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

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	''[[On Physical Lines of Force]] → [[On Physical Lines of Force/Modern derivations\|Modern derivations]] → The displacement current''		''[[On Physical Lines of Force]] → [[On Physical Lines of Force/Modern derivations\|Modern derivations]] → The displacement current''

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	''Plain-language version: [[On Physical Lines of Force/Explainers/The missing term\|The missing term]]''		''Plain-language version: [[On Physical Lines of Force/Explainers/The missing term\|The missing term]]''

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

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

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''[[On Physical Lines of Force]] → [[On Physical Lines of Force/Modern derivations|Modern derivations]] → The displacement current''

== What Maxwell had ==

Equation '''(112)''', from [[On Physical Lines of Force/Part III/Prop. XIV|Part III, Prop. XIV]], under the modest heading "To correct the equations (9) of electric currents for the effect due to the elasticity of the medium":

<math display="block">4\pi\!\left(p+\frac{df}{dt}\right) = \frac{d\gamma}{dy}-\frac{d\beta}{dz}, \qquad\text{and cyclically} \qquad\qquad\mathrm{(112)}</math>

''Provenance: the equation number is attested three independent ways; the body is a reconstruction in Maxwell's notation. See [[On Physical Lines of Force/Sources and editorial method|Sources]].''

The total current is the conduction current <math>p,q,r</math> plus the rate of change of the displacement <math>f,g,h</math>. In modern notation:

<math display="block">\mathbf{J}_{\text{total}} = \mathbf{J} + \frac{\partial\mathbf{D}}{\partial t}</math>

The second term is the '''displacement current'''. It is not a current of anything in the ordinary sense: no charge crosses the gap of a capacitor. It is a rate of change of the electric field that has the same magnetic effect a current would have.

== Maxwell's route: elasticity ==

His argument occupies three lines and is worth having in front of you.

# Equation '''(105)''' of [[On Physical Lines of Force/Part III/Prop. XIII|Prop. XIII]] relates the electromotive force to the displacement, <math>R=-4\pi E^{2}h</math>.
# Differentiate with respect to time: a varying electromotive force produces a varying displacement.
# A variation of displacement is particles in motion; particles in motion are a current; so it must be added to <math>p,q,r</math>.

His own phrase: '''"a variation of displacement is equivalent to a current."'''

And from [[On Physical Lines of Force/Part III/Conduction and induction|the introductory pages of Part III]], stated even before the propositions:

<blockquote>
This displacement does not amount to a current, because when it has attained a certain value it remains constant, but it is the commencement of a current, and its variations constitute currents in the positive or negative direction, according as the displacement is increasing or diminishing.
</blockquote>

[[File:OPL-fig-cell-elasticity.svg|thumb|center|620px|The cell yields, and pushes back]]

== The modern route: consistency ==

The argument textbooks give today has nothing to do with elasticity.

Charge is conserved. That is the continuity equation:

<math display="block">\nabla\cdot\mathbf{J} + \frac{\partial\rho}{\partial t} = 0</math>

Ampère's uncorrected law says <math>\nabla\times\mathbf{H}=\mathbf{J}</math>. Take the divergence; the left side vanishes identically; so the law requires <math>\nabla\cdot\mathbf{J}=0</math>, which contradicts continuity whenever charge accumulates.

Now use Gauss's law, <math>\rho=\nabla\cdot\mathbf{D}</math>:

<math display="block">\nabla\cdot\mathbf{J} + \frac{\partial}{\partial t}\big(\nabla\cdot\mathbf{D}\big) = \nabla\cdot\!\left(\mathbf{J}+\frac{\partial\mathbf{D}}{\partial t}\right) = 0</math>

The combination <math>\mathbf{J}+\partial\mathbf{D}/\partial t</math> is divergence-free unconditionally. '''It is the only object that can sit on the right of Ampère's law without contradicting the conservation of charge.'''

The term is therefore not a hypothesis. Given Gauss's law and charge conservation, it is forced.

== The capacitor ==

[[File:OPL-fig-displacement-current.svg|thumb|center|680px|The variation of displacement is itself a current]]

The standard illustration, and the one that makes the necessity concrete.

