Slow Motion of Charges Interacting Through the Maxwell Field

Details Slow Motion of Charges Interacting Through the Maxwell Field Slow Motion of Charges Interacting Through the Maxwell Field

2000-01-03

arxiv.org/abs/math-ph/0001002v1

Physics

Mathematical Physics

Scientific paper

10.1007/s002200000219

We study the Abraham model for $N$ charges interacting with the Maxwell field. On the scale of the charge diameter, $R_{\phi}$, the charges are a distance $\eps^{-1}R_{\phi}$ apart and have a velocity $\sqrt{\eps} c$ with $\eps$ a small dimensionless parameter. We follow the motion of the charges over times of the order $\eps^{-3/2}R_{\phi}/c$ and prove that on this time scale their motion is well approximated by the Darwin Lagrangian. The mass is renormalized. The interaction is dominated by the instantaneous Coulomb forces, which are of the order $\eps^{2}$. The magnetic fields and first order retardation generate the Darwin correction of the order $\eps^{3}$. Radiation damping would be of the order $\eps^{7/2}$.

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