On Physical Lines of Force/Modern derivations/The constitutive relations

On Physical Lines of Force → Modern derivations → Constitutive relations

The gap the four equations leave

The four field equations relate $𝐄$, $𝐁$, $𝐃$ and $𝐇$ to one another and to the charges and currents. They do not say how $𝐃$ is related to $𝐄$, or $𝐁$ to $𝐇$, and left as they stand they cannot be solved. It is worth counting to see why. The four vector fields have twelve components between them, while the four equations supply only eight scalar equations. There are more unknowns than equations, and no ingenuity with the field equations will close the gap, because what is missing is not a fact about fields at all.

It is a fact about matter. The relations that close the system are called the constitutive relations, and they describe the substance the field passes through, not the field.

What Maxwell had

He has two, one from each end of the paper.

The magnetic one

From Part I, where $\mu$ is introduced as the capacity for magnetic induction and given a mechanical reading as the density of the substance of the vortices:

$$𝐁=\mu 𝐇$$

Maxwell takes $\mu =1$ in air, greater in paramagnetic bodies, less in diamagnetic ones. In components it is the trio $\mu \alpha ,\mu \beta ,\mu \gamma$ that appears wherever the induction is meant.

The electric one

From Part III, Prop. XIII:

$$R=-4\pi E^{2}h\mathrm{(}\mathrm{105}\mathrm{)}$$

Provenance: the equation number is attested; the body is a reconstruction in Maxwell's notation. See Sources.

$R$ is the electromotive force along $z$, $h$ the displacement, and $E$ a coefficient depending on the dielectric — the elasticity of the cells. The relation is Hooke's law for the medium, with the minus sign recording that the medium pushes back.

In modern terms this is $𝐃=\epsilon 𝐄$, and comparing constants gives

$$\epsilon =\frac{1}{E^2},\text{that is}E=\sqrt{\frac{1}{\epsilon }}$$

The elasticity of a cell: force against displacement

Why these are different in kind

The field equations are, so far as anyone knows, exact and universal: they hold in vacuum, in iron, in a plasma, in the interior of a star. The constitutive relations are empirical descriptions of particular materials, and every one is an approximation with a limited domain.

• They can be non-linear. $𝐃=\epsilon 𝐄$ assumes doubling the field doubles the displacement. In a strong enough field it does not, and the whole of non-linear optics lives in the correction terms.
• They can fail to be functions at all. In a ferromagnet $𝐁$ is not determined by $𝐇$; it depends on the history of the sample, the same $𝐇$ giving different $𝐁$ according to how you got there. This is hysteresis, it is why a permanent magnet is permanent, and there is no $\mu$ to write down.
• They can be directional. In a crystal the displacement need not be parallel to the field. Then $\epsilon$ and $\mu$ are not numbers but tensors, and the material responds differently along different axes.
• They depend on frequency. The same substance has a different $\epsilon$ for a static field and for one oscillating at optical frequencies. More on this below, because it is the trap.

The third relation, and two independent qualities

A third relation of the same character belongs with Maxwell's two — Ohm's law, in which $\sigma$ is the conductivity:

$$𝐉=\sigma 𝐄$$

Maxwell is clear that conductivity and inductive capacity are separate properties of a body, and the introductory pages of Part III give the analogy:

A conducting body may be compared to a porous membrane which opposes more or less resistance to the passage of a fluid, while a dielectric is like an elastic membrane which may be impervious to the fluid, but transmits the pressure of the fluid on one side to that on the other.

That is a distinction between two kinds of response, which is exactly what a constitutive relation describes. The porous membrane dissipates: fluid gets through, and the resistance turns work irreversibly into heat — that is $\sigma$. The elastic membrane stores: nothing gets through, but pressure is transmitted and returned — that is $\epsilon$. Nothing requires the two coefficients to have anything to do with each other, and in real materials they do not.

