Influence of a High-Frequency Field on the Interaction of Charged Particles

Abstract—

The influence of an external ac electric field on the interaction of two charged particles is investigated. The corrections to Coulomb’s law, which result from averaging over high-frequency oscillations, are calculated. The character of the change in the orbital motion for two attracting particles is determined.

APPENDIX

1.1 GENERATING FUNCTIONS

In the case of linearly polarized external field (4), the higher terms of the generating function expansion have the form

$$\begin{gathered} {{S}_{1}} = - {{{\bar {p}}}_{z}}\cos (t) - \frac{t}{4} + \frac{1}{8}\sin (2t), \\ {{S}_{2}} = \sigma \sin (t)\frac{z}{{{{r}^{3}}}}, \\ {{S}_{3}} = \frac{\sigma }{{8{{r}^{5}}}}\left( {{{x}^{2}} + {{y}^{2}} - 2{{z}^{2}}} \right)\left( {8{{{\bar {p}}}_{z}} + 6t - 3\sin (2t)} \right) \\ - \,\,\frac{{3\sigma z\cos t}}{{{{r}^{5}}}}\left( {x{{{\bar {p}}}_{x}} + y{{{\bar {p}}}_{y}}} \right). \\ \end{gathered} $$

(20)

For circular polarization (5), the terms of the generating function expansion are

$$\begin{gathered} {{S}_{0}} = \bar {J}t + y\left( {{{{\bar {p}}}_{y}} + \eta \cos (t)} \right) + x\left( {{{{\bar {p}}}_{x}} - \sin (t)} \right) + {{{\bar {p}}}_{z}}z, \\ {{S}_{1}} = - {{{\bar {p}}}_{x}}\cos (t) + \eta \,{{{\bar {p}}}_{y}}\sin (t), \\ {{S}_{2}} = \frac{\sigma }{{{{r}^{3}}}}\left( {\sin (t)x - \eta \cos (t)y} \right), \\ {{S}_{3}} = - \frac{\sigma }{{4{{r}^{5}}}} \\ \times \,\,\left[ {6yx\left( {2\eta \,{{{\bar {p}}}_{x}}\sin (t) + 2{{{\bar {p}}}_{y}}\cos (t) + 3\eta {{{\cos }}^{2}}(t)} \right)} \right. \\ + \,\,{{x}^{2}}\left( {\cos (t)\left( {8{{{\bar {p}}}_{x}} - 9\sin (t)} \right) - 4\eta \,{{{\bar {p}}}_{y}}\sin (t)} \right) \\ + \,{{y}^{2}}\left( {8\eta \,{{{\bar {p}}}_{y}}\sin (t) + \cos (t)\left( {9\sin (t) - 4{{{\bar {p}}}_{x}}} \right)} \right) \\ - \,\,4{{z}^{2}}\left( {\eta \,{{{\bar {p}}}_{y}}\sin (t) + {{{\bar {p}}}_{x}}\cos (t)} \right) \\ + \,\,\left. {12{{{\bar {p}}}_{z}}z\left( {\eta {\kern 1pt} y\sin (t) + 12x\cos (t)} \right)} \right]. \\ \end{gathered} $$

(21)