RADIATION OF ELECTROMAGNETIC WAVES

Transcription

1 Chapter RADIATION OF ELECTROMAGNETIC WAVES. Introduction In the preceding Chapters, some general properties of TEM waves have been discussed. A question arises as to how to excite electromagnetic waves. We know that a charge q creates the Coulomb eld given by E c 4 0 q r e r; but a stationary charge cannot radiate electromagnetic waves which are necessarily accompanied by energy ow in the form of the Poynting vector. For a stationary charge, the magnetic eld is absent. Even a charge drifting with a constant velocity (not speed) cannot radiate, since the electric eld due to a drifting charge is still of Coulombic nature being proportional to r. Because of energy conservation, the radiation electric eld due to a localized source (such as point charge) must be proportional to r, so that the radiation power through a spherical surface with radius r is independent of the radius r, P c 0 E r d where d is the di erential solid angle. Therefore, the radiation electric eld (/ r) should be entirely di erent from the Coulomb eld (/ r ). Electromagnetic radiation occurs only when charges are under acceleration or deceleration. In

2 antennas, electrons are forced to oscillate back and forth by a generator, and they are under periodic acceleration and deceleration. A heated body emits infrared and visible light, and the origin of light emission can also be explained by the vibrational motion of electrons. In this Chapter, radiation of electromagnetic waves from an accelerated charge will be discussed in detail. This is followed by analysis on radiation from a macroscopic object (such as antennas) in which many charges are collectively involved. In material media, a charged particle can have a velocity larger than the velocity of electromagnetic waves. In this case, Cherenkov radiation, which does not require acceleration on charges, occurs.. Qualitative Picture of Radiation from an Accelerated Charge Let a charge q be accelerated from rest with an acceleration a (m/sec ) for a short duration t. The charge acquires a velocity v at after the acceleration, and starts drifting with velocity. Before and after the acceleration, the electric eld due to the charge is of Coulombic nature and radially outward from the charge. However, the electric eld lines before acceleration are radially outward from the original stationary position of the charge, while those after acceleration originate from the position at vt att from the origin. From the continuity of the electric ux (Gauss law), the electric eld lines before and after acceleration must be somehow connected. The only way to make such a connection is to bend the eld line at the radial position r ' ct, the distance travelled by the disturbance in the electric eld lines at the speed of light, c. At the kink, there is indeed an electric eld component perpendicular to the distance r, as well as the radial Coulomb eld, E c. The tangential component, E t, is the desired radiation electric eld. The ratio between the two elds is E t E c since the Coulomb eld at the kink is given by E c vt at sin sin (.) ct c q 4" 0 (ct) q 4" 0 r (.) we nd E t qa 4" 0 c sin (.3) r

3 Et Ec q vt r ct θ Figure.: The Coulomb eld E c and the radiation eld E t due to an accelerated charge. The radiation eld is maximum in the direction perpendicular to the acceleration a. Indeed, at 0 and, there are no kinks in the electric eld lines. Vectorially, the radiation electric eld E R due to an acceleration a can be written as E R q 4" 0 c n (n a) (.4) r where n r r is the unit vector in the radial direction. It should be cautioned that the radiation eld given in Eq. (.3) is valid only if the charge is nonrelativistic, v c. Also, the acceleration a appearing in Eqs. (.3) and (.4) is the acceleration rc seconds earlier than the observing time t, because it takes the electromagnetic disturbance rc seconds to travel over the distance r. The acceleration at t (rc) is called the acceleration at the retarded time and denoted by a ret Similarly, other variables, r and n, are, to be precise, those at the retarded time. If a charge undergoes harmonic oscillation (continuous acceleration and deceleration), the radiation eld is also harmonic with the same frequency. (Again, this is valid only in non-relativistic cases.) The radiation magnetic eld associated with the radiation electric eld is perpendicular to 3

4 both E R and r which is the direction of the Poynting vector or energy ow, B R c n E R (.5) The magnitude of the Poynting vector is S r E R p q a sin " 0 0 (4" 0 c ) r Then, the total radiation power can be readily found, q a sin 4" 0 4c r (W/m ) (.6) P q a 4" 0 4c 3 sin d (.7) where d sin dd is the di erential solid angle. Performing integration, we nd P q a 4" 0 4c 3 sin 3 d d 0 0 q a 4" 0 3c 3 (W) (.8) This is known as the Larmor s formula for radiation power emitted by nonrelativistic charge v c:.3 Lienard-Wiechert Potentials and Consequent Fields If the particle velocity is large and approaches the speed of light, the radiation electric eld in Eq. (.4) will be modi ed signi cantly. To see how relativistic e ects modi es the radiation eld, we need to nd how the scalar and vector potentials are a ected by relativistic velocity. As a preparation, let us rst convince ourselves that a rod moving toward (away from) us appears longer (shorter) than its actual length l. (This has nothing to do with the celebrated relativistic length contraction, which was formulated by Lorentz well after the work by Lienard-Wiechert.) This is because light emitted from the rear end of the rod takes a longer time than that emitted from the front end to reach an observer, and for an observer to be able to measure the length of the rod, he/she needs the two signals arriving at the same instant. Let the rod move toward an observer with a velocity v. Light emitted by the front end at the instant when the front end is at a distance 4

