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

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

Maxwell's statement

Prop. I. — If in two fluid systems geometrically similar the velocities and densities at corresponding points are proportional, then the differences of pressure at corresponding points due to the motion will vary in the duplicate ratio of the velocities and the simple ratio of the densities.

"Duplicate ratio" is the older phrase for ratio of the squares. Put in modern words: pressure difference scales as density times velocity squared.

Why this proposition comes first

Maxwell is about to fill space with eddies of unspecified size. A reader is entitled to object at once: eddies how big? If the answer matters, the theory is empty until it is supplied, and Maxwell has no way to supply it. Prop. I removes the objection before it is made.

The argument is one of similarity, and it needs no fluid dynamics beyond dimensional bookkeeping. Take two geometrically similar systems, and let

• $l$ be the ratio of their linear dimensions,
• $m$ the ratio of their velocities at corresponding points,
• $n$ the ratio of their densities,
• $p$ the ratio of the pressures arising from the motion.

Corresponding volumes are as $l^3$, so corresponding masses are as $n{l}^3$, and momenta as $n{l}^{3}m$. The time taken to traverse corresponding distances is as $l/m$, so the impulse required — force times time — is as $n{l}^{3}m÷(l/m)=n{l}^2{m}^2$. But force is pressure times area, and corresponding areas are as $l^2$. Hence

$$p{l}^2\propto n{l}^2{m}^2,\text{so}p\propto n{m}^2$$

and $l$ has cancelled out entirely.

$$\Delta P=\text{const}\times \rho v^2$$

Provenance: this equation is a modern restatement of the proposition's content, not a transcription. The printed equation numbers in Prop. I could not be established. See Sources.

What it licenses

The size of a vortex never again appears in Part I, II or III. Maxwell can say "molecular" and mean it as a claim about smallness with no numerical commitment, because the stress his vortices generate is fixed entirely by

• the density of the substance in them — which he will identify with the capacity for magnetic induction, $\mu$; and
• the circumferential velocity — which he will identify with the intensity of the magnetic force.

This is the first and cleanest instance of a habit that runs through the whole paper: Maxwell finds the property of the model that the observations can actually constrain, and quietly declines to specify anything else.

A parameter deliberately left free. The one place where vortex size does come back is Part IV, result (E) — the amount of magneto-optical rotation depends on the mean radius of the vortices. Maxwell notes there that optical measurements are then "probably the only data for determining their relative size in different substances." Having got rid of the parameter in Part I, he finds a way to measure it in Part IV.

The specialisation to a circular vortex

Applying the result to a single circular vortex of uniform angular velocity and density, Maxwell obtains the three pressures quoted in the introductory discussion:

where	pressure
at the axis	$p$
at the circumference	$p+\rho v^2$
mean, parallel to the axis	$p+\frac{1}{2}\rho v^2$

so that the excess of the equatorial over the axial pressure is $\frac{1}{2}\rho v^2$.

For a vortex whose angular velocity and density are not uniform through its substance, the same form holds with a different numerical coefficient — some multiple of $\rho v^2$ depending on the distribution. Maxwell keeps the coefficient general and never needs to fix it, because it is absorbed into the definition of the magnetic units.

Then a bundle of them

Parallel vortices make an anisotropic medium

Set many such vortices side by side with their axes parallel and you have a medium that is not an ordinary fluid: its pressure across the axes exceeds its pressure along them. Left to itself it would spread sideways and draw together lengthwise.

That is precisely the behaviour required to make two unlike magnetic poles attract and two like poles repel — see the plain-language version. Prop. II turns it into a stress tensor.

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Next: Prop. II · Previous: The introductory discussion