On Physical Lines of Force/Part I/Prop. III

On Physical Lines of Force → Part I → Prop. III

Maxwell's statement

Prop. III. — To find the resultant force on an element of the medium, arising from the variation of internal stress.

This is the central proposition of Part I, and one of the most economical arguments in nineteenth-century physics.

The idea

A uniform stress moves nothing. A block of material squeezed equally on all sides stays where it is. Force appears only where the stress changes from place to place: the medium pushes harder on one face of a small element than on the opposite face, and the difference is a net force.

So take the six stresses of Prop. II and differentiate. The force per unit volume is the divergence of the stress:

$$X=\frac{d{p}_{xx}}{dx}+\frac{d{p}_{xy}}{dy}+\frac{d{p}_{xz}}{dz}$$

and similarly for $Y$ and $Z$. Substituting Prop. II and sorting the result gives Maxwell's equations (12), (13), (14).

The result

$$X=\underset{\text{1: on magnetic poles}}{\underbrace{\frac{\alpha }{4\pi }(\frac{d(\mu \alpha )}{dx}+\frac{d(\mu \beta )}{dy}+\frac{d(\mu \gamma )}{dz})}}-\underset{\text{2: on magnetizable bodies}}{\underbrace{\frac{{\alpha }^2+{\beta }^2+{\gamma }^2}{8\pi }\frac{d\mu }{dx}}}+\underset{\text{3 and 4: on currents}}{\underbrace{\mu \gamma q-\mu \beta r}}-\underset{\text{5: simple pressure}}{\underbrace{\frac{d{p}_0}{dx}}}\mathrm{(}\mathrm{12}\mathrm{)}$$

$$Y=\frac{\beta }{4\pi }(\frac{d(\mu \alpha )}{dx}+\frac{d(\mu \beta )}{dy}+\frac{d(\mu \gamma )}{dz})-\frac{{\alpha }^2+{\beta }^2+{\gamma }^2}{8\pi }\frac{d\mu }{dy}+\mu \alpha r-\mu \gamma p-\frac{d{p}_0}{dy}\mathrm{(}\mathrm{13}\mathrm{)}$$

$$Z=\frac{\gamma }{4\pi }(\frac{d(\mu \alpha )}{dx}+\frac{d(\mu \beta )}{dy}+\frac{d(\mu \gamma )}{dz})-\frac{{\alpha }^2+{\beta }^2+{\gamma }^2}{8\pi }\frac{d\mu }{dz}+\mu \beta p-\mu \alpha q-\frac{d{p}_0}{dz}\mathrm{(}\mathrm{14}\mathrm{)}$$

Provenance: the equation numbers (12), (13), (14) are attested — Maxwell cites them by number in Part I and again in Part II. The bodies above are reconstructions in his notation. Their structure is fixed by Maxwell's own five-term gloss, quoted below, which the reconstruction reproduces term for term. See Sources.

Maxwell reads his own answer

We may now write down the expressions for the components of the resultant force on an element of the medium per unit of volume.

and then, 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.

This is the moment the paper earns its title. One state of stress, differentiated once, produces:

term	what it is	modern name
1	force on a magnetic pole, via "imaginary magnetic matter" of density $\frac{1}{4\pi }\mathrm{\nabla }\cdot 𝐁$	the force on a magnetic charge, $𝐇{\rho }_m$
2	force where the inductive capacity varies from point to point	the gradient force, $-\frac{1}{8\pi }{H}^2\mathrm{\nabla }\mu$
3 and 4	force transverse both to the current and to the lines of force	$𝐉\times 𝐁$ — the Laplace force
5	ordinary pressure gradient	not electromagnetic at all

The sign of the second term is the one the divergence of the Prop. II stress gives; the printed 1861 sign could not be checked, and it depends on how much of the vortex pressure Maxwell absorbed into $p_0$. The physics is not in dispute: integrated over a body, it draws material of greater inductive capacity into the stronger part of the field. See the modern derivation.

Nothing has been put in by hand. Maxwell assumed a stress; the forces on poles, on iron, and on currents all fell out of differentiating it. That is a genuine unification, and it is the reason the vortex hypothesis was worth the trouble of the next three Parts.

Where equations (9) come from

The third and fourth terms contain $p,q,r$, which have not yet been defined. Maxwell defines them here, in the course of interpreting the terms: they are

the quantity of electric current per unit of area perpendicular to the axes of x, y, and z respectively

and the definition that makes the terms come out right is

$$4\pi p=\frac{d\gamma }{dy}-\frac{d\beta }{dz},4\pi q=\frac{d\alpha }{dz}-\frac{d\gamma }{dx},4\pi r=\frac{d\beta }{dx}-\frac{d\alpha }{dy}\mathrm{(}\mathrm{9}\mathrm{)}$$

From work round a circuit to the curl

In modern notation this is $4\pi 𝐉=\mathrm{\nabla }\times 𝐇$: the circulation of the magnetic force round a closed curve measures the current threading it. Maxwell states the same law verbally at the opening of Part IV:

The connexion between the distribution of lines of magnetic force and that of electric currents may be completely expressed by saying that the work done on a unit of imaginary magnetic matter, when carried round any closed curve, is proportional to the quantity of electricity which passes through the closed curve.

A definition now, a result later. Note what has not happened. In Part I equations (9) are a definition — the arrangement of currents that makes Prop. III come out right. It is Prop. V of Part II that turns them into a derived result, obtained from the transference of particles between vortices. Maxwell is explicit about this when he gets there: equation (33) "will be identical with the first of equations (9)".

The case of no currents

Setting $p=q=r=0$ removes the third and fourth terms, and the magnetic force becomes derivable from a potential. Maxwell then identifies $\mu$ as the

capacity for magnetic induction

equal to 1 in air, and shows that in a medium of constant $\mu$ his field is identical with the field of Coulomb's "imaginary matter" acting at a distance. The two theories are observationally indistinguishable in that case — which is exactly why he needs the cases in the applications, where they are not.

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Next: The applications · Previous: Prop. II

Modern derivation: The Maxwell stress tensor · Ampère's law and its correction