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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 [itex] \frac{e^2}{r} \left [1 - \frac{v_1 v_2}{c^2} \right ][/itex].

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

1.) [itex]\mathbf{F} = e (\mathbf{v}_1 \times \mathbf{B})[/itex] and

2.) [itex] \mathbf{B} = \frac{e}{c^2} \frac{\mathbf{v}_2 \times \mathbf{r}}{r^3} [/itex],

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 [itex] \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}) [/itex], 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

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# Classical magnetic energy of two electrons

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