5 A v B pulse A pulse B l l' Figure.: A rod of length l moving toward an observer appears to be longer l 0 should not be confused with relativity e ect. l This x from the observer reaches the observer after xc sec. Light emitted by the rear end at the same instant reaches the observer at (x + l)c sec, that is lc sec later. If we denote the apparent length seen by the observer by l 0, the extra time needed for the light leaving the rear end is l 0 c During this interval, the rod has moved a distance l 0 l with a velocity v. Therefore, l 0 c l0 l v : Solving for l 0, we nd l 0 l v (.9) c In general, if an object is moving with a velocity v, its dimension toward an observer appears to change by a factor n (.0) where vc, and n is the unit vector toward the observer. If the object has a volume dv, the apparent volume is dv 0 dv n (.) For a charge density, the apparent di erential charge is therefore dq 0 dv n dq n (.) 5

6 The current density J v is also modi ed as JdV 0 JdV n (.3) Therefore, the scalar and vector potentials due to a charge moving at velocity v (t) are to be evaluated according to 4" 0 q jr r p (t 0 )j A 0 qv (t 0 ) 4 jr r p (t 0 )j (.4) (.5) where r p (t 0 ) is the instantaneous location of the charge at the retarded time and v (t 0 ) dr p (t 0 )dt 0 is the particle velocity at the retarded time t 0 : The dimensionless quantity t 0 n (.6) also depends on time t 0 : The potentials given in Eqs. (.4,.5) are called the Lienard-Wiechert potentials which were formulated in The electromagnetic elds to be derived from these potentials properly contain all relativistic e ects. It should be noted that all variables in those potentials are to be evaluated at the retarded time which is related to the observing time t through t 0 t jr r p (t 0 )j c (.7) where r 0 (t 0 ) is the instantaneous position of the charge. For example, the gradient operation r with respect to the observing coordinates r is to be performed as follows: r r R + 0 r R + @t (.8) where R r r p (t 0 ), R jrj; n RR. Similarly, the time derivative follows the chain rule, 0 (.9) t + n v c 6 jr r p (t 0 )j

7 In this derivation, note that v(t 0 Then, we nd an important relationship between t and t n (.0) Figure.3: The change in the unit vector n is caused by the perpendicular velocity v? : The gradient of the retarded time rt 0 can be calculated similarly, from which it follows rt 0 r t rt 0 jr r p (t 0 ) j c c rjr r p t 0 j n + (n )rt0 c n n c n c (.) The transformations in Eqs. (.0) and (.) allow us to evaluate the electric and magnetic elds due to a moving charge. For example, the electric eld is to be found from (.) 7

8 The gradient of the scalar potential becomes q r r R + rt 4" 0 n R q r R + rt 4" 0 n 0 q 6 n(n ) n 4 4" 0 ( n ) R n R q 4" 0 n R n c ( n ) n n(n ) ( n ) R ( n ) >: ( n ) R + n However, the time variation of the unit vector n RR can be caused only by the velocity component perpendicular to n, as seen in Fig..3. From the two similar triangles in the gure, n v R 93 > 7 5 >; (.3) we nd or Therefore, r q 4" 0 dn v?dt 0 v?dt 0 R (n ) ( n ) + n ( n ) (n ) n? 3 R + n c 3 R (.4) n _ (.5) where The time derivative of the vector potential 0 v 0 R q _ 4" 0 c R q 4" 0 0 (R) " _ R + R (n _) # c (R) ( n ) (.7) where 0 has been eliminated in favour of 0 through c 0 0. Substituting Eqs. (.5) and (.7) into (.), we thus nd E (r; t) q 4" 0 3 (n ) R ret + q h i 4" 0 c 3 R n (n ) _ ret (.8) 8