What the mechanical model bought

In a purely phenomenological theory $\epsilon$ and $\mu$ are two numbers you go into the laboratory and measure. They are unrelated to each other and to anything else, and there is nothing more to say about them. On Maxwell's model they are mechanical properties of a mechanical medium:

• $\mu$ is a density — the density of the substance of the vortices.
• $E$ is an elasticity — the stiffness of the cells against being deformed.

And here is what follows. Any elastic solid of known density and known rigidity carries transverse waves, at a speed fixed by those two properties alone: the square root of the rigidity divided by the density. That is elementary continuum mechanics, known long before 1861 and owing nothing to electricity.

So Maxwell can do something no phenomenological theory permits. He can take two constants measured in electrical experiments — one from magnets, one from Leyden jars — and compute a wave speed from them. He does it in Prop. XVI, and the number is the velocity of light.

This is the single most important thing the mechanical model supplied — not the equations, most of which can be reached other ways, but the interpretation of the constants. Without a physical reading of $\mu$ and $E$ as density and elasticity there is no reason to combine them at all, no reason to expect the combination to have the dimensions of a velocity, and no reason to compare the result with anything. The model is what makes the question askable. See The model and the equations and Units and the constant c.

In modern SI the same content is carried by

$$c^2=\frac{1}{{\mu }_0{\epsilon }_0}$$

which every textbook writes down and few pause over. It is the descendant of Maxwell's density-and-elasticity argument, with the mechanics quietly removed.

A caution for the modern reader

$\epsilon$ and $\mu$ are frequency-dependent, and forgetting it produces a spectacular-looking refutation of Maxwell that is not one.

The theory predicts that the dielectric constant of a transparent substance should equal the square of its refractive index. Check it against water:

quantity	value for water
static dielectric constant	about 80
refractive index (visible light)	about 1.33
square of the refractive index	about 1.77

Eighty against one point eight: the prediction appears to fail by a factor of forty-five. It does not. The two numbers are measured at wildly different frequencies. Water molecules are strongly polar, and the static value is dominated by whole molecules rotating to line up with the field — a reorientation that cannot keep pace with a field reversing some $10^{14}$ times a second. At optical frequencies the response comes from the electrons alone. Maxwell's relation holds when both quantities are measured at the same frequency, and works well for non-polar substances, where there is no reorientation to lose.

Maxwell goes part of the way towards this in Prop. XVII. He expects the coefficient to differ along different axes of a crystal — the tensor character, in modern language — and ties the value along each axis to the velocity of light polarised along it, already treating the constant as an optical quantity rather than a static one. What he does not have is frequency dependence as such; the 1862 text does not state it.

Where the conventions differ

	Maxwell 1861	Modern
magnetic relation	$𝐁=\mu 𝐇$, $\mu$ the density of the vortex substance	$𝐁=\mu 𝐇$ — unchanged in form
$\mu$ in vacuum	$\mu =1$ (in air)	$1$ in Gaussian; ${\mu }_0$ in SI, within about a part in a billion of $4\pi \times 10^{-7}$
electric relation	eq. (105), $R=-4\pi E^{2}h$	$𝐃=\epsilon 𝐄$
the constant	$E$, an elasticity, with the dimensions of a velocity	$\epsilon$, a permittivity
conversion	—	$\epsilon =1/E^2$
conduction	an independent quality of a body; the porous membrane	$𝐉=\sigma 𝐄$
anisotropy	anticipated in Prop. XVII, per-axis values of $E$	$\epsilon ,\mu$ as tensors
frequency	not treated	$\epsilon (\omega ),\mu (\omega )$; dispersion
status of the relations	mechanical properties of the model medium, and hence explanatory	empirical properties of matter, taken as given

That final row is the trade the twentieth century made. Dropping the mechanical model made the theory more general and more secure — it no longer depends on an ether nobody could find. It also gave up the explanation of where the constants come from, handing that question to atomic physics, which answered it far better than vortices and idle wheels ever could. See What survived.

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Next: The Ampère–Maxwell law · Previous: Gauss's law for electricity

In the paper: Prop. XIII · Prop. XVII · Conduction and induction