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Edmund T. Whittaker published an electrodynamics theory in 1904

Proceedings of the London Mathematical Society, Vol. 1, 1904, p. 367-372. ON AN EXPRESSION OF THE ELECTROMAGNETIC FIELD DUE TO ELECTRONS BY MEANS OF TWO SCALAR POTENTIAL FUNCTIONS By E. T. Whittaker.

which uses only two scalars potentials F and G to derive the electric and magnetic fields, in the form of dielectric displacement $\vec{d}$ and "magnetic force" $\vec{h}$ as

$$ d_x = \frac{\partial^2F}{\partial x \partial z} + \frac{1}{c}\frac{\partial^2G}{\partial y \partial t} \\ d_y = \frac{\partial^2F}{\partial y \partial z} - \frac{1}{c}\frac{\partial^2G}{\partial x \partial t} \\ d_z = \frac{\partial^2F}{\partial z^2} - \frac{1}{c^2}\frac{\partial^2G}{\partial t^2} \\ h_x = \frac{1}{c}\frac{\partial^2F}{\partial y\partial t} - \frac{\partial^2G}{\partial x\partial z} \\ h_y = -\frac{1}{c}\frac{\partial^2F}{\partial x\partial t} - \frac{\partial^2G}{\partial y\partial z} \\ h_z = \frac{\partial^2G}{\partial x^2} + \frac{\partial^2G}{\partial y^2} \\ $$

F and G are determined by summing over all electrons. Symbolism 116 years ago were different from today, where primed coordinates x'(t), y'(t), z'(t) denote position of the electron at time t, and $\bar{x}$ is retarded position $= x(t-r/c)$ etc.

$$ F(x, y, z, t) = \sum \frac{e}{4\pi}sinh^{-1}\frac{\bar{z}' - z}{((\bar{x}'-x)^2 + (\bar{y}' -y)^2)^½} \\ G(x, y, z, t) = \sum \frac{e}{4\pi}tan^{-1}\frac{\bar{y}' - y}{\bar{x}'-x} \\ $$

In a continuous form the two scalar potentials F and G can be

$$ F = \int_0^\pi \int_0^{2\pi}f(x \space sin(u) cos(v) + y \space sin(u)sin(v) + z \space cos(u) + ct, u, v) du dv \\ G = \int_0^\pi \int_0^{2\pi}g(x \space sin(u) cos(v) + y \space sin(u)sin(v) + z \space cos(u) + ct, u, v) du dv \\ $$

where f and g are arbitrary functions of their arguments.

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    $\begingroup$ @Urb I notice that you replaced the linked paper with the DOI. I understand that it makes sense to not simply link a PDF of the paper, since these links may die later, however the OP included a free PDF online and I see removing it as counterproductive, since you have placed the paper behind a paywall. I for one do not have access to it any more. In the future, I would suggest adding the DOI separately, but not removing the links provided by the OP. $\endgroup$
    – Philip
    Aug 23, 2020 at 12:17
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    $\begingroup$ Whittaker also mentions this idea of his in his monumental book archive.org/details/AHistoryOfTheTheoriesOfAetherAndElectricity/… but without any further comments anywhere else implying that this maybe a "dead-end". $\endgroup$
    – hyportnex
    Aug 23, 2020 at 12:27
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    $\begingroup$ Dear @Urb, 1) The link does work for me. 2) You're right, I mentioned in my comment that adding a DOI is great and I fully support it. The linked discussion however, also mentions that one may cite "a pdf of a paper or similar that is not on the arXiv or a similar repository". 3) I understand that you feel it necessary to remove these papers. I do not. I'm not sure it is our job as a community to police these things, but I may be wrong. I will ask a question on Meta to see what the rest of the community thinks. $\endgroup$
    – Philip
    Aug 26, 2020 at 18:20
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    $\begingroup$ @hyportnex Your link doesn't work anymore. Can you remove it, or update it with archive.org/details/historyoftheorie00whitrich ? $\endgroup$ Aug 9, 2021 at 19:20
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    $\begingroup$ @hyportnex Can you remove your comment and rewrite it and link to the relevant page at en.wikisource.org/wiki/… ? $\endgroup$ Sep 3, 2021 at 16:13

2 Answers 2

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Whittaker discusses this subject in the revised enlarged 1951 edition by Thomas Nelson & Son of his book to which the "archive" link disappeared. Here is a long quote from its Vol I, Chapter XIII CLASSICAL THEORY IN THE AGE OF LORENTZ, pages 409-410:

Any electromagnetic field is thus expressed in terms of the four functions $\phi, a_x, a_y, a_z,$ (the scalar-potential and the three components of the vector-potential), and these are given by the above formulae in terms of the positions and velocities of the electrons which generate the field. It was however shown in 1904 by E. T. Whittaker, Proc. Lond. Math. Soc. 121, i (1904), p.367, that only two functions are actually necessary (in place of the four), namely, functions F and G defined by the equations

$$F(x,y,z,t)= \frac{1}{2}\sum_e \mathrm{log} \frac{\bar r +\bar z' - z}{\bar r -(\bar z'-z)}\\ G(x,y,z,t)= -\mathfrak i \frac{1}{2}\sum_e \mathrm{log} \frac{\bar x' -x+ \mathfrak i(\bar y' - y)}{\bar x'-x -\mathfrak i(\bar y'-y)} $$

where the summation is taken over all the electrons in the field, and where $x'(t), y'(t), z'(t)$ denote the position of the electron at the instant t, and $\bar x'(t)$ signifies the value of $x'$ at an instant such that a light-signal sent from the electron at this instant reaches the point $(x, y, z)$ at the instant $t$, so that $\bar x' = x'(t- \bar r/c)$. Thus $\bar x', \bar y',\bar z'$ are known functions of $(x, y, z, t)$ when the motions of the electrons are known, and $$ \bar r^2 = (\bar x'-x)^2+(\bar y'-y)^2+(\bar z'-z)^2$$

