Tangential force, Frictional Torque and Heating Rate of a Small Neutral Rotating Particle Moving through the Equilibrium Background Radiation
TTangential force, Frictional Torque and Heating Rate of a Small Neutral Rotating Particle Moving through the Equilibrium Background Radiation
G.V. Dedkov and A.A. Kyasov Nanoscale Physics Group, Kabardino-Balkarian State University, Nalchik, 360004, Russia For the first time, based on the fluctuation-electromagnetic theory, we have calculated the drug force, the radiation heat flux and the frictional torque on a small rotating particle moving at a relativistic velocity through the equilibrium background radiation (photon gas). The particle and background radiation are characterized by different temperatures corresponding to the local thermodynamic equilibrium in their own reference frames. PACS 42.50 Wk; 41.60.-m; 78.70.-g
1. Introduction
Interaction of a small neutral polarizable body with vacuum electromagnetic fields is the long-standing classical problem of the fluctuation and quantum electrodynamics [1]. Of particular interest is the case when the particle moves, rotates or simultaneously rotates and moves with relativistic velocity. The first dynamical problem of such a kind has been solved by Mkrtchian et. al. [2], relating to the case of frictional drug acting on a nonrelativistic particle moving through an equilibrium photon gas. This photon gas (heat bath) is formed in an oven or it may represent a cosmic microwave background. It is worth noting that the well-known formula [3] for the radiation force exerted by the photon gas on a relativistic sphere is not adequate in the case under consideration, namely W R l<< , with R and W l being the particle radius and the characteristic wave-length of thermal radiation. The problems of relativistic frictional drug and particle-vacuum radiation (for small particles W R l<< ) were first examined by us in [4,5]. The results [2] and [4,5] have been also confirmed by other authors [6,7], using a fully covariant calculation method. Quite recently, we and other authors [8-10] have calculated frictional torque and thermal radiation intensity on a particle rotating in vacuum [8], near the surface [9,10], and for two rotating particles in vacuum [11]. The purpose of this work is to consider the more general problem relating to the particle and the background electromagnetic radiation, assuming the Corresponding author e-mail: [email protected]
2. Theory
We will use the same calculation method as that in [4,5,9], considering a small spherical particle of radius R with the dipole electric polarizability )( wa and temperature T , moving with the velocity )0,0,( V = V and simultaneously rotating with the angular velocity WWW W )0,0,( W= (Fig. 1a) or WWW W ),0,0( W= (Fig. 1b) in the space filled by the equilibrium electromagnetic radiation (photonic gas) of temperature T . Therefore, the angular velocity W is defined in the frame S ¢ which moves with the velocity V relative to the system S related to the equilibrium radiation. The values )( wa and T are defined in the particle rest frame S ¢¢ rotating with the angular velocity W relative to the frame S ¢ . Assuming that all the physical quantities are defined in S , the starting equations for the tangential force x F , the frictional torque x M and the particle heating rate Q & are given by ( ) ( ) )2()1(spindspindindspindsp xxxxx FFF +”(cid:215)+(cid:215)(cid:209)+(cid:215)+(cid:215)(cid:209)=
BmEdBmEd (1) ( ) ( ) )2()1(spindspindindspindsp
QQQ &&&&&&& +”(cid:215)+(cid:215)+(cid:215)+(cid:215)=
BmEdBmEd (2) )2()1(spindspindindspindsp xxxxx
MMM +”·+·+·+·=
BmEdBmEd (3) In Eqs. (1)-(3), the superscripts “sp” , “ind” denote the spontaneous and induced components of the fluctuating dipole electric and magnetic moments d, m, and electromagnetic fields E, B , the points above md ,, Q denote the time differentiation, while the angular brackets denote complete quantum and statistical averaging. It is worth noting that within the relativistic statement of the problem, even the particle with zero magnetic polarizability in its rest frame S ¢¢ has the fluctuating magnetic moment in S .
