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	''[[On Physical Lines of Force]] → [[On Physical Lines of Force/Part III\|Part III]] → Prop. XVII''		''[[On Physical Lines of Force]] → [[On Physical Lines of Force/Part III\|Part III]] → Prop. XVII''

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	''Next: [[On Physical Lines of Force/Part IV\|Part IV]] · Previous: [[On Physical Lines of Force/Part III/Prop. XVI\|Prop. XVI]]''		''Next: [[On Physical Lines of Force/Part IV\|Part IV]] · Previous: [[On Physical Lines of Force/Part III/Prop. XVI\|Prop. XVI]]''

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''[[On Physical Lines of Force]] → [[On Physical Lines of Force/Part III|Part III]] → Prop. XVII''

== Maxwell's statement ==

<blockquote>
'''Prop. XVII.''' — To find the electric capacity of a Leyden jar composed of any given dielectric placed between two conducting surfaces.
</blockquote>

Printed on pp. 22–24, and closing Part III.

A Leyden jar is a capacitor: two conducting surfaces with an insulator between them. Maxwell's result is the one still taught:

* capacity varies '''directly as the area''' of the surfaces;
* capacity varies '''inversely as the thickness''' of the dielectric;
* capacity varies '''inversely as <math>E^{2}</math>''' — that is, directly as the specific inductive capacity of the material.

''Provenance: no equation from Prop. XVII could be recovered. The three proportionalities are attested in the surviving prose. See [[On Physical Lines of Force/Sources and editorial method|Sources]].''

== The step to optics ==

Having related capacity to <math>E^{2}</math>, and having just shown in [[On Physical Lines of Force/Part III/Prop. XVI|Prop. XVI]] that <math>E</math> is the velocity of propagation, Maxwell can relate the inductive capacity of a dielectric to its '''refractive index''' — because the refractive index is the ratio of the velocity of light in vacuum to its velocity in the substance.

The prediction, in modern terms, is that the dielectric constant of a transparent substance should equal the square of its refractive index. This is a genuine, checkable consequence of identifying the two media, and it is the first quantitative test of the electromagnetic theory of light that anyone could attempt.

<blockquote>
'''How the prediction fared.''' It works well for simple non-polar substances and fails badly for others — water has a refractive index near 1.33 but a static dielectric constant near 80. The discrepancy is real and its resolution took decades: the two quantities are measured at wildly different frequencies, and the dielectric constant of a polar liquid is dominated by molecular reorientation that cannot follow an optical-frequency field. Maxwell's relation holds when both are measured at the same frequency. This is not stated in the 1862 paper; it is noted here because a reader who checks the prediction against a table of constants will otherwise conclude, wrongly, that it simply fails.
</blockquote>

== Anisotropic media, and the limits Maxwell draws ==

The closing pages extend the treatment to crystals — and are also where Maxwell is most explicit about how far he thinks the theory can be pushed in dense matter.

<blockquote>
In dense media, however, the optical, electric, and magnetic phenomena may be modified in different degrees by the particles of gross matter; and their mode of arrangement may influence these phenomena differently in different directions. The axes of optical, electric, and magnetic properties will probably coincide; but on account of the unknown and probably complicated nature of the reactions of the heavy particles on the ætherial medium, it may be impossible to discover any general numerical relations between the optical, electric, and magnetic ratios of these axes.

It seems probable, however, that the value of E, for any given axis, depends upon the velocity of light whose vibrations are parallel to that axis, or whose plane of polarization is perpendicular to that axis.

In a uniaxal crystal, the axial value of E will depend on the velocity of the extraordinary ray, and the equatorial value will depend on that of the ordinary ray. In "positive" crystals, the axial value of E will be the least and in negative the greatest.
</blockquote>

''Provenance: this closing passage rests on a single extraction and its symbols are unreliable. See [[On Physical Lines of Force/Sources and editorial method|Sources]].''

Note the concession in the first paragraph. Maxwell expects the axes of the optical, electric and magnetic properties of a crystal to coincide — but he expects the ''numerical'' relations between them to be beyond reach, because of "the unknown and probably complicated nature of the reactions of the heavy particles on the ætherial medium." He is drawing a boundary around what his theory can be expected to deliver: it is a theory of the medium, and matter complicates it in ways he cannot compute.

== The torque on a suspended sphere ==

The final result of Part III is a small, concrete, testable one:

<blockquote>
If a spherical portion of a crystal, radius = ''a'', be suspended in a field of electric force which would act on unit of electricity with force = ''I'', and if [''D''₁] and [''D''₂] be the coefficients of dielectric induction along the two axes in the plane of rotation, then if θ be the inclination of the axis to the electric force, the moment tending to turn the sphere will be [proportional to the difference of the two coefficients and to sin 2θ], and the axis of greatest dielectric induction will tend to become parallel to the lines of electric force.
</blockquote>

The <math>\sin 2\theta</math> dependence, and the conclusion that the axis of greatest inductive capacity aligns with the field, are exactly right, and describe the dielectric torque used in laboratories today. Part III ends, as Part I did, not with a grand claim but with a laboratory prediction.

== The state of the theory at the end of Part III ==

{| class="wikitable"
! modern equation !! where it appears in this paper
|-
| <math>\nabla\cdot\mathbf{D}=\rho</math> || Part III, eq. (113)
|-
| <math>\nabla\cdot\mathbf{B}=0</math> || Part II, eq. (56)
|-
| <math>\nabla\times\mathbf{E}=-\partial\mathbf{B}/\partial t</math> || Part II, eq. (54)
|-
| <math>\nabla\times\mathbf{H}=\mathbf{J}+\partial\mathbf{D}/\partial t</math> || Part III, eq. (112)
|}

All four. Plus the constitutive relations <math>\mathbf{B}=\mu\mathbf{H}</math> and <math>\mathbf{D}=\varepsilon\mathbf{E}</math>, the vector potential, the energy densities, the stress tensor, the motional term of the Lorentz force, and the identification of light.

What remains is [[On Physical Lines of Force/Part IV|Part IV]], which is not about any of this. It is about Faraday's rotation of polarized light, and about why Maxwell thought magnetism ''had'' to be rotation.

----

''Next: [[On Physical Lines of Force/Part IV|Part IV]] · Previous: [[On Physical Lines of Force/Part III/Prop. XVI|Prop. XVI]]''

On Physical Lines of Force/Part III/Prop. XVII - Revision history

BFP Rauch: Add DISPLAYTITLE: title the page by its section rather than its path

BFP Rauch: 1 revision imported: Maxwell, On Physical Lines of Force — initial import

opl>Import: Imported from the DokuWiki edition