The mass a magnetic field has to be held in

Where the energy of a field is settled that a magnetic field stores energy in the space it fills, $B^2/2\mu_0$ per cubic metre, and the force read off a surface that touches nothing found that the same quantity is a stress. Field lines behave like stretched elastic that also pushes sideways: a tension $B^2/2\mu_0$ along them and a pressure $B^2/2\mu_0$ across them. Energy per unit volume and force per unit area have the same units, and for a field they are the same number.

That identity has a blunt practical consequence. Every current-carrying winding sits in the field it makes, and the field pushes on it with that pressure. One tesla pushes with about four atmospheres, which is easily held. But the pressure grows as the square of the field, and at a few tens of teslas it is pushing as hard as the strongest metals can resist. A high-field magnet is a pressure vessel whose contents are field, and the question of how strong a field can be made turns into the question of what can hold it.

Why the energy is the pressure #

The identity of energy density and pressure can be checked with a thought experiment that needs nothing but conservation of energy. Take a long solenoid, its winding a thin sleeve of radius $R$, and let the sleeve expand outward by a small distance $\delta R$ while the flux through it is held fixed — as it would be if the winding were superconducting, or if the expansion were fast compared with the circuit’s time constant. The flux is $\Phi = B\pi R^2$, so holding it fixed while the area grows lowers the field: $\delta B/B = -2\,\delta R/R$. The stored energy per unit length is $W = (B^2/2\mu_0)\pi R^2$, and with $B^2 R^2$ falling as $1/R^2$ times a constant, the energy falls by

$$\delta W = -\frac{B^2}{2\mu_0}\,2\pi R\,\delta R.$$

Energy lost by the field is work done on the winding, and $2\pi R\,\delta R$ is the volume swept out per unit length. So the force per unit area on the winding is $B^2/2\mu_0$, outward. The pressure is the energy density because moving a wall that confines a field changes the field’s energy by exactly the energy density times the volume swept.

The same argument, run along the axis instead, shows the tension along the lines: lengthening a solenoid at fixed current adds field energy, and the windings are pulled towards each other by the field just as the ends of a stretched band pull together. The same force whichever way the surface faces found the pressure turning up in a surface current regardless of which way the surface was oriented; the virtual-work argument shows why it has the size it has. And the field that cannot get out met the most direct case, a superconductor that excludes a field from its interior, so that the field presses on its surface with exactly this pressure. A superconducting sphere in a field of a few teslas is squeezed with tens of atmospheres, which is the force that levitates a superconductor over a magnet, and which tears apart superconducting samples placed in fields too strong for them.

There is a pleasing corollary. A gas exerts a pressure equal to two-thirds of its kinetic energy density, and radiation one-third of its energy density. A magnetic field pressing sideways exerts a pressure equal to its whole energy density, more for its energy than any gas — which is why magnetic fields are such effective containers, and also why the containers that hold them have to be so strong.

When the field pushes harder than metal holds #

The comparison is simple to draw: the field’s pressure against the strength of the materials that might carry the current.

The crossing points are the fields at which a conductor with nothing else to support it would yield under its own field: $B = \sqrt{2\mu_0\sigma}$. For hard-drawn copper, with a working strength of about 300 megapascals, that is 27 T. Copper alloys strengthened with fine niobium filaments, developed for exactly this purpose, reach about a gigapascal and 50 T. Above that, no known conductor can hold its own field unaided, and the magnet has to be built with reinforcement — steel, or fibre composites wound round the conductor — whose job is purely mechanical. In the most powerful continuous magnets, more of the volume is reinforcement than conductor.

These are not exotic numbers. Laboratory resistive magnets producing steady fields of 30 to 45 teslas, such as the hybrid magnets at the National High Magnetic Field Laboratory in Tallahassee, operate within a factor of two of these limits, and their design is dominated by managing stress. The current density and the heat it generates matter, but the reason they cannot simply be wound more tightly is that the windings would burst.

The theorem for any shape #

It is natural to wonder whether a clever geometry could avoid the problem, arranging the windings so that the forces balance each other and little structure is needed. A theorem says no, and it sets a minimum that no arrangement can beat.

