# Is Leigh Page's "electrodynamics from the standpoint of the electron theory" valid?

I am interested in this approach, and came accross this book by Leigh Page ("Chairman of Mathematical Physics at the Sloane Physics Laboratory of Yale University for three dacades" by Wikipedia). This book is available from archive.org to read for free ( published by 1922; https://archive.org/details/ost-chemistry-introductiontoel00pagerich/mode/2up )

After deriving Lorentz transformation from relative principles in Chapter 1, He used about 12 pages to derive the Lienard-Wiechert field from electrostatics and Lorentz transformation (page 20-29, page 42-44). However, I found his logic hard to follow, and found this approach not popular nowadays. My question is that whether his approach is valid. Has anybody studied this approach in detail and had a comment? I can't find discussions online.

More specifically, on page 21 he said: "Consider an electric field which is being observed from the two reference systems S and S' of the previous chapter. The principle of relativity requires that the velocity of the moving elements comprising the field shall have the same numerical value no matter whether observations are carried on in S, S', or some other system reciprocal to one of these." I think here "the velocity of the moving elements comprising the field" may be understood as the "bullet" analogy of the field, or the propagation of the field outward at light speed. But I do not understand why they "shall have the same numerical value" from his chapter 1.

Another example is also on page 21, "Suppose a charged particle to be permanently at rest in S. Although the moving elements constituting its field are in motion with the velocity of light, the lines of force themselves are stationary. Hence the motion must be entirely along these lines." I do not understand how this "Hence" is reached. I think maybe we can deduce that from symmetry, but I do not understand his logic.

Also on the same page, "As the charge carries its field along with it, the velocity of a moving element will be along the lines of force only at points in the line of motion." I do not understand why the velocity the moving element should not be along the lines of force at other places. I know the results of mordern approaches, but I can't follow his logic.

• Can you be a bit more specific than "I got lost somewhere within 12 pages of this book?" What conceptual problems did you have? For what it is worth I skimmed the first few pages of the text and it seems to be legitimate (at least it didn't set off any immediate alarm bells), and the Lienard-Wiechard potentials are obviously a well-known result so there's a good chance the derivation is fine. Can I ask why you want to follow the logic of this book, instead of a more modern treatment? Commented Feb 19, 2021 at 3:05
• @Andrew Thanks. I edited the question to be more specific. I have this question because I think magnetic force maybe deduced from electric force by some kind of argument better than those in Purcell's textbook or in Schwartz's textbook. Commented Feb 19, 2021 at 3:27
• I haven't gone through it in detail, but they seem to get everything from the principle that the number of field lines normal to a surface per unit area should be constant in different frames. I'd have to think about this, but basically this is (in my opinion) a rather old-fashioned way to look at things. For what it is worth I think the argument in Purcell is excellent, do you have an issue with it? Commented Feb 19, 2021 at 3:33
• @Andrew I do have some issue with Purcell's. The main one is that it can't explain the magnetic force when the particle is moving toward the wire. The other is that in the frame of the moving current-carrying particles of the wire (the same frame with the single particle outside the wire, suppose they have the same speed as Purcell did), the fixed oppositely charged ions are moving backwards. This effect was not counted I think. Commented Feb 19, 2021 at 3:40
• Fair enough, although I've always taken Purcell's main point to be that logically magnetic fields must exist if you combine electrostatics and special relativity, rather than saying all magnetic fields can be explained as a boost of an electrostatic situation. Anyway, to me the clearest logic to derive the LW potentials is to start by demanding a local, Lorentz invariant theory of a spin-1 field (which leads you to Maxwell's equations), rather than muck around with lines of force. To me this is a more modern, elegant approach. Commented Feb 19, 2021 at 4:27