9 and the vector identity A (B C) B (A C) C (A B) (.9) has been used. Eq. (.8) is the desired general expression for the electric eld due to a moving charge, and valid for arbitrarily large particle velocity. The electric eld above was rst formulated by Heaviside. [ ] ret means that all quantities in the brackets are retarded, that is, for evaluation of the electric eld at time t; [ ] evaluated at t 0 t r rp t 0 c should be used. The reason the formula, Eq. (.8), remains valid at relativistic velocities even though it has been derived from the Lienard-Wiechert potentials formulated before the discovery of the relativity theory (Einstein 905) is due to the obvious fact that electromagnetic waves propagate at the speed c irrespective of the source speed once they are emitted. Sound waves in air also propagate with the sound velocity after being emitted irrespective of source speed. A major di erence between electromagnetic waves in vacuum and sound waves in air is that the speed of electromagnetic waves remains c even when the observer is moving, while the sound speed appears to change if the observer is moving relative to the wave medium, that is, the air. The magnetic eld is to be calculated from B r A c n E (.30) where E is the electric eld given in Eq. (.8). Derivation of Eq. (.3) is left for an exercise. The electric eld in Eq. (.8) has two terms. The rst term is inversely proportional to R, and does not contain the acceleration _. This is essentially the Coulomb eld corrected for relativistic e ects (). The second term is inversely proportional to R and proportional to the acceleration _: This term is the desired radiation electric eld. Note that at large R, the radiation eld (/ R) becomes predominant over the Coulomb eld (/ R ). In nonrelativistic limit,, we recover the radiation electric eld worked out in Eq. (.4) from qualitative arguments where all quantities, n, R, v are to be evaluated at the retarded time. We will return to radiation problems associated with relativistic particles in Sec..5. 9

10 .4 Radiation from a Charge under Linear Acceleration If the acceleration is parallel (or anti-parallel) to the velocity, _ k ; the particle trajectory remains linear, as in linear accelerators. (The reason high energy electron accelerators are linear rather than circular as for proton accelerators is because radiation loss in circular electron accelerators becomes intolerably large.) The radiation electric eld in this case is given by E (r; t) q 4 n 3 n _ k 4" 0 cr 3 5t0 (.3) where all quantities at the retarded time should be used. If the angle between and n is ; the Poynting ux is S (r; t) 0 je (r; t)j q _ k sin 4" 0 r 4c ( cos ) 6 t 0 (.3) If the Poynting ux is integrated over the spherical surface of radius r; one gets a radiation power at the observing time t: However, what is more meaningful is the radiation power at the retarded time. Since the Poynting ux at the retarded time t 0 is dt 0 dt cos Integration over the solid angle yields S r; t 0 ( cos ) S (r; t) q _ k sin 4" 0 r 4c ( cos ) 5 : (.33) P t 0 k q _ 4" 0 4c 4" 0 3 q _ c k 0 sin 5 sin d ( cos ) 6 (.34) where p (.35) 0

11 E r q θ v, dv/dt Figure.4: Linear acceleration k _ with nonrelativistic velcoity : In highly relativistic case, ; the radiation is emitted predominantly along the beam velocity. (Fig. 7) is the relativity factor. Relevant integral is x ( x) 5 dx (.36) The radiation power is independent of the particle energy mc since the parallel acceleration is inversely proportional to 3 as can be seen from the equation of motion mc d dt 0 B _ k q k k q k F k _ k + k mc 3 _ k F 3 C A F Therefore, the radiation power in Eq. (.34) is independent of the particle energy. This is the main advantage of linear electron accelerators. The angular distribution of the radiation power is proportional to the function f () sin ( cos ) 5 (.37) In nonrelativistic limit ; the radiation intensity peaks in the direction (perpendicular to the velocity and acceleration _ : In relativistic case! ; ; the radiation intensity pro le becomes narrow with an angular spread about the velocity of order ' : The angle at which the radiation intensity peaks can be found from df () d 0

12 which yields In the limit ' ( ) ; 3 cos + cos 5 0 cos q cos q 3 s 3 ' ! + Since ; cos ' ; we nd r 5 ' 4 The radiation is essentially con ned in the angle ' about the velocity vector. This alignment with the velocity is in fact independent of the acceleration direction and ' about the velocity holds even for acceleration perpendicular to the velocity. sin ( 0: cos ) 5 y x.0 sin ( 0:9 cos ) 5

13 y x 0:9: sin ( 0:99 cos ) 5 y e+6 e+6 5e+6 e+7.5e+7 x Figure.5: Radiation pattern when, from top, 0:; 0:90 and 0:99: Note the large radiation intensity as appoaches unity...5 Radiation due to Acceleration Perpendicular to the Velocity _? In this case, the radiation electric eld is E (r; t) 4" 0 e c 3 r n h (n ) _? i (.38) 3