Add up the stresses in a static, self-contained system — the field together with the structure holding it — and the total force on any part is zero. Weighing what cannot be put on a scale used the same bookkeeping, the virial theorem, to weigh galaxies from their motions. For a static magnet, multiplying the balance of forces by position and integrating over all space gives an identity: the integral over the structure of the trace of its stress — the sum of its three principal stresses, tension positive — equals the energy stored in the field,

$$\int_{\text{structure}} (\sigma_1 + \sigma_2 + \sigma_3)\, dV = W_B.$$

The field’s stresses, a tension along the lines and a pressure across them, have a trace that integrates to exactly $-W_B$ over all space, and the material must supply the rest. Compression counts negative in the sum, so any member in compression has to be paid for by extra tension elsewhere. The structure as a whole must therefore carry tension amounting to at least the stored energy.

If the material can work at a stress $\sigma$, the volume of material needed is at least $W_B/\sigma$, and its mass at least

$$M \ge \frac{\rho\,W_B}{\sigma}.$$

The bound depends on nothing but energy and the material’s specific strength, $\sigma/\rho$. A magnetic resonance scanner storing ten megajoules needs at least about seventy kilograms of steel in perfect tension. The toroidal-field coils of ITER, the large tokamak being built in France, store about 41 gigajoules; the bound is 270 tonnes of steel, and the real coils and their casings weigh many thousands of tonnes, because their structure carries enormous compressive loads where the coils press inward on one another at the centre of the machine. Every one of those compressive members has to be balanced by more tension elsewhere, which is the theorem’s prediction in hardware.

The bound is the same theorem that limits flywheels and pressure tanks. A flywheel stores kinetic energy at most $\sigma/\rho$ per kilogram of rim, and a gas tank at most a similar amount per kilogram of wall. A magnet, a flywheel and a tank are all ways of storing energy held in by stress, and all are limited by the same property of the material. Put as a number per kilogram, steel working at 1.2 gigapascals can hold at most about 150 kilojoules of field, roughly the chemical energy in three grams of petrol, and a magnet is therefore a poor way to store energy in bulk, whatever its other merits. Superconducting magnetic energy storage, proposed for smoothing electricity grids, runs into this limit directly: the energy it can hold is fixed by the tonnage of its reinforcement, not by the superconductor. That is why carbon fibre, with fourteen times the specific strength of steel, transformed flywheels and pressure vessels, and why it is wound round the coils of the strongest pulsed magnets.

The solenoid’s walls #

For the commonest magnet, a long solenoid, the stress has a familiar form. The field inside pushes outward on the winding with pressure $p = B^2/2\mu_0$, exactly as gas pushes on the wall of a pipe, and the winding develops a hoop stress

$$\sigma_\theta \approx \frac{pR}{t} = \frac{B^2 R}{2\mu_0 t},$$

for a winding of radius $R$ and thickness $t$.

A thin solenoid is weak: with a wall a twentieth of its radius it fails at 11 teslas even in the best conductor. Thickening the wall helps, but slowly, since the field the wall can hold goes as the square root of its thickness. Eventually the wall is as thick as the bore is wide, and beyond that the outer layers, where the field has fallen almost to nothing, carry less and less of the load — the same diminishing return that limits a thick-walled gun barrel. The strongest continuous magnets are built as nested coils, each reinforced separately so that its own stress is carried by its own structure, rather than relying on outer layers to hold in inner ones.

Above a hundred teslas: only inertia #

Above about a hundred teslas, no static structure survives, and the record steady fields have stalled near 45 teslas for continuous operation and about a hundred for pulses that last a fraction of a second. Higher fields are made, but only by abandoning the idea of holding them.

A single loop of copper connected to a large capacitor bank carries a current pulse of millions of amperes for a few microseconds. The field inside reaches two or three hundred teslas, the pressure exceeds anything the copper can hold, and the loop explodes outward. But metal accelerated by a pressure $p$ takes a time $\sqrt{2\rho t\delta/p}$ to move a distance $\delta$, and for a loop two millimetres thick under the pressure of 300 teslas that is a microsecond to move a millimetre. The measurement inside the loop is made in that microsecond, before the walls have moved enough to matter. The sample usually survives, because the loop explodes away from it. The field is held by nothing but the time metal takes to get out of the way.

Flux compression pushes the same idea further. A field is trapped inside a metal cylinder, and the cylinder is crushed inward by explosives, compressing the trapped flux into a smaller area and multiplying the field. Fields of over two thousand teslas have been reached this way, in Russia and the United States, for a few microseconds, at the cost of the apparatus and often the sample. The virial bound has not been broken; it has been bypassed, by replacing a static structure with a moving one, and the momentum of the moving metal takes the place of the tension that no material could supply.

