It is unclear to me, if an electron in motion in a conductor emits a stationary (Coulomb field) or a dynamic (retarded) field. The motivation for this question, is the following paradoxical observation:
- If the field is given by the static equation, I am unable to derive the correct magnetic forces in a relativistic analysis. See my question on this topic:
Is magnetism an electrostatic or electrodynamic effect in the rest frame of the affected charge?
- If the field is given by the dynamic equation, the wire will not be electrically neutral in its rest frame. This can easily be seen by computing a segment of a wire containing a positive particle at rest, and a negative particle in motion.
$E_{segment}=E_++E_-=\frac{q}{(4\pi\epsilon_0 r^2 )} \hat{r} +\frac{-q(1-v^2/c^2 )}{4\pi \epsilon_0 r^2 (1-v^2/c^2 sin^2 (\theta))^{3/2}} \hat{r}$
Where $\theta$ is the angle between the velocity vector of the negative particle and r. If the negative particle has nonzero velocity then $E_{segment}\neq 0$.
I analyzed a conductor with a kink, and two effects were at play. 1. Due to retardation, the field of the electron was displaced out of the wire at the kink. 2. The angular dependency of the field enhanced or weakened the field at given angles.
Considerations.
The wave function of an electron in a conducting circuit, is distributed over the conductor, and its center of mass is at rest in the lab frame. (This is in favor of the Coulomb field)
In a classical analysis, an electron in a curved wire is being accelerated to follow the curvature of the wire, and thus should radiate as predicted by the acceleration term of the general electric field expression of a point charge. But we know from superchargers that this is not the case. (Much as an electron in an atom)
So, in short, if the field is static, how do we explain the magnetic forces, and if it is dynamic, why don't we see an electric field in the lab frame?
Any input would be appreciated!
Thanks.
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