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

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			{{DISPLAYTITLE:Part II — Prop. XI}}
	''[[On Physical Lines of Force]] → [[On Physical Lines of Force/Part II\|Part II]] → Prop. XI''		''[[On Physical Lines of Force]] → [[On Physical Lines of Force/Part II\|Part II]] → Prop. XI''

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	''Modern derivation: [[On Physical Lines of Force/Modern derivations/The Lorentz force\|The motional term and the Lorentz force]]''		''Modern derivation: [[On Physical Lines of Force/Modern derivations/The Lorentz force\|The motional term and the Lorentz force]]''

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''[[On Physical Lines of Force]] → [[On Physical Lines of Force/Part II|Part II]] → Prop. XI''

== Maxwell's statement ==

<blockquote>
'''Prop. XI.''' — To find the electromotive forces in a moving body.
</blockquote>

And his opening sentence, which is the whole plan of the proposition:

<blockquote>
The variation of the velocity of the vortices in a moving element is due to two causes — the action of the electromotive forces, and the change of form and position of the element.
</blockquote>

== The assembly ==

Everything needed is already in hand.

* The change from '''electromotive force''' — [[On Physical Lines of Force/Part II/Prop. VIII|Prop. VIII]], equation (54).
* The change from '''deformation and rotation''' of the moving element — [[On Physical Lines of Force/Part II/Props. IX and X|Props. IX and X]].

Add them, require that the total account be consistent, and the electromotive force acquires a term it did not have before. The result is equation '''(77)''', the climax of Part II:

<math display="block">P = \mu\!\left(\gamma\frac{dy}{dt}-\beta\frac{dz}{dt}\right) - \frac{dF}{dt} - \frac{d\psi}{dx} \qquad\qquad\mathrm{(77)}</math>

<math display="block">Q = \mu\!\left(\alpha\frac{dz}{dt}-\gamma\frac{dx}{dt}\right) - \frac{dG}{dt} - \frac{d\psi}{dy}</math>

<math display="block">R = \mu\!\left(\beta\frac{dx}{dt}-\alpha\frac{dy}{dt}\right) - \frac{dH}{dt} - \frac{d\psi}{dz}</math>

''Provenance: the equation number (77) is attested and independently corroborated as the complete electromotive-force relation including the velocity-dependent term. The body is a reconstruction in Maxwell's notation. The numbering between (58) and (77) could not be resolved. See [[On Physical Lines of Force/Sources and editorial method|Sources]].''

== Reading the three terms ==

{| class="wikitable"
! term !! mechanically !! modern form !! name today
|-
| <math>\mu(\gamma\,\dot y - \beta\,\dot z)</math> || the element is carried ''through'' the field || <math>\mathbf{v}\times\mathbf{B}</math> || the motional term
|-
| <math>-\,dF/dt</math> || the electromagnetic momentum of the medium is changing || <math>-\partial\mathbf{A}/\partial t</math> || the induced term
|-
| <math>-\,d\psi/dx</math> || the particles press on one another || <math>-\nabla\varphi</math> || the electrostatic term
|}

In vector notation the whole of (77) is

<math display="block">\mathbf{E} \;=\; \mathbf{v}\times\mathbf{B} \;-\; \frac{\partial\mathbf{A}}{\partial t} \;-\; \nabla\varphi</math>

== What the first term is ==

<math>\mathbf{v}\times\mathbf{B}</math> is the magnetic part of what is now called the '''Lorentz force''' — the force felt by a charge moving through a magnetic field. It appears here, in 1861, thirty-odd years before Lorentz, and it appears not as a postulate about charges but as a consequence of carrying a piece of the medium about.

The distinction Maxwell is making is one that would trouble physics for the next forty years. Consider a magnet and a loop of wire, in relative motion:

* '''Move the loop, hold the magnet still.''' The element of the medium is transported through a field that is not changing. The first term of (77) supplies the electromotive force.
* '''Move the magnet, hold the loop still.''' The field at the loop is now changing in time. The second term supplies it.

The measured current is the same in both cases. But the ''explanation'' on this model is different in the two cases, and that asymmetry is exactly what Einstein opens the 1905 relativity paper by complaining about — his first paragraph is about a magnet and a conductor, and about a theory that gives two accounts of one phenomenon. Special relativity dissolves the distinction by making <math>\mathbf{E}</math> and <math>\mathbf{B}</math> two aspects of one object seen from different frames.

Maxwell could not have that resolution. What he had instead was a medium — and in a medium there is a fact of the matter about what is moving, so two accounts are not a defect but a feature. It is worth appreciating that the model behaves entirely correctly here; the trouble is philosophical, not numerical, and it took a change in the concept of space to remove it.

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

{| class="wikitable"
! result !! where !! modern form
|-
| currents and the curl of the magnetic force || [[On Physical Lines of Force/Part II/Prop. V|Prop. V]], eq. (33) = (9) || <math>\nabla\times\mathbf{H}=4\pi\mathbf{J}</math>
|-
| induction || [[On Physical Lines of Force/Part II/Prop. VIII|Prop. VIII]], eq. (54) || <math>\nabla\times\mathbf{E}=-\partial\mathbf{B}/\partial t</math>
|-
| no magnetic sources || [[On Physical Lines of Force/Part II/Prop. VIII|Prop. VIII]], eq. (56) || <math>\nabla\cdot\mathbf{B}=0</math>
|-
| the vector potential || [[On Physical Lines of Force/Part II/Prop. VIII|Prop. VIII]], eq. (55) || <math>\mathbf{B}=\nabla\times\mathbf{A}</math>
|-
| electromotive force, complete || Prop. XI, eq. (77) || <math>\mathbf{E}=\mathbf{v}\times\mathbf{B}-\partial\mathbf{A}/\partial t-\nabla\varphi</math>
|}

Everything on that list was already known experimentally. Part II is a triumph of ''explanation'', not of prediction: it accounts mechanically for laws that Ørsted, Ampère and Faraday had established. Nothing in it would have surprised a well-read physicist of 1861 except the manner of its arrival.

That changes in [[On Physical Lines of Force/Part III|Part III]].

----

''Next: [[On Physical Lines of Force/Part II/Induced currents|Induced currents]] · Previous: [[On Physical Lines of Force/Part II/Props. IX and X|Props. IX and X]] · On to [[On Physical Lines of Force/Part III|Part III]]''

''Modern derivation: [[On Physical Lines of Force/Modern derivations/The Lorentz force|The motional term and the Lorentz force]]''

On Physical Lines of Force/Part II/Prop. XI - 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