The force read off a surface that touches nothing

There are two ways to compute the force between two charged objects. The first is to add up the force each piece of one exerts on each piece of the other, which requires knowing where every piece is. The second is to draw a surface between them, measure the field on that surface, and integrate. The second way does not require knowing what is inside at all.

That is not a trick of symmetry or a special case. It is the content of Maxwell’s stress tensor, and it says that the electromagnetic field transmits force across empty space in the way a stressed solid transmits it across a plane — with a value at each point, a direction, and a bookkeeping that is exact rather than suggestive.

Faraday’s two words, made into a number #

Faraday described the lines of force as though they were physical objects: pulling along their length like stretched cords, and pushing sideways as though crowded. It was a picture rather than a theory, and Maxwell turned it into arithmetic that either works or does not.

The tensor is

$$T_{ij} = \varepsilon_0\!\left(E_iE_j - \tfrac{1}{2}\delta_{ij}E^2\right) + \frac{1}{\mu_0}\!\left(B_iB_j - \tfrac{1}{2}\delta_{ij}B^2\right)$$

and $T_{ij}$ is the $i$-component of force transmitted per unit area across a surface whose normal points along $j$. It reads more clearly split in two: an isotropic pressure of $\varepsilon_0E^2/2$ in every direction, plus a tension of $\varepsilon_0E^2$ along the direction the field points. Add them and a surface perpendicular to the field feels a net tension of $\varepsilon_0E^2/2$, while a surface parallel to it feels a net pressure of the same size.

The size of the two pieces is the same number, and it is the energy density. That is not a coincidence to be admired but an identity forced by dimensions: energy per volume and force per area are the same units, and a field carrying energy at a density $u$ transmits stresses of order $u$. Where where the energy of a field actually is puts a number on the energy, this puts the same number to work.

Where the expression comes from #

The tensor is not a guess that happens to work. It is what is left when the force on a lump of charge and current is rewritten using Maxwell’s equations and nothing else.

Start with the force per unit volume on matter carrying charge density $\rho$ and current density $\mathbf{J}$, which is $\rho\mathbf{E} + \mathbf{J}\times\mathbf{B}$ and is not in dispute. Then eliminate the matter: Gauss’s law replaces $\rho$ by $\varepsilon_0\nabla\cdot\mathbf{E}$, and the Ampère–Maxwell law replaces $\mathbf{J}$ by a combination of $\nabla\times\mathbf{B}$ and $\partial\mathbf{E}/\partial t$. What remains contains only the fields, and it rearranges — with some work and one use of Faraday’s law to keep the time derivatives together — into the divergence of a quantity plus the time derivative of another.

The quantity whose divergence appears is the stress tensor. The one being differentiated in time is $\varepsilon_0\mathbf{E}\times\mathbf{B}$, the momentum density of the field. So the rearrangement is a conservation law: the momentum lost by the matter in a region is either carried out through its boundary or stored in the field inside, and the tensor is the flux term of that law rather than a separate postulate.

This is worth having because it settles two questions at once. It says why the surface integral gives a force — a flux of momentum is a force, by definition. And it says why the static case is special: the stored term has no time derivative to take, so everything that leaves through the boundary came from the matter. Every caveat later in this essay is a case where that second term is not zero.

The same rewriting is what makes the term that made light necessary, incidentally. Without the displacement current the algebra does not close, and the resulting expression fails to be a conservation law at all — momentum would appear and disappear in regions where a current was changing.

Two charges, and the plane between them #

The cleanest test of the claim is the case where the answer is already known, so take two equal charges and ask the plane halfway between them what force it is carrying.

On that plane, symmetry does something useful. The two charges’ fields have z-components that cancel exactly, so the field there lies entirely in the plane. A surface whose normal is perpendicular to the field feels the pressure and not the tension, and the pressure pushes the two halves of space apart. For opposite charges the situation reverses precisely: the in-plane components cancel and the field is perpendicular to the plane, so the plane feels tension and the halves are pulled together.