A capacitor is charging. Current <math>I</math> flows in the wires; nothing crosses the gap. Take a closed loop <math>C</math> encircling one of the wires. Ampère's law in integral form says the circulation of <math>\mathbf{H}</math> round <math>C</math> is fixed by the current through ''any'' surface bounded by <math>C</math>. So take two:

{| class="wikitable"
! surface !! pierced by !! uncorrected answer
|-
| <math>S_{1}</math>, flat, cutting the wire || the wire, carrying <math>I</math> || <math>I</math>
|-
| <math>S_{2}</math>, bulging out to pass between the plates || nothing at all || <math>0</math>
|}

Two answers for one loop. The law as stated is inconsistent.

Now count the displacement term. Between the plates, with charge <math>Q</math> on a plate of area <math>A</math>, the displacement is <math>D=Q/A</math>, so

<math display="block">\int_{S_{2}}\frac{\partial\mathbf{D}}{\partial t}\cdot d\mathbf{S} = A\,\frac{d}{dt}\!\left(\frac{Q}{A}\right) = \frac{dQ}{dt} = I</math>

exactly the current missing from <math>S_{2}</math>. The two surfaces now agree, and they agree for every surface, because the total current density is divergence-free.

== Two routes, one term ==

This deserves to be stated plainly, because it is the most remarkable fact about the paper.

{| class="wikitable"
!   !! Maxwell, 1862 !! modern textbook
|-
| starting point || the cell substance is elastic || charge is conserved
|-
| motivation || to explain how rotation reaches the inside of a cell || to make Ampère's law self-consistent
|-
| the term obtained || <math>\partial\mathbf{D}/\partial t</math> || <math>\partial\mathbf{D}/\partial t</math>
|}

Maxwell was patching a mechanical difficulty about spinning cells. He was not repairing a continuity equation — he does not raise the consistency problem at all. That the patch is precisely the term consistency requires is either remarkable luck or evidence that the mechanical analogy was tracking something structural. Historians have argued both; Daniel Siegel's ''Innovation in Maxwell's Electromagnetic Theory'' (1991) is the standard treatment and comes down on the side of the analogy doing real work.

== Why everything depends on it ==

Without the term, the equations do not couple:

<math display="block">\nabla\times\mathbf{E} = -\frac{\partial\mathbf{B}}{\partial t}, \qquad \nabla\times\mathbf{H} = \mathbf{J}</math>

A changing magnetic field makes an electric field. A changing electric field makes nothing. The chain has one link and stops.

With the term:

<math display="block">\nabla\times\mathbf{E} = -\frac{\partial\mathbf{B}}{\partial t}, \qquad \nabla\times\mathbf{H} = \mathbf{J} + \frac{\partial\mathbf{D}}{\partial t}</math>

Each field's change generates the other. Take the curl of one, substitute the other, and a wave equation appears. That is [[On Physical Lines of Force/Modern derivations/The electromagnetic wave equation|the next page]], and it is light, radio, radar, and every wireless signal that has ever been sent.

== Where the conventions differ ==

{| class="wikitable"
!   !! Maxwell 1861 !! modern SI
|-
| displacement || <math>f,g,h</math>, with <math>R=-4\pi E^{2}h</math> || <math>\mathbf{D}=\varepsilon\mathbf{E}</math>; Maxwell's triple is <math>\mathbf{D}/4\pi</math> up to sign
|-
| the sign || his own convention for the direction of displacement || fixed by <math>\mathbf{D}=\varepsilon\mathbf{E}</math>
|-
| the coefficient || <math>E</math>, an elasticity, with <math>\varepsilon = 1/E^{2}</math> || <math>\varepsilon</math>, a permittivity
|-
| status of the term || a correction for elasticity || required by charge conservation
|}

----

''Next: [[On Physical Lines of Force/Modern derivations/The unit ratio|The unit ratio, and why it is a velocity]] · Previous: [[On Physical Lines of Force/Modern derivations/The constitutive relations|The constitutive relations]]''

''Plain-language version: [[On Physical Lines of Force/Explainers/The missing term|The missing term]]''

On Physical Lines of Force/Modern derivations/The displacement current - 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