The electric vector $(d_x, d_y, d_z)$ and the magnetic vector $(h_x, h_y, h_z)$ are then given by the formulae

$$\begin{matrix} d_x=\frac{\partial ^2F}{\partial x \partial z}+\frac{1}{c}\frac{\partial ^2 G}{\partial y \partial t}, & h_x=\frac{1}{c}\frac{\partial ^2 F}{\partial y \partial t}-\frac{\partial ^2 G}{\partial x \partial z}\\ d_y=\frac{\partial ^2F}{\partial y \partial z}-\frac{1}{c}\frac{\partial ^2 G}{\partial x \partial t}, & h_y=-\frac{1}{c}\frac{\partial ^2 F}{\partial x \partial t}-\frac{\partial ^2 G}{\partial y \partial z}\\ d_z=\frac{\partial ^2F}{\partial^2 z}-\frac{1}{c^2}\frac{\partial ^2 G}{\partial^2 t}, & h_z=\frac{\partial ^2 G}{\partial^2 x}+\frac{\partial ^2 G}{\partial^2 y} \end{matrix}$$

It will be noted that F and G are defined in terms of the positions of the electrons· alone, and do not explicitly involve their velocities. Since in the above formulae for $\mathbf d$ and $\mathbf h$ an interchange of electric and magnetic quantities corresponds to a change of G into F and of F into G, it is clear that the two functions F and G exhibit the duality which is characteristic of electromagnetic theory : thus an electrostatic field can be described by F alone, and a magnetostatic field by G alone ; again, if the field consists of a plane wave of light, then the functions F and G correspond respectively to two plane-polarised components into which it can be resolved. Since there are an infinite number of ways of resolving a plane wave of light into two plane-polarised components, it is natural to expect that, corresponding to any given electromagnetic field, there should be an infinite number of pairs of functions F and G capable of describing it, their difference from each other depending on the choice of the axes of co-ordinates-as is in fact the case. Thus there is a physical reason why any particular pair of functions F and G should be specially related to one co-ordinate, and cannot be described by formulae symmetrically related to the three co-ordinates $(x, y, z)$.

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    $\begingroup$ Man is this ending up identical to jefimenko? Very similar. Probably a couple definitions would do it, and link the two. He derives E and B in two equations in terms of current and charge and their derivatives all as rhs, only. All Ive ever read about jefimenko never mentioned whitaker at all. I thought whole idea was new $\endgroup$
    – Al Brown
    Sep 3, 2021 at 17:10
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    $\begingroup$ Did Jefimenko or Whittaker ever include the advanced position at the advanced time $x_a = x(t + r/c)$? Classical electrodynamics does. en.wikipedia.org/wiki/Retarded_time#Retarded_and_advanced_times $\endgroup$ Sep 3, 2021 at 18:13
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    $\begingroup$ @AlBrown Whittaker is even simpler than Jefimenko since Whittaker only uses charge positions and retarded charge positions. There is no need for currents if these two positions are known, and which could be understood as currents, and are currents. The problem with classical electrodynamics is that they didn't know about charge quantisation when it was established. $\endgroup$ Sep 3, 2021 at 18:21
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    $\begingroup$ I have typed the relevant two pages from Whittaker without any change in content or notation, vectors are denoted by bold letters. $\endgroup$
    – hyportnex
    Sep 3, 2021 at 18:31
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    $\begingroup$ @AlBrown the same stuff that is called "Jefimenko's Equations" were written down before he was born; engineers learned it from Stratton & Chu (1939) then from Panofsky & Phillips (1960) and both refer back to Ignatowsky (1907) $\endgroup$
    – hyportnex
    Sep 3, 2021 at 21:28
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Classical electrodynamics is based on electrical current which is much more simple to measure than electrical charge and speed separately and retarded/advanced position of electrical charge separately, which is used in Whittaker electrodynamics. The use of electrical currents instead of charges and charge positions is a kind of degeneracy of two degrees of freedom into one, and thus introducing an artificial degree of freedom in the form of gauge freedom. The question of arbitrarily small charge carriers was discussed by Maxwell [citation needed] but since they didn't know about the electron at that time the theory was developed to allow for unspecified charge carriers and/or perfectly continuous charge flow.

Bulk electrodynamics did not adapt at that time (1897-1905) and as a consequence got the associated theory of relativity (which is connected with electrodynamic gauge freedom).

Gauge freedom is also associated with arbitrariness of the gauge of electrical voltage - only voltage differences or gradients are considered physical. I have not been able to relate that to Whittaker electrodynamics yet.

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