Classical magnetic energy of two electrons

[Syrius]

Cheers everybody,

I've got a question about an equation in the famous paper "The Effect Of Retardation On The Interaction Of Two Electrons" by G. Breit. There on the first page, it is said, that a first guess for a two electron relativistic wave equation is made by constructing the interaction potential analogously to the classical one $\frac{e^2}{r} \left [1 - \frac{v_1 v_2}{c^2} \right ]$.

If I want to derive this equation by myself I use the formulas

1.) $\mathbf{F} = e (\mathbf{v}_1 \times \mathbf{B})$ and

2.) $\mathbf{B} = \frac{e}{c^2} \frac{\mathbf{v}_2 \times \mathbf{r}}{r^3}$,

where the first formula describes the Lorentz force that is experienced by the first electron due the B-field that is created by the second electron (Eq. 2). If I plug in Eq. 2 in Eq. 1 and use a vector identity for the cross product I get $\mathbf{F} = \frac{e^2}{c^2 r^3} ((\mathbf{v}_1 \mathbf{r})\mathbf{v}_2-(\mathbf{v}_1 \mathbf{v}_2) \mathbf{r})$, which as a conclusion asks for a second term in the interaction potential above. It seems that this term has been neglected or that the velocities of the electrons are assumed to be parallel, neither of which I understand why. Why should the electrons move with parallel velocity in a classical treatment. Do you have any suggestions?

Greetings, Syrius

 

[Meir Achuz]

Gregory must have been thinking of simple geometry to simplify the algebra.
Are you interested in history of physics to be studying that paper?
The relativistic force between two moving charges is in (advanced) EM textbooks.

 

[Syrius]

Hello Achuz,

I am studying this paper, because I'd like to know the origin of the Breit interaction, since it appears frequently in atomic physics calculations.

Greetings,
Syrius