TL;DR
This paper investigates the dynamics of a classical charge under a magnetic field, focusing on radiative damping effects and comparing different approximation methods to derive the friction equation.
Contribution
It provides a detailed analysis of the Sommerfeld-Page approximation and contrasts it with standard Taylor expansion methods for small magnetic fields.
Findings
Derivation of the friction equation from the Sommerfeld-Page approximation.
Comparison showing differences between the linear memory equation and the second order Taylor expansion.
Insights into the physical relevance of the extended charge distribution model.
Abstract
We are interested in the motion of a classical charge coupled to the Maxwell self-field and subject to a uniform external magnetic field, B. This is a physically relevant, but difficult dynamical problem, to which contributions range over more than one hundred years. Specifically, we will study the Sommerfeld-Page approximation which assumes an extended charge distribution at small velocities. The memory equation is then linear and many details become available. We discuss how the friction equation arises in the limit of "small" B and contrast this result with the standard Taylor expansion resulting in a second order equation for the velocity of the charge.
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