3 For definiteness, let us consider configuration 1 (Fig. 1a). In subsequent calculations, one should take into account that in the presence of rotation the fluctuation-dissipation theorem (FDT) for the spontaneous fluctuating dipole moment of the particle in frame
S ¢ takes the form Tkdd
Bxspxsp wwawwdpww hh ¢¢¢+=¢¢¢ (4) ¢¢+¢¢(cid:215) (cid:215)¢+=¢¢¢=¢¢¢ --++ TkTk dddd
BB zspzspyspysp hh h (5) ¢¢-¢¢(cid:215) (cid:215)¢+-=¢¢¢-=¢¢¢ --++ TkTk dddd
BB yspzspzspysp hh h (6) where
W–= – ww . Moreover, in the case of rotation, relationships between the induced dipole and magnetic moments of rotating particle and the spontaneous fluctuating electromagnetic field of the equilibrium background radiation in frame S take the form ( ) ( ) tVkEVkkddtd xxspxxin )-i(exp),()()2(1)( ∫ k (7) ( )( ){ ( ) ( ) [ ] ( ) ( ) ( ) [ ] } (cid:215) +--W--+ +--+(cid:215)W+-(cid:215) (cid:215)-= ∫ ),(i),(),(i),()( ),(i),(),(i),()(21 )-i(exp)2()( kkkk kkkk yspzspzspyspx yspzspzspyspx xyin BBEEVk BBEEVk tVkkddtd (8) ( )( ){ ( ) ( ) [ ] ( ) ( ) ( ) [ ] } (cid:215) -++W--+ +++-(cid:215)W+-(cid:215) (cid:215)-= ∫ ),(i),(),(i),()( ),(i),(),(i),()(21 )-i(exp)2()( kkkk kkkk zspyspyspzspx zspyspyspzspx xzin BBEEVk BBEEVk tVkkddtd (9) )()(),()(,0)( tdtmtdtmtm yinzinzinyinxin bb -=== (10) where )1(,/ - -== bgb cV .
4 In the case of configuration 2, the correlators of dipole moments are obtained from (4)—(6) by a cyclic permutation xzyx fififi , while induced dipole moments take the form ( ) ( ) [ ] ),(),(),(i),(),( )-i(exp)2()(
211 43 kkk wbwwawwag wpw zspyspxsp xxin
BEE tVkkddtd -W+W(cid:215) (cid:215)-= - ∫ (11) ( ) ( ) [ ] ),(),(),(),(),(i )-i(exp)2()(
12 43 kkk wbwwgawwa wpw zspyspxsp xyin
BEE tVkkddtd -W+W-(cid:215) (cid:215)-= ∫ (12) ( ) ( ) [ ] ),(),()()-i(exp)2()( kk wbwwgawpwg yspzspxxzin BEVktVkkddtd +--= ∫ (13) ( ) ))(())((21),( W--–W+-=W
VkVk xx wgawgawa (14) The corresponding magnetic moments are given by Eqs. (10) combined with (11)—(13). Using (4)—(10), the calculations in (1)-(3) are performed very similar to [5] and [12].
3. Results
The obtained final expressions for the tangential force x F , the heating rate Q & and the frictional torque x M ( z M ) on a particle are given by ( )1(,/ - -== bgb cV ) a) configuration 1 ( WWW W ,0,0)( W= ) ( ) ( ) [ ] ( ) ( ) W++-(cid:215)W++¢¢++++ + +-(cid:215)+¢¢-- (cid:215)-= ∫∫ -+¥¥- Tk xTkxxx Tk xTkxx dxxdcF
BB BBx bwgwbgwabb bwgwbgwab wwp g hhhhh (15) ( ) ( ) [ ] ( ) ( ) W++-(cid:215)W++¢¢++++ + +-(cid:215)+¢¢-- (cid:215)+= ∫∫ -+¥¥- Tk xTkxxx Tk xTkxx xdxdcdtdQ
BB BB bwgwbgwabb bwgwbgwab bwwp g hhhhh (16) [ ] ( )( )
W++-(cid:215) (cid:215)W++¢¢+++-= ∫∫ -+¥¥-
12 221133
Tk xTk xxxdxdcM
BBx bwgw bgwabbwwp g hhh (17) b) configuration 2 (
WWW W ),0,0( W= [ ] ( )( ) [ ] ( )( ) +-(cid:215) +¢¢++++ W++-(cid:215) (cid:215)W++¢¢+--+++ (cid:215)-= ∫∫ -+¥¥-
12 2212 2222 1144
Tk xTk xxxTk xTk xxxx dxxdcF
BB BBx bwgw bgwabbbwgw bgwabbb wwp g hh hhh (18) [ ] ( )( ) [ ] ( )( ) +-(cid:215) (cid:215)+¢¢++++
W++-(cid:215) (cid:215)W++¢¢+--+++ +(cid:215)= ∫∫ -+¥¥-
12 2212 2222 1143
Tk xTk xxxTk xTk xxxx xdxdcdtdQ
BB BB bwgw bgwabbbwgw bgwabbb bwwp g hh hhh (19) ( ) ( )( )
W++-(cid:215) (cid:215)W++¢¢+--= ∫∫ -+¥¥-
12 11 233
Tk xTk xxxdxdcM
BBz bwgw bgwabwwp hh h (20)
