On Physical Lines of Force/Modern derivations/The unit ratio

On Physical Lines of Force → Modern derivations → The unit ratio

What Maxwell had

Prop. XV of Part III fixes the constant relating the two measures of a quantity of electricity, and his own summary of what he had done:

I have deduced from this result the relation between the statical and dynamical measures of electricity, and have shown, by a comparison of the electro-magnetic experiments of MM. Kohlrausch and Weber with the velocity of light as found by M. Fizeau, that the elasticity of the magnetic medium in air is the same as that of the luminiferous medium, if these two coexistent, coextensive, and equally elastic media are not rather one medium.

The constant is his $E$, and the point of this page is why it had to have the dimensions of a speed.

Two ways to define a unit of charge

The coincidence that settled it

The electrostatic measure

Define the unit charge by the force between two charges at rest. In Gaussian-style units Coulomb's law is written with no constant at all:

$$F=\frac{q_1{q}_2}{r^2}$$

so the unit charge is the one that repels an equal charge at unit distance with unit force. Dimensionally, $[q_{\text{es}}{]}^2=[F][L{]}^2$.

The electromagnetic measure

Define the unit instead by the force between two currents. For two long parallel wires a distance $r$ apart, the force per unit length is

$$\frac{F}{\ell }=\frac{2I_1{I}_2}{r}$$

and the unit current is the one giving unit force per unit length at unit separation. Since current is charge per time, $[q_{\text{em}}{]}^2=[F][L][T{]}^2$.

The ratio

Divide:

$$\frac{[q_{\text{es}}{]}^2}{[q_{\text{em}}{]}^2}=\frac{[F][L{]}^2}{[F][L][T{]}^2}=\frac{[L{]}^2}{[T{]}^2}$$

so

$$\frac{q_{\text{es}}}{q_{\text{em}}}=\text{a velocity}$$

No motion appears in either definition. One is a statement about charges sitting still; the other is a statement about steady currents in wires. Yet the conversion factor between the two units is a speed. Nothing in the argument says which speed, or that it should be a speed of anything in particular. It is a fact about dimensions, and it is forced.

In SI

The same fact wears different clothes. SI writes Coulomb's law and the force between currents as

$$F=\frac{1}{4\pi {\epsilon }_0}\frac{q_1{q}_2}{r^2},\frac{F}{\ell }=\frac{{\mu }_0}{2\pi }\frac{I_1{I}_2}{r}$$

Working through the dimensions of the two constants gives

$$\frac{1}{\sqrt{{\epsilon }_0{\mu }_0}}=\text{a velocity}$$

which is why $c=1/\sqrt{{\epsilon }_0{\mu }_0}$ in the wave equation. The dimensional fact is the same one; SI just hides it inside the two constants.

The measurement

Wilhelm Weber and Rudolf Kohlrausch measured the ratio in the 1850s. The method is straightforward in principle: charge a Leyden jar and measure its charge electrostatically, by the force it exerts; then discharge it through a galvanometer and measure the same charge electromagnetically, by the magnetic effect of the current. Divide.

The number Maxwell uses is 310,740,000,000 millimetres per second.

Fizeau had measured the velocity of light in 1849 with a toothed wheel and a distant mirror. Maxwell quotes his result as "70,843 leagues per second (25 leagues to a degree)", giving 314,858,000,000 millimetres per second, and adds the footnote:

Comptes Rendus, vol. xxix. (1849), p. 90. In Galbraith and Haughton's 'Manual of Astronomy,' M. Fizeau's result is stated at 169,944 geographical miles of 1000 fathoms, which gives 193,118 statute miles; the value deduced from aberration is 192,000 miles.

	mm/s	km/s	method
Weber and Kohlrausch	310,740,000,000	310,740	a Leyden jar, twice
Fizeau	314,858,000,000	314,858	a toothed wheel and a mirror
modern value	299,792,458,000	299,792.458	exact, by definition since 1983

The two 1860s figures differ from each other by about 1.3 per cent, and both run about 4 to 5 per cent above the modern value. That both were high is not a coincidence of the theory but of the measurements: Fizeau's toothed-wheel result was known to be somewhat too large, and the Weber–Kohlrausch determination carried its own systematic errors.

Provenance: the two numerals are strongly corroborated but were not read from a page scan. Maxwell restates the same two figures in metres per second in the 1865 Dynamical Theory, §96. See Sources.

Why the agreement mattered so much

Dimensional analysis says the ratio is a velocity. It does not say which velocity, and there is no reason on the face of it why a fact about Leyden jars should have anything to do with light.

Maxwell's model supplied the missing link. The elasticity and the density of the medium both have electrical meanings; a wave in an elastic medium travels at $\sqrt{m/\rho }$; the algebra gives $V=E$, where $E$ is precisely the unit ratio. So on this theory the unit ratio is not an arbitrary velocity — it is the speed at which disturbances travel through the electromagnetic medium.

And that speed, measured with no optical instrument whatever, came out equal to the measured speed of light.

The velocity of transverse undulations in our hypothetical medium, calculated from the electro-magnetic experiments of MM. Kohlrausch and Weber, agrees so exactly with the velocity of light calculated from the optical experiments of M. Fizeau, that we can scarcely avoid the inference that light consists in the transverse undulations of the same medium which is the cause of electric and magnetic phenomena.

What survived the model

Everything structural, and none of the mechanism.

• The unit ratio has the dimensions of a velocity — a fact about definitions, independent of any theory.
• That velocity is $1/\sqrt{{\epsilon }_0{\mu }_0}$ — a fact about the field equations, independent of the vortices.
• It equals the speed of light — the physical claim, which the wave equation establishes without any mechanical medium.

Maxwell later took the measurement itself seriously enough to redo it: his work on electrical standards for the British Association, beginning in 1863, was in part about pinning down exactly this ratio. Bruce Hunt has argued that this standards work, more than any disillusionment with vortices, is what turned him toward a physics of measurable relations rather than mechanisms — see the aftermath page.

A modern footnote

The relationship has since been inverted. Since 1983 the metre has been defined by fixing $c$ at 299,792,458 m/s exactly, and since the 2019 SI redefinition ${\mu }_0$ is a measured quantity rather than a defined one. What was for Maxwell an astonishing empirical coincidence between two laboratory numbers is now, in part, a matter of definition — which is what tends to happen to a coincidence once everyone believes it.

Where the conventions differ

	Maxwell 1861	modern
the ratio	$E$, also the elasticity of the cells	$1/\sqrt{{\epsilon }_0{\mu }_0}=c$
units	millimetres per second	metres per second
status	measured, and compared with light	defined exactly since 1983
the reason it is a velocity	dimensional, and unchanged	dimensional, and unchanged

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Plain-language version: Two ways to measure electricity