Quadrupole Radiation of an Uncharged Droplet That Oscillates in the Presence of Uniform Electric Field

Abstract

Capillary oscillations of an uncharged spheroid droplet of nonviscous conducting uncompressible liquid in the presence of uniform electric field are analyzed in the first order of smallness with respect to the ratio of the oscillation amplitude to a linear size of the droplet and in the second order of smallness with respect to the squared ratio of the linear size to the radiation wavelength. It is shown that the electric quadrupole moment of the droplet is time-dependent due to surface oscillations, which lead to the emission of quadrupole electromagnetic waves. A mathematical model of the quadrupole electromagnetic radiation of an uncharged droplet that oscillates in the presence of electrostatic field is constructed, and the radiation intensity and frequency are estimated.

APPENDIX

Coefficients k₁–k₆ and p₁–p₆ in expressions (10) and (13) are given by the following expressions:

$${{k}_{1}} = \frac{{j(j - 1)}}{{(2j - 1)}},\quad {{k}_{2}} = \frac{{(j + 1)(j - 1)}}{{(2j + 3)}},$$

$${{k}_{3}} = \frac{{j(j - 1)(j - 2)(2{{j}^{2}} + 3j + 7)}}{{2(2j - 1)(2j - 3)(2j - 5)}},$$

$${{k}_{4}} = \frac{{j(100{{j}^{5}} + 116{{j}^{4}} - 165{{j}^{3}} - 205{{j}^{2}} + 75j - 81)}}{{30{{{(2j - 1)}}^{2}}(2j - 3)(2j + 3)}},$$

$${{k}_{5}} = \frac{{(j + 1)(100{{j}^{5}} + 765{{j}^{4}} + 2159{{j}^{3}} + 2591{{j}^{2}} + 704j - 600)}}{{30(2j - 1){{{(2j + 3)}}^{2}}(2j + 5)}},$$

$${{k}_{6}} = \frac{{(j + 1)(j + 2)(j + 3)(2{{j}^{2}} + 11j + 21)}}{{2(2j + 3)(2j + 5)(2j + 7)}},$$

$${{p}_{1}} = \frac{{j(j + 3)}}{{(2j - 1)}},\quad {{p}_{2}} = \frac{{(j + 1)(j + 3)}}{{(2j + 3)}},$$

$${{p}_{3}} = - \frac{{j(j - 1)(j - 2)(9j - 1)}}{{2(2j - 1)(2j - 3)(2j - 5)}},$$

$${{p}_{4}} = - \frac{{j(40{{j}^{5}} + 274{{j}^{4}} - 265{{j}^{3}} - 475{{j}^{2}} + 375j - 9)}}{{30{{{(2j - 1)}}^{2}}(2j - 3)(2j + 3)}},$$

$${{p}_{5}} = \frac{{(j + 1)(135{{j}^{9}} + 990{{j}^{8}} + 2990{{j}^{7}} + 4924{{j}^{6}} + 6145{{j}^{5}} + 8251{{j}^{4}} + 7564{{j}^{3}} + 1927{{j}^{2}} - 886j - 360)}}{{30(2j - 1){{{(2j + 1)}}^{2}}{{{(2j + 3)}}^{2}}(2j + 5)}},$$

$${{p}_{6}} = - \frac{{j(j + 1)(j + 2)(j + 3)(4{{j}^{2}} + 7j - 5)}}{{2(2j + 3)(2j + 5)(2j + 7)}}.$$