6 As we can see, effects of relativistic motion and rotation form universal combinations in the frequency arguments. The modified frequencies in the polarizability )( wa and in the terms involving the particle temperature T appear due to the Lorentz and rotation transformations of the electromagnetic field and dipole moments from reference frame of vacuum to the reference frame of the particle. The lack of this dependence in the arguments of cotangent involving vacuum temperature T is the characteristic mark of the reference frame at rest. One can also see that the integrands in Eqs. (15),(16) and in Eqs. (18),(19) are completely identical with the replacement x by )1( x b+ . The difference is due to the differentiation over the - x coordinate in (1) and over the time t in (2). At 0 =b , Eq. (16),(17) and (19),(20) are reduced to the results by Manjavacas et. al. [1] for the particle rotating in vacuum, while at =W we obtain 0 = x M from (17) and = z M from (19). In addition, Eqs.(15),(18) are reduced to [4,5]. Finally, one can see that the value of the frictional moment in configuration 1 is a factor g higher than that in configuration 2. For a particle with the magnetic polarizability )( wa m , formulas (15)-(20) remain the same with the replacement )()( wawa m fi , while in general, when the particle has both electric and magnetic moments, we have to take the sum of the polarizabilities. Conclusions
We have performed a generalization of the theory of the fluctuation-electromagnetic interaction relating to a small polarizable particle rotating with the angular velocity W and uniformly moving through the equilibrium electromagnetic radiation with relativistic velocity V . The particular cases of spinless moving particle and particle with spin at rest follow from the obtained formulas in a simple way. References [1] L. D. Landau and E. M. Lifshitz, Course of Theoretical Physics, Vol. 9: Statistical Physics , Part 2, Butterworth–Heinemann, Oxford, 1998. [2] V.E. Mkrtchian, V.A. Parsegian, R. Podgornik, and W.M. Saslow, Phys. Rev. Lett. 91 (22) (2003) 220801(4pp). [3] G. R. Henry, R. B. Feduniak, J. E. Silver, and M.A. Peterson, Phys. Rev. 176 (1968) 1451. [4] G.V. Dedkov and A.A. Kyasov, Phys. Lett. A339 (2005) 212. [5] G.V. Dedkov and A.A. Kyasov, Nucl. Instr. Meth. B268 (2010) 599. [6] F. Intravaia, C. Henkel, and M. Antezza, in
Casimir Physics , Vol.834 of
Lecture Notes in Physics , ed. By D.A.R. Dalvit, P.W. Milonni, D. Rorberts, and. F.da Rosa (Springer, Berlin Heidelberg, 2011), p.345. [7] G. Pieplow and C. Henkel, arXiv: 1209.6511v1; New J. Phys. (in print). [8] A. Manjavacas, F. Garcia de Abajo, Phys. Rev. Lett. 105 (2010) 113601. [9] R. Zhao, A. Manjavacas, F. Garcia de Abajo, and J.B. Pendry, Phys. Rev. Lett. 109 (2012) 123604 [10] G.V. Dedkov, A.A. Kyasov, Europhys. Lett. 99 (2012) 6302. [11] A.A. Kyasov, G.V. Dedkov, arXiv: 1210.6957. [12] G.V. Dedkov, A.A.Kyasov, Phys. Solid State 51/1 (2009) 3., ed. By D.A.R. Dalvit, P.W. Milonni, D. Rorberts, and. F.da Rosa (Springer, Berlin Heidelberg, 2011), p.345. [7] G. Pieplow and C. Henkel, arXiv: 1209.6511v1; New J. Phys. (in print). [8] A. Manjavacas, F. Garcia de Abajo, Phys. Rev. Lett. 105 (2010) 113601. [9] R. Zhao, A. Manjavacas, F. Garcia de Abajo, and J.B. Pendry, Phys. Rev. Lett. 109 (2012) 123604 [10] G.V. Dedkov, A.A. Kyasov, Europhys. Lett. 99 (2012) 6302. [11] A.A. Kyasov, G.V. Dedkov, arXiv: 1210.6957. [12] G.V. Dedkov, A.A.Kyasov, Phys. Solid State 51/1 (2009) 3.