What a tonne of steel can afford #

The bound can be turned round to ask how much field a given investment in material can hold.

That is a severe law. A field’s energy rises as the square of its strength, the structure must rise with the energy, and so the structure needed for a given volume of field rises as the square of the field. Doubling the field of a magnet with the same bore needs four times the reinforcement at least, in practice more. It is the reason that fusion devices are designed around a trade between field and size: the plasma pressure a magnetic bottle can hold is a fraction of the magnetic pressure, as the product a fusion plasma has to beat set out, so a stronger field allows a smaller machine — but only if the coils, and the structure that holds them, can be made to bear the square of that field. The recent push towards compact fusion devices rests on high-temperature superconductors that carry current at high fields, and their designs are limited, in the end, by the stress in the coils.

Fields that hold themselves, and fields that cannot #

The theorem has a striking consequence for magnetic fields with no material at all. A field whose stresses balance everywhere — a force-free field, in which the current flows along the field lines and there is no force on it anywhere — needs no structure locally. But the virial identity, applied to such a field alone, says that its stress trace integrates to $-W_B$, which must vanish. A force-free field confined to a finite region, with nothing outside, would have to have zero energy. So it cannot exist. Every force-free field must be anchored somewhere: to a star’s surface, to a conducting wall, to a current at infinity.

That result follows from the virial identity in two lines and was recognised in the 1950s, when force-free fields were first studied for stars; it is why the twisted, force-free magnetic structures that the twist that outlives the turbulence followed in laboratory plasmas and the solar corona are always rooted in something: they are anchored in the Sun’s surface, or held in by a conducting vessel. A magnetic bubble floating free in empty space, holding itself together by its own field, is forbidden by the same arithmetic that sets the minimum steel for a scanner. The current that squeezes what carries it found the same balance in a plasma pinch, where the virial theorem fixed the relation between current and temperature independently of the pinch’s profile.

What the picture cannot show #

The working strengths in the figures are representative values, not the limits of particular products. Real high-field magnets are designed to stresses below a material’s ultimate strength by a margin for fatigue, since a pulsed magnet is loaded and unloaded thousands of times, and for the cold: superconducting coils run at a few kelvin, where steels are stronger but more brittle and composites behave differently. The figure of 270 tonnes for 41 gigajoules uses a working stress of 1.2 gigapascals for steel; a more conservative stress would raise it proportionally.

The virial bound is a minimum that real designs do not approach. It assumes the structure is everywhere in uniform tension at its working stress, which no real coil is. Some parts are in compression, some in bending, some lightly loaded; the actual structure is usually several to many times the bound. The bound is useful as a floor and as an argument that no design trick avoids it, not as an estimate of what a magnet will weigh.

The hoop-stress formula for a solenoid is the thin-wall approximation, with a crude halving for thick walls. Thick coils need a full elastic analysis, with the field falling through the winding and the stress varying through the thickness; and long solenoids also carry axial compression from the end fields pulling the windings together. The inertial time is for a free conductor under a constant pressure, ignoring the strength of the metal and the field’s rise and fall during the pulse; real single-turn coils are modelled with full magnetohydrodynamic codes.

Still open: how strong a steady field can become #

The steady-field record has advanced slowly: from the low thirties of teslas some decades ago to 45 teslas in hybrid magnets combining superconducting outer coils with resistive inserts, and to around 45 teslas in all-superconducting test coils using high-temperature superconductors, whose ability to carry current at very high fields makes the conductor no longer the limit. The limit then is stress, and the open question is how close to the virial bound real designs can be pushed with stronger reinforcements and better ways of distributing the load. Laboratories are pursuing steady fields of 60 teslas and beyond, and each step requires new materials or new ways of supporting the windings. Whether a continuous hundred-tesla magnet can ever be built depends on finding a structure that can hold four gigapascals indefinitely, and no conductor-and-reinforcement system has yet come close.

A magnetic field pushes on its conductors with $B^2/2\mu_0$ — the strength of hard copper at 27 T, 4.0 GPa at 100 T — and by the virial theorem any static structure holding a field must carry tension equal to the stored energy, a mass of at least ρW/σ whatever the shape: 270 tonnes of steel for 41 GJ, and four times the steel for twice the field; above about a hundred teslas only inertia holds a field, for a microsecond. The habit worth carrying is to ask what holds a stored energy in, because when the energy is large enough, the holding is what costs.