14 We consider a charge undergoing circular motion with radius and normalized velocity in the x z plane. At t 0; the charge passes the origin. At this instant, e z (.39) and _? _? e x (.40) Substituting e x rx r (r sin cos ) sin cos n + cos cos e sin e (.4) and e z r (r cos ) cos n sin e (.4) into n h(n e z ) _ i? e x we obtain and n h(n e z ) _ i? e x _? [( cos ) cos e + ( cos ) sin e ] (.43) The retarded di erential power is E (r; t) e _? 4" 0 c 3 r [( cos ) cos e + ( cos ) sin e ] (.44) dp (t 0 ) d r 0 jej _? e 4" 0 4c ( cos ) 5 ( cos ) sin cos (.45) and the radiation power is P t 0 dp (t 0 ) d d e _? 4 (.46) 4" 0 4c 4

15 Relevant integrals are dx ( x) 3 ( ) 4 (.47) x ( x) 5 dx 4 3 ( ) (.48) Example: Radiation power emitted by 3 GeV electron undergoing circular motion with radius R 0 m. The relativity factor is The acceleration is Then the radiation power is P E mc 3 GeV 0:5 MeV 5900 a? v R ' c R 9 05 m/s e _? 4 4" 0 4c e ja? j 4" 0 4c : (3 0 8 ) 3 (5900) 4 6: W 4:7 0 ev/s/electron Electron rapidly loses its energy to radiation (synchrotron radiation). To reduce radiation power, the orbit radius R must be increased and must be decreased. Circular particle accelerators are therefore practical only for protons. (For proton energy of 00 GeV, the relativity factor is rather mild, 06:).6 Synchrotron Radiation Charged particles emit radiation whenever they are subjected to acceleration. Synchrotron radiation is emitted by relativistic electrons. The classical radiation mechanism is simply bending electron trajectory by a magnetic eld. The acceleration due to trajectory bending is a? ' c R 5

16 where R is the curvature radius of the trajectory. The radiation power due to perpendicular acceleration is with peak frequency components around where P 4" 0 3 (ea? ) c 3 4! ' 3 eb m e eb m e! ce eb m e is the relativistic cyclotron frequency. If B 0:5 T and 5000; the dominant radiation frequency is 3: Hz. Modern synchrotron light sources are equipped with wigglers and undulators to cover wider radiation spectrum and provide higher radiation intensities. In wigglers, pairs of NS magnets are placed periodically and electron beam going through such structure experiences periodic kicks in the direction perpendicular to the beam. The wavelength of emitted radiation is approximately given by w where w is the spatial period of the wiggler. In contrast to the radiation by bending magnet, wiggler radiation is a result of maser or laser action, namely, ampli cation of electromagnetic waves in an electron beam. Wiggler radiation is thus more coherent than bending magnet radiation..7 Radiation from Antennas In antennas used for broadcasting and communication, a large number of conduction electrons are collectively accelerated by a harmonic generator. In fact, any unshielded transmission lines can e ectively become an antenna and they radiate electromagnetic waves. For signal and power transmission purposes, this is an undesirable feature since energy loss inevitably occurs. If, however, a transmission line is carefully shielded except at its end, the open end becomes an e ective antenna. At an open end, current standing waves are formed. For an antenna much shorter than the 6

17 Figure.6: In a wiggler, an electron beam is modulated by a periodic magnetic eld. Electrons acquire spatially oscillating perpendicular displacement x(z) and velocity v x (z) which together with the radiation magnetic eld B Ry produces a ponderomotive force v x (z) B Ry (z) directed in the z direction. The force acts to cause electron bunching required for coherent radiation (as in lasers). wavelength, the radiation electric eld can be found from that due to an accelerated charge where qa can be replaced by E R with l the antenna length. The radiation power is given by P The radiation resistance may be de ned by qa sin 4" 0 c r ; (.49) qa I!l (.50) (I!l) 4" 0 3 c (kl) I ; (W) (.5) P R rad I ; (.5) and in the case of short antenna kl ; it is given by R rad 6 0 (kl) ; () (.53) 7

18 daz r z'cosθ feeder I(z')dz' z' θ I cos(kz') 0 r Figure.7: Center-fed half wavelength dipole antenna. Figure.7 shows the case of center-fed antenna. Since the antenna length is comparable with the wavelength, the radiation electric eld should be calculated taking into account the phase di erence of the antenna current. To nd the radiation eld, we assume the following standing wave, I (z; t) I 0 cos (kz) e j!t ; The retarded radiation vector potential at kr is 4 < z < 4 : (.54) A z (r; ) and the radiation magnetic eld is I 0 e j(!t r cos 0I 0 r kr) I 0 e j(!t kr) r 0 cos k sin e jkz0 cos cos kz 0 dz 0 cos kz 0 cos cos kz 0 dz 0 (.55) e j(!t kr) (.56) H ' 0 jk A z (.57) which yields H j I cos 0 cos e j(!t kr) (.58) kr sin 8

