On an expression of the EM field by two scalar potential functions
E. T. Whittaker
Summary and citation · read the original at the source
In one page
Read to the London Mathematical Society in November 1903, Whittaker’s short paper proves something clean about Maxwell’s theory. Everyone before him had described an electromagnetic field with a scalar potential plus a vector potential — four functions in all. Whittaker shows that for any number of electrons moving in any manner, all six components of the field can be written as derivatives of just two scalar functions, which he calls F and G. He evaluates them explicitly from the charges and positions of the electrons, and then points out the payoff: F and G depend only on where the electrons are, while the electric and magnetic vectors depend in a complicated way on their velocities and accelerations. Away from any electron, F and G each obey the wave equation, and the general solution gives the most general possible disturbance in the ether as a superposition of waves running in every direction. He closes with a special case that has been argued over ever since.
Why it matters hereChapter 10 turns on whether the potentials are bookkeeping or physics, and this is the paper that shows how much structure lives in them: two scalars, depending only on where the charges are, carry the entire field — and Whittaker’s zero-net-charge case is the origin of the scalar-wave tradition that chapter follows.
What it claims
01For any number of electrons moving in any manner, the six functions that define the resulting field — three components of dielectric displacement and three of magnetic force at every point — can be expressed in terms of the derivatives of two scalar potential functions, where previous writers required a scalar potential plus a vector potential, equivalent to four scalar functions.Section 1, Object of Paper, p. 367
Settled physics02The two potentials F and G are evaluated explicitly in terms of the charges and coordinates of the electrons, and the substitution makes the ordinary scalar potential drop out automatically.Section 3, Introduction and Evaluation of the two Scalar Potentials, pp. 369–370
Settled physics03The gain is in what each description depends on: F and G are simple functions of the coordinates of the electrons, whereas the electric and magnetic vectors are complicated functions of the electrons’ velocities and accelerations. In vector form the two potentials can be taken everywhere and at all times parallel to a single fixed direction, which is why two scalar quantities suffice.Section 4, Discussion of the Apparent Asymmetry of the preceding Result, p. 371
Settled physics04The apparent asymmetry of the result is real and unavoidable: the vector equations are themselves invariant and can be written in vector notation, but they do not possess invariant solutions, so no perfectly symmetrical pair of potentials exists.Section 4, p. 371
Settled physics05At every point not actually occupied by an electron, F and G each satisfy the wave equation, and writing down the general solution gives the most general type of electrodynamic disturbance in the ether as a double integral over all directions of arbitrary functions — a superposition of plane waves travelling in every direction at once.Section 5, Deduction of the General Functional Form of an Electrodynamic Disturbance in the Aether, pp. 371–372
Settled physics06Whittaker’s closing special case: any number of electrons whose total charge is zero, moving so as to remain in the vicinity of a given point — what he calls stationary motion — generate a field of the type where G vanishes and F is an arbitrary function of time minus distance over the speed of light, divided by that distance. He remarks that this may be of interest given the view, then advocated by some physicists, that atoms consist of sets of electrons of zero total charge in stationary motion.End of Section 3, p. 370
Settled physics
The way in
https://londmathsoc.onlinelibrary.wiley.com/doi/abs/10.1112/plms/s2-1.1.367Published in 1904 and long out of copyright, though the publisher still gates its own scan. The complete paper is free to read in the bound volume digitised by the Internet Archive as ‘Proceedings of the London Mathematical Society 1904: Vol 1’, where it begins at page 367.
How to cite it
E. T. Whittaker (1904) On an expression of the EM field by two scalar potential functions. doi:10.1112/plms/s2-1.1.367
Where it sits in the curriculum
Scalar waves and the field behind the fieldsWhat the vacuum is