividing a string’s tension by its mass per length gives the square of the speed a wave runs along it — which is the medium deciding the speed in the one setting where the medium is a field. The next rung is that wave. What matters here is only the consequence: a bend of radius $R$ straightens in a time $R/v_A$, and every rate in a magnetised fluid is that time or a multiple of it.

This is the force that flings plasma out of a reconnection layer at the Alfvén speed, which is the step in the Sweet–Parker balance that was asserted rather than derived: the reconnected lines are sharply bent, the bend pulls, and the outflow speed is fixed by the geometry rather than by the resistivity. It is also why a coronal loop is a loop rather than a tangle. A tangled field is a field full of small radii of curvature, each pulling hard, and the pulling stops only when the curvature does.

The tube that cannot stay put #

Now put a horizontal bundle of field lines — a flux tube — inside a stratified atmosphere, and ask what holds it up.

Across the tube’s wall the total pressure has to balance, or the wall moves. Total pressure is gas plus magnetic:

$$p_{\text{in}} + \frac{B^2}{2\mu_0} = p_{\text{out}}.$$

So the gas pressure inside is lower than outside by exactly the magnetic pressure, and the atmosphere it sits in is one where pressure knows only depth. If the tube is at the same temperature as its surroundings — which it will be, given time, since heat conducts along field lines very efficiently — then the density follows the gas pressure, and the tube is lighter.

The deficit is $1/(1+\beta)$, and the shape of that expression is the whole result. It is small when $\beta$ is large — a deeply buried tube is only very slightly lighter than the gas around it — and it is never, for any finite $\beta$, zero. There is no field strength and no depth at which a horizontal flux tube is neutrally buoyant. It can only be slowly buoyant, and given the four hundred thousand kilometres of a convection zone to rise through, slowly is fast enough.

That is magnetic buoyancy, and it is the reason the Sun has a magnetic cycle rather than a magnetic field. A toroidal field wound up by differential rotation near the base of the convection zone cannot stay there. It is lighter than its surroundings, it rises, it breaks through the surface, and the two places where a rising loop’s ends cut the photosphere are the two spots of a sunspot pair — which is why they have opposite polarities, why the leading one sits slightly closer to the equator, and why the whole arrangement reverses sign every eleven years. None of that is put in; all of it follows from a tube being lighter than what it displaces, which follows from a pressure balance, which follows from the field having a stress at all.

The stability question runs the same way as a parcel in a stratified column: the tube’s fate depends on how its density deficit changes as it moves against how the environment’s density changes with height. A tube can be held down by an environment that thins faster than the tube does, and released by one that does not, which is why the field does not simply leak out of the Sun continuously.

What a sunspot is, arithmetically #

The last figure is the one where the numbers stop being illustrative. Sunspot umbral fields are measured — the Zeeman splitting of an infrared iron line is large enough to resolve directly — and they come out near 0.3 tesla. That is 35.8 kilopascals of magnetic pressure. The gas pressure at the level the ordinary photosphere is seen at is about 12.

The immediate consequence is that a sunspot cannot exist at the level of the photosphere at all. There is not enough gas pressure there to be displaced. The spot’s interior must therefore sit lower, where the surrounding gas pressure has risen enough to match, and the depth is a prediction: on a scale height of 150 kilometres, matching 35.8 kilopascals means going down $150 \ln(35.8/12) = 164$ kilometres.

This has been measured, by an argument older than any of the physics. A spot near the limb of the Sun is seen almost edge-on, and if its floor is depressed the near wall hides part of it — so the spot looks lopsided in a way that depends on the depression. Alexander Wilson noticed the effect in 1769. Modern versions of the measurement give four to eight hundred kilometres.

So the pressure balance gets the scale and not the number, and the honest reading is that something has been left out. What has been left out is opacity: cooler gas is more transparent, so the level at which the umbra becomes visible is not the level at which its pressure matches, but a level further down still. Including that closes some of the gap and not all of it, and the residual is currently an argument about the umbral atmosphere’s structure rather than about the field.

The darkness itself is a separate consequence and a cleaner one. The field suppresses convection — a parcel trying to move across the lines has to bend them, and bending them costs $B^2/\mu_0 R$ per unit volume — the same prohibition that stops a frozen-in field changing its connectivity, acting on a convective cell instead of on a flare — so the umbra loses the main channel by which heat reaches the solar surface, and cools until radiation alone can carry what is left. It settles near 4,100 K against the surroundings’ 5,780. The fourth power turns that into a brightness ratio of 0.253, and the measured umbral intensity is about a fifth. A sunspot is not dark because it is magnetic. It is dark because it is cool, and it is cool because a field with a stress can stop convection.

Where this stops being right #

The electric part of the stress was dropped, and dropping it is a statement about speed. The full Maxwell tensor has an electric term of the same form, and it was discarded because in a conductor moving at $v$ the electric field is of order $vB$, so its stress is smaller by $(v/c)^2$. For the solar photosphere that is $10^{-10}$. For a pulsar magnetosphere it is one, and none of this arithmetic survives there.

A stress tensor says what force crosses a surface, not where the force is applied. The energy of a field can be booked in more than one place and so can its momentum; what is unambiguous is the total across a closed surface. Every figure here draws a traction on a plane and none of them shows a mechanism, because there is no mechanism to show at this level of description — the tensor is a summary of what the Lorentz force on the currents adds up to.

The buoyancy argument assumed equal temperature and a thin tube. A tube thick enough for its own curvature to matter has a tension term in its balance as well, and a tube that has not had time to reach the temperature of its surroundings can be denser than them. Both corrections have the wrong sign for the simple story often enough to matter.

And β is not a single number for a real object. It varies by four orders of magnitude between the base and the top of one coronal loop, and quoting a value is quoting a place. The figure’s rows are each a representative point in a distribution, not a property of a region.

What the drawings leave out #

The traction figures draw a surface as a point with an arrow on it, which is exactly the abstraction the tensor performs and exactly what makes it unintuitive. There is no surface in a plasma. The planes drawn are imaginary cuts, and the forces drawn across them are internal forces that cancel in pairs everywhere except at a real boundary. Nothing on those pictures would be measured by an instrument placed there.

The β figure draws seven points and hides seven distributions. And the sunspot figure draws pressure against a depth axis as though “depth” were a well-defined thing in an object with no surface — the photosphere is a place where the opacity changes quickly, not a place where the Sun stops, and every number on that axis is measured relative to a level defined by how far light gets rather than by where anything is.

What the tension is about to become #

Three rungs stand on flux-freezing now. The first is a conservation law; the second is the prohibition inside it; this one is what the conserved thing does mechanically once it is admitted that it has a stress.

The habit worth carrying away is about names that arrive in pairs. When a subject offers two effects with two names and one formula between them, check whether the formula distinguishes them. Here it does not: magnetic pressure and magnetic tension are the same traction of the same size, sorted by the angle it happens to point at, and the 45° case that belongs to neither is the evidence. The same test applied elsewhere separates the real pairs from the bookkeeping ones — electric and magnetic fields are genuinely one object seen from different frames, while a centrifugal force and a Coriolis force really are two distinct terms with different structures.

What is left on this ladder is the wave. The tension supplies a restoring force and the plasma supplies an inertia, which is a string; the speed that came out of the curvature argument uninvited is the speed a disturbance runs at; and the wave that results carries no compression at all, which makes it the only wave in classical physics whose speed contains no thermodynamic quantity.