19 Figure.8: The Poynting ux on the surface of antenna of nite radius a is approximated by that on the cylindrical surface of radius a surrounding a thin antenna. The radiation Poynting ux is and the radiation power is S r 0 jhj I0 cos cos 0 (r) sin ; W/m (.59) I 0 P 0 () 0 I cos cos sin cos cos sin d d 0 d (.60) The integral is approximately. and we nd the radiation resistance of the half wavelength dipole antenna, R rad 0 : 73: (.6) If the Poynting ux on the antenna surface is directly integrated, the reactive power can be estimated as well. When the antenna length is ; the reactance is about j40 (inductive). However, it sensitively depends on the length. For l 0:49 ; the reactance vanishes. 9

20 The calculation presented above entirely ignores the reactive power which may exist due to storage of electric and/or magnetic energy in the vicinity of the antenna. To account for such reactive power, we must deviate from the far- eld analysis and integrate the Poynting ux directly on a surface close to the antenna surface. Instead of calculating the Poynting ux on the antenna surface of nite radius a, we calculate the Poynting ux on a cylindrical surface of radius a surrounding an ideally thin antenna of length as shown in Fig..8. In this approximation, the magnetic eld on the antenna surface may be replaced by the static form without retardation, H (z) I(z) a I 0 a cos(kz); 4 < z < 4 : (.6) The electric eld on the antenna surface is zero within our assumption of ideally conducting antenna except at the gap at the midpoint. However, the electric eld due to the current lament assumed at the axis is nite at the surface a distance a away. It can be calculated E r B r(r A) r A; (.63) where A is the vector potential on the surface. It can be found from the integral, where A z (z) 4 0 I 0 cos kz R(z; z 0 ) e jkr(z;z0) dz 0 ; (.64) R(z; z 0 ) p (z z 0 ) + a ; (.65) is the distance between a point on the surface ( a; z) and a current segment I(z 0 )dz 0 at z 0 : With this approximation, Eq. (.63) reduces to i! c E A + k A z ; (.66) since in current-free region the vector potential satis es the Helmholtz equation r + k A z 0: (.67) The integral in E z (z) j c 0 I 0 4! 4 cos(kz 0 e jkr(z;z 0 ) + k R(z; z 0 ) dz0 ; (.68) 0

21 can be performed by e jkr(z;z0 R(z; z 0 ) and by integrating by parts twice with the result where R s z E z (z) j 0I 0 e jkr(z;z0) R(z; z 0 ) ; (.69) e jkr R + a 4 ; R + e jkr ; (.70) R s z + + a 4 : (.7) The radial outward Poynting ux on the antenna surface is therefore given by S E z H j 0I0 e jkr 8 a R + e jkr cos(kz); (.7) R and the power leaving through the antenna surface is 4 P a j 0I S dz 4 e jkr R + e jkr cos(kz)dz: (.73) R The integral has to be performed numerically. Introducing x 4z ; A 4a ; we rewrite the integral in the form 0 h exp j p i ( x) + A p + ( x) + A h exp j p i ( + x) + A p A cos ( + x) + A x dx; (.74) which is shown in Fig..9 as a function of the normalized antenna radius A 4a: For A < 0:0; the radiation impedance is constant and approximately equal to rad ' 73: + j4:0 () which is inductive. The real part agrees with the radiation resistance calculated earlier. The reactive part of the impedance is inductive due to dominant magnetic energy compared with the electric capacitive energy. However, the reactance is a very sensitive function of the antenna length.

22 x Figure.9: Real (lower curve) and imaginary (upper curve) parts of the integral. It vanishes if the antenna length is chosen at l ' 0:49 and further decrease in the length makes the reactance capacitive. Radiation from a center-fed antenna can be analyzed by assuming a standing wave form, I(z) I 0 sin[k(l jzj)]; and is left for exercise.the axial electric eld E z (z) in Eq. (.70) can be alternatively (perhaps more conveniently) found from where is the retarded scalar potential given by A z; (.75) (z) e ijr r 0 j 4" 0 jr r 0 j r0 dv 0 e ir 4" 0 R l z 0 dz 0 ; (.76) with R q (z z 0 ) + a ; and l the linear charge density that can be found 0; l (z) i c I 0 sin kz; (C m ): (.77)

23 Then, ii 0 4" 0 c ii 0 4" 0 A 0 e jkr R e jkr R + e jkr R sin kz 0