20 Further Applications of the Classical Radiation Theory 20.1 RELATIVISTIC RADIATION AND THE STOKES VECTOR FOR A LINEAR OSCILLATOR In previous chapters we have considered the emission of radiation by nonrelativistic moving particles In particular, we determined the Stokes parameters for particles moving in linear or curvilinear paths Here and in Section 20.2 and 20.3 we reconsider these problems in the relativistic regime It is customary to describe the velocity of the charge relative to the speed of light by ¼ v=c For a linearly oscillating charge we saw that the emitted radiation was linearly polarized and its intensity dependence varied as sin2  This result was derived for the nonrelativistic regime ð ( 1Þ We now consider the same problem, using the relativistic form of the radiation field Before we can this, however, we must first show that for the relativistic regime ð $ 1Þ the radiation field continues to consist only of transverse components, E and E , and the radial or longitudinal electric component Er is zero If this is true, then we can continue to use the same definition of the Stokes parameters for a spherical radiation field The relativistic radiated field has been shown by Jackson to be ! e n _ Eðx, tÞ ¼ ð20-1aÞ Â fðn À Þ Â ð Þg 4"0 c2 3 R ret where ¼1ÀnÁ ð20-1bÞ The brackets ½Á Á ÁŠret means that the field is to be evaluated at an earlier or retarded time, t0 ¼ t À Rðt0 Þ=c where R/c is just the time of propagation of the disturbance from one point to the other Furthermore, c is the instantaneous velocity of the particle, c _ is the instantaneous acceleration, and n ¼ R=R The quantity  ! for nonrelativistic motion For relativistic motion the fields depend on the velocity as Copyright © 2003 by Marcel Dekker, Inc All Rights Reserved Figure 20-1 Coordinate relations for an accelerating electron P is the observation point and O is the origin (From Jackson.) well as the acceleration Consequently, as we shall soon clearly see the angular distribution is more complicated In Fig 20-1 we show the relations among the coordinates given in (20-1a) We recall that the Poynting vector S is given by S ¼ c"0 jEj2 n ð20-2Þ Thus, we can write, using (20-1a), 2 e2  ½ S Á nŠ ¼ n  ½ðn À _ފ À ½ur  ð  _ފ 4"0 c  R The triple vector product relation can be expressed as a  ðb  cÞ ¼ bða Á cÞ À cða Á bÞ so (20-11) can be rewritten as  e Eðr, tÞ ¼ ur ður Á _Þ À _ður Á ur Þ À _ður Á _Þ þ _ður Á ފ 4"0 c  R ð20-12Þ ð20-13Þ In spherical coordinates the field Eðr, tÞ is Eðr, tÞ ¼ Er ur þ E u þ E u ð20-14Þ Taking the dot product of both sides of (20-13) with ur and using (20-14), we see that Er ¼ ður Á _Þ À ður Á _Þ À ður Á Þður Á _Þ þ ður Á _Þður Á Þ ¼ Copyright © 2003 by Marcel Dekker, Inc All Rights Reserved ð20-15Þ [...]... ! 20- 3Þ ret There are two types of relativistic effects present The first is the effect of the specific spatial relationship between and _, which determines the detailed angular distribution The other is a general relativistic effect arising from the transformation from the rest frame of the particle to the observer’s frame and manifesting itself by the presence of the factor  in the denominator of (20- 3)... ¼T2 ðS Á nÞ t0 ¼T 1 dt 0 dt dt0 20- 4Þ À Á The quantity ðS Á nÞ dt=dt0 is the power radiated per unit area in terms of the charge’s own time The terms t0 and t are related by t0 ¼ t À Rðt0 Þ c 20- 5Þ Furthermore, as Jackson has also shown, ¼1þ 1 dRðt0 Þ c dt0 Copyright © 200 3 by Marcel Dekker, Inc All Rights Reserved 20- 6Þ Differentiating (20- 5) yields dt ¼ dt0 20- 7Þ The power radiated per unit solid... apply these results to various problems of interest, we must demonstrate that the definition of the Stokes parameters (20- 9) is valid for relativistic motion That is, the field is transverse and there is no longitudinal component ðEr ¼ 0Þ We thus write (20- 1a) as " # à n  ðn  _Þ À ½n  ð  _ފ e 20- 10Þ Eðx, tÞ ¼ 3 4"0 c2 R ret Because the unit vector n is practically in the same direction as ur , (20- 10)... The components E and E are readily found for the relativistic regime We have x_ i þ y_ j þ z_k c € € x i þ y j þ z€k _ ¼ c ¼ 20- 16aÞ 20- 16bÞ The Cartesian unit vectors in (20- 16a) and (20- 16b) can be replaced with the unit vectors in spherical coordinates, namely, i ¼ sin ur þ cos u 20- 17aÞ j ¼ u 20- 17bÞ k ¼ cos ur À sin u 20- 17cÞ In (20- 17) the azimuthal angle has been set to zero because... ¼ 20- 11Þ ½ur  ður  _ފ À ½ur  ð  _ފ 2 3 4"0 c  R The triple vector product relation can be expressed as a  ðb  cÞ ¼ bða Á cÞ À cða Á bÞ so (20- 11) can be rewritten as  e Eðr, tÞ ¼ ur ður Á _Þ À _ður Á ur Þ À _ður Á _Þ þ _ður Á ފ 2 3 4"0 c  R 20- 12Þ 20- 13Þ In spherical coordinates the field Eðr, tÞ is Eðr, tÞ ¼ Er ur þ E u þ E u 20- 14Þ Taking the dot product of both sides of (20- 13)... R2 S Á n d dt 20- 8Þ These results show that we will obtain a set of Stokes parameters consistent with (20- 8) by defining the Stokes parameters as  à 1 S0 ¼ c"0 R2 E Eà þ E Eà 2  à 1 S1 ¼ c"0 R2 E Eà À E Eà 2  à 1 S2 ¼ c"0 R2 E Eà þ E Eà 2  à 1 S3 ¼ c"0 R2 iðE Eà À E EÃ Þ 2 20- 9aÞ 20- 9bÞ 20- 9cÞ 20- 9dÞ where the electric field Eðx, tÞ is calculated from (20- 1a) Before we... and using (20- 14), we see that Er ¼ ður Á _Þ À ður Á _Þ À ður Á Þður Á _Þ þ ður Á _Þður Á Þ ¼ 0 Copyright © 200 3 by Marcel Dekker, Inc All Rights Reserved 20- 15Þ so the longitudinal (radial) component is zero Thus, the radiated field is always transverse in both the nonrelativistic and relativistic regimes Hence, the Stokes parameters definition for spherical coordinates continues to be valid The components... in the denominator of (20- 3) For ultrarelativistic particles the latter effect dominates the whole angular distribution In (20- 3), S Á n is the energy per unit area per unit time detected at an observation point at time t due to radiation emitted by the charge at time t0 ¼ t À Rðt0 Þ=c To calculate the energy radiated during a finite period of acceleration, say from t0 ¼ T1 to t0 ¼ T2 , we write Z Z t¼T2... ¼ sin ur þ cos u 20- 17aÞ j ¼ u 20- 17bÞ k ¼ cos ur À sin u 20- 17cÞ In (20- 17) the azimuthal angle has been set to zero because we assume that we always have symmetry around the z axis Substituting (20- 17) into (20- 16) yields c