That is the whole of the mechanism, and it explains something the naive picture gets wrong. Two like charges have no field line running between them. Their lines curve away from each other and never connect, so there is nothing to pull, and a reader who has only been told about tension has no account of why they repel at all. The account is that the lines lying flat against the plane press on it — the pressure term, doing the work the tension cannot.

The number that comes out is Coulomb’s law, and getting it back is the point rather than the discovery. The integral is over a plane passing through nothing but empty space; nothing about what the charges are made of enters it; and the answer agrees with the law derived from the charges themselves to the accuracy of the quadrature. If it did not, the tensor would be wrong.

The slow tail is a real practical cost. Half the force has arrived within a couple of separations of the axis, and collecting the last per cent takes the rest of the infinite plane. That is why the method is used where the surface can be chosen to make the tail small — inside a machine, around a pole piece, on a boundary the geometry already supplies — and why a brute-force application to an arbitrary problem is usually the wrong tool.

Any surface, and why #

The stronger claim is that the surface does not matter. Any closed surface enclosing the same charge and no other must give the same force, because the difference between two such surfaces is a closed surface enclosing no charge at all, over which the integral vanishes.

The three surfaces that return nothing are the more interesting half of the figure. A surface enclosing both charges reports the force the outside world exerts on the pair, which is nothing, because there is no outside world in this problem. The internal repulsion is invisible to it — as it must be, since Newton’s third law makes the pair’s internal forces cancel, and a method that reported otherwise would be reporting a system able to push itself along.

That is the structural reason the method is trustworthy and also the reason it must be used carefully: the answer depends on what the surface encloses and on nothing else, so a surface drawn through a body rather than around it gives the force on the part enclosed, and a surface accidentally cutting a wire gives an answer nobody asked for.

There is a practical corollary that is easy to miss. Because every enclosing surface agrees, the surface can be chosen for convenience rather than for meaning, and the good choice is almost always the one where the field is simplest rather than the one closest to the object. A sphere hugging an oddly shaped electrode is the worst possible surface: the field on it varies over every square millimetre. A large sphere far away, or a flat plane placed where symmetry makes one field component vanish, turns the same integral into a few lines of arithmetic. Choosing the surface is the whole of the skill, and it is the same skill a symmetry argument uses.

The argument here is the same one that makes counting what comes out work: a statement about a closed surface, indifferent to how the contents are arranged, provable by noting that the difference of two surfaces encloses nothing. Gauss’s law counts charge that way; the stress tensor counts momentum.

The case the method was built for #

Recovering Coulomb’s law is a check. The reason the tensor exists is the cases where nothing simpler is available.

Consider a dielectric slab being pulled into the gap of a charged capacitor. The force on it is a sum over every polarised molecule of the force the local field exerts on its induced dipole — a calculation requiring a model of the polarisation, of the field each molecule sees rather than the average, and of what happens at the ragged edge where the slab enters. The force that lives where the model is not is what that calculation runs into.

Draw a surface in the vacuum around the slab instead, and none of that is needed. The field on the surface is measurable or computable, and its integral is the total force whatever is happening inside — including in the ragged edge, which is where the force actually comes from. Magnetic circuits are computed this way as a matter of routine: the pull of an electromagnet on its armature comes out as $B^2A/2\mu_0$ over the pole face, which is the tensor evaluated on a surface in the air gap, and the iron never enters the calculation.

The same argument reaches into problems with no charges in them at all. The charge that has to be somewhere else computes the attraction of a charge to a grounded plane by inventing an image charge behind it. The stress integral over the plane itself gets the same answer using only the real field on a real surface, and it makes clear what the image is: a way of writing down a field that satisfies the boundary condition, not a thing.

Where the numbers start to matter #

The electric stresses in the figures above are tiny. A field of three million volts per metre — enough to spark across a centimetre of air — carries about forty pascals, which is a four-thousandth of atmospheric pressure and less than the wind on a windy day. Electrostatic forces are famously feeble in this sense, and the reason the pressure a charge puts on its own metal is a subtle effect rather than an obvious one is precisely this.

Magnetic fields are a different matter.

At 10 tesla the field inside a magnet is pushing its own windings apart at four hundred atmospheres — comparable with the pressure inside a scuba cylinder, applied outward over the whole bore. A large research magnet is therefore a pressure vessel first and an electrical device second, and its limit is set by the yield strength of the steel and the composite holding the coil together rather than by the current the conductor will carry. Fusion magnets, high-field solenoids and pulsed magnets all fail structurally before they fail electrically, and pulsed magnets above about a hundred tesla destroy themselves on every shot as a matter of design.

The scaling is what makes this abrupt. Stress goes as the square of the field, so doubling the field quadruples the load: a 1 tesla laboratory electromagnet holds four atmospheres and needs no thought, a 5 tesla magnet holds a hundred and needs a proper former, and a 20 tesla magnet holds sixteen hundred and is at the limit of what any material can be asked to contain continuously. There is no gradual region. A designer moving from one field to the next is not scaling a structure but changing which failure decides the answer.

In that regime the field stops being a bookkeeping device and becomes the load. The engineer computing hoop stress in a magnet coil is integrating the same tensor over the same kind of surface, and calling the result a pressure is not a metaphor.

The same statement is why a magnetic field resists being bent. The tension along a line, $B^2/\mu_0$ in the decomposition above, gives a restoring force to a bent field line exactly as tension in a string does — which is the origin of the wave that runs along a magnetic field in a conducting fluid, and of the way the field that cannot get out resists being tangled by the flow it is frozen into.

Where the model stops #

The static statement is not the general one. In full, the surface integral equals the force on the contents plus the rate of change of the electromagnetic momentum stored inside. In statics the second term is zero and the integral is the force. It is not zero for a radiating system, and it is the whole content of the momentum of something that is not moving, where a static-looking arrangement of charge and current turns out to be storing momentum in its own field.

Inside matter the tensor is not unique. There is a long-running disagreement about how to split the momentum of a field inside a polarisable medium between the field and the matter it is polarising, with two well-known answers and a century of experiments that mostly measure the sum. Everything here is in vacuum, where the ambiguity does not arise; a version written for the interior of a dielectric has to say which convention it is using, and pairs of conventions that disagree about the split agree about every measurable total.

A surface has to be in a region where the field is known. The method converts a hard interior problem into an easier exterior one, and it converts nothing at all if the field on the surface is as hard to find as the force was. Its value is exactly the value of the boundary being simple.

And the tensor says nothing about where the force is applied. It gives the total, and for a rigid body the total and the torque are all that matter. For a deformable one — a stretched dielectric, a coil — the distribution matters, and the surface integral does not supply it. That is a real limitation of the method rather than a caveat about it.

What the pictures cannot show #

The stress is a tensor and every figure here plots one of its components. What is not drawn is that the same field at the same point transmits different forces across differently oriented surfaces — a pull across one plane and a push across the plane at right angles to it — and no scalar picture can carry that. The two curves in the second figure are the same field seen by two different orientations of surface, which is the closest a plot gets to showing it.

Nor do the figures show the field lines of the two-charge configurations they compute. That is deliberate: the argument is exactly that the answer comes from the field on a surface rather than from the topology of the lines, and drawing the lines invites the reading that they are what is doing the pulling. They are a way of seeing the field’s direction, as field lines are a choice says at more length, and the direction is one of the two things the tensor needs.

Where the ladder goes next #

The field-energy ladder began with where the energy of a field actually is, asking where the energy of a charged capacitor sits and finding it in the space between the plates. It went on to the angular momentum that is in nothing at all and to the momentum of something that is not moving, where the field turns out to keep not only energy but momentum, in configurations where nothing is going anywhere. This rung completes the set: the field also transmits force, and the transmission has a value at every point and across every orientation.

The rung after it is the one where all three are a single object. Energy density, momentum density and stress are the components of one tensor in four dimensions, and its conservation is a single equation that contains the energy theorem, the momentum theorem and the force law together. The habit worth carrying forward is the one this rung is built on: when a force is hard to compute where it acts, look for a surface where the field is simple, and ask the surface instead.