On Physical Lines of Force/Modern derivations/The Maxwell stress tensor

On Physical Lines of Force → Modern derivations → The stress tensor

What Maxwell had

Prop. II of Part I resolves the state of the medium into two pieces — a hydrostatic pressure acting equally in all directions, and a tension acting along the axis of the vortices — and reports the result as six stresses on the coordinate planes, three normal and three tangential:

$p_{x x} = \frac{μ}{4 π} α^{2} - p_{0}, p_{y y} = \frac{μ}{4 π} β^{2} - p_{0}, p_{z z} = \frac{μ}{4 π} γ^{2} - p_{0}$

$p_{x y} = \frac{μ}{4 π} α β, p_{y z} = \frac{μ}{4 π} β γ, p_{z x} = \frac{μ}{4 π} γ α$

Provenance: the printed equation numbers for Prop. II could not be established; the bodies are reconstructions in Maxwell's notation. See Sources.

Resolving the stress on an element of the medium

Prop. III then differentiates that stress and obtains the force per unit volume, equations (12), (13), (14). Maxwell reads his own answer term by term:

The first term of each expression refers to the force acting on magnetic poles. The second term to the action on bodies capable of magnetism by induction. The third and fourth terms to the force acting on electric currents. And the fifth to the effect of simple pressure.

Five terms, from one stress, differentiated once. The business of this page is to write that stress as a modern reader would, take the derivative in modern notation, and see the same five things come out.

Six numbers, one object

A stress at a point is not a single number and not a vector. To say what the medium is doing at a point you must say, for every orientation of an imaginary small surface through that point, what force per unit area the material on one side exerts on the material on the other. That is nine numbers — three force components for each of three surface orientations — and Maxwell's six are the whole of them, because the array is symmetric: $p_{x y} = p_{y x}$ , and so on, as his three tangential stresses already show.

Write $H_{1}, H_{2}, H_{3}$ for Maxwell's $α, β, γ$ and let the indices $i$ and $j$ each run over $1, 2, 3$ . Then every one of Maxwell's six lines is the single statement

$p_{i j} = \frac{μ}{4 π} H_{i} H_{j} - p_{0} δ_{i j}$

where $δ_{i j}$ is 1 when $i = j$ and 0 otherwise, so that the pressure $p_{0}$ appears in the three normal stresses and nowhere else.

The modern magnetic stress tensor differs from this in one respect only: it fixes the isotropic part instead of leaving it open. In Gaussian units, with $𝐁 = μ 𝐇$ ,

$T_{i j} = \frac{1}{4 π} (B_{i} H_{j} - \frac{1}{2} δ_{i j} 𝐁 \cdot 𝐇)$

and the full electromagnetic tensor, with the electric part alongside, is

$T_{i j} = \frac{1}{4 π} (E_{i} D_{j} + B_{i} H_{j} - \frac{1}{2} δ_{i j} (𝐄 \cdot 𝐃 + 𝐁 \cdot 𝐇))$

Maxwell's Part I has only the magnetic half. The electric half arrives with the elastic cells of Part III.

Tension along, pressure across

Point the field along $x$ , so $𝐁 = B \hat{x}$ , and read off the diagonal in vacuum:

$T_{x x} = \frac{1}{4 π} (B^{2} - \frac{1}{2} B^{2}) = + \frac{B^{2}}{8 π}, T_{y y} = T_{z z} = \frac{1}{4 π} (0 - \frac{1}{2} B^{2}) = - \frac{B^{2}}{8 π}$

A positive normal stress is a pull, a negative one a push. So the medium is in tension of $B^{2} / 8 π$ along the line of force and under pressure of $B^{2} / 8 π$ across it — equal in magnitude, opposite in sense, perpendicular in direction.

Tension along the lines, pressure across them

That is word for word the sentence Maxwell set out to justify, and it is the whole physical content of the tensor. Two magnets attract because the lines of force running between them are shortening ropes; two like poles repel because the lines crowded between them are pushing sideways. Every later reformulation keeps this sentence and changes only what is supposed to be doing the pulling.

Taking the divergence

A uniform stress moves nothing: push a block equally on every face and it sits still. Force appears only where the stress varies from place to place, and the rate of that variation is the divergence. For a vector field the divergence $\nabla \cdot 𝐁 = \partial B_{x} / \partial x + \partial B_{y} / \partial y + \partial B_{z} / \partial z$ measures net outflow per unit volume. For a stress you do the same thing once for each force direction: the force per unit volume in direction $i$ is

$f_{i} = \sum_{j} \frac{\partial T_{i j}}{\partial x_{j}} = \frac{\partial T_{i 1}}{\partial x} + \frac{\partial T_{i 2}}{\partial y} + \frac{\partial T_{i 3}}{\partial z}$

which is exactly the calculation Maxwell performs in Prop. III. Carrying it out on the magnetic tensor, and using $\nabla \times 𝐇 = 4 π 𝐉$ from Ampère's law, gives

$\sum_{j} \frac{\partial T_{i j}}{\partial x_{j}} = \underset{on magnetic poles}{\underset{⏟}{\frac{H_{i}}{4 π} (\nabla \cdot 𝐁)}} \underset{on magnetizable bodies}{\underset{⏟}{- \frac{H^{2}}{8 π} \frac{\partial μ}{\partial x_{i}}}} + \underset{on currents}{\underset{⏟}{{(𝐉 \times 𝐁)}_{i}}}$

Three lines of algebra do it. Expand the derivative of the product; use the identity $(𝐇 \cdot \nabla) 𝐇 - \frac{1}{2} \nabla H^{2} = - 𝐇 \times (\nabla \times 𝐇)$ ; and let $\nabla \times 𝐇$ become the current.

Compare with Maxwell's list. His first term is a force on "imaginary magnetic matter" of density $\frac{1}{4 π} \nabla \cdot 𝐁$ — and the pole term above is precisely $𝐇$ times that density. His third and fourth terms are $μ γ q - μ β r$ and its cyclic partners, which is $(𝐉 \times 𝐁)_{x}$ written out in components: the Laplace force, the sideways push on a current-carrying wire that turns every electric motor. His fifth term, $- \nabla p_{0}$ , is ordinary hydrostatic pressure and is not electromagnetic at all; the modern tensor has already spent its isotropic part on $- \frac{1}{8 π} 𝐁 \cdot 𝐇$ , so no such term survives.

The second term deserves a word of caution. Its sign and its exact form depend on what has been absorbed into the isotropic part, and Maxwell's $p_{0}$ is an unspecified pressure where the modern $- \frac{1}{8 π} δ_{i j} 𝐁 \cdot 𝐇$ is fixed. The reconstruction of equation (12) on this wiki prints this term with the opposite sign to the one above; which sign the 1861 page carries could not be checked. The physics is not in dispute either way — a body of higher $μ$ than its surroundings is drawn toward the stronger field, which is why iron flies to a magnet.

What it says today

Maxwell built a stress because he had a material to put it in. Take the material away and the mathematics does not care, which is fortunate, because the modern reading of $T_{i j}$ is not "stress" at all. It is momentum flux density.

The electromagnetic field carries momentum, of density

$𝐠 = \frac{1}{4 π c} 𝐄 \times 𝐁$

and the conservation law for the total momentum, field plus matter, is

$\frac{\partial}{\partial t} (g_{i} + g_{i}^{mech}) = \sum_{j} \frac{\partial T_{i j}}{\partial x_{j}}$

Read that as an accounting statement. $T_{i j}$ is the rate at which momentum in direction $i$ flows across a surface facing direction $j$ ; its divergence is the net momentum delivered into a small volume per unit time; and what that momentum does on arrival is either sit in the field or push on matter. Maxwell's "force per unit volume" is the second option written on its own, which is all he needed in a paper where the field was a material and momentum in it was momentum of the material.

This is what makes radiation pressure intelligible. A plane wave absorbed head-on delivers momentum at a rate equal to its energy density, so sunlight pushes; a laser beam focused to a point can hold a bacterium still, which is what an optical tweezer does. None of that is in the 1861 paper. Maxwell had a stress in a medium, not momentum in a field, and the reinterpretation belongs to the decades after the ether was abandoned. But the tensor was not rewritten to accommodate it. The same six numbers, in the same arrangement, changed their meaning entirely and stayed correct — which is a good example of what it means to say that Maxwell's equations outlived Maxwell's model. See The model and the equations.

Where the conventions differ

	Maxwell 1861	modern
the object	six stresses, written out	one tensor $T_{i j}$ , indices $i, j = 1, 2, 3$
isotropic part	an unspecified pressure $p_{0}$ , subtracted	fixed at $- \frac{1}{8 π} δ_{i j} 𝐁 \cdot 𝐇$
sign convention	pressure counted positive	tension counted positive
the factor 4π	explicit, as always in his system	explicit in Gaussian; absorbed in SI, where the vacuum tensor is $\frac{1}{μ_{0}} (B_{i} B_{j} - \frac{1}{2} δ_{i j} B^{2})$
tension along a line	$\frac{μ}{8 π} H^{2}$	$B^{2} / 8 π$ Gaussian, $B^{2} / 2 μ_{0}$ SI
the μ-gradient term	sign as printed could not be verified	$- \frac{1}{8 π} H^{2} \nabla μ$ with the modern split
what it is	a stress in a material medium	momentum flux density of a field

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In the paper: Prop. II · Prop. III · Plain-language version: Why magnets pull