Classical magnetic energy of two electrons

In summary, Syrius is studying the famous paper "The Effect Of Retardation On The Interaction Of Two Electrons" by G. Breit and has a question about an equation on the first page regarding the construction of a two electron relativistic wave equation. They mention using formulas for the Lorentz force and magnetic field, but have noticed a missing term in the interaction potential. They wonder if this term has been neglected or if the velocities of the electrons are assumed to be parallel. They also mention studying the paper to understand the origin of the Breit interaction for atomic physics calculations.
  • #1
Syrius
8
0
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 [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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  • #2
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.
 
  • #3
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
 

1. What is classical magnetic energy?

Classical magnetic energy is the energy associated with the interaction between two magnetic fields. It is a form of potential energy that arises from the alignment of magnetic dipoles.

2. How is classical magnetic energy calculated?

The classical magnetic energy between two electrons is calculated using the equation E = -μ0/4π(r1•r2)/r^3, where μ0 is the permeability of free space, r1 and r2 are the position vectors of the two electrons, and r is the distance between them.

3. What are the units of classical magnetic energy?

The units of classical magnetic energy are joules (J), the same as any other form of energy. In some cases, it may also be expressed in electron volts (eV) or ergs (erg).

4. How does classical magnetic energy affect the behavior of two electrons?

Classical magnetic energy is a conservative force, meaning it only influences the motion and behavior of the electrons when they are in the presence of a magnetic field. It can cause the electrons to align with the magnetic field or experience a force that changes their trajectory.

5. Can classical magnetic energy be converted into other forms of energy?

Yes, classical magnetic energy can be converted into other forms of energy, such as kinetic energy or thermal energy. This can occur when the aligned electrons experience a force and begin to move, transferring the energy associated with their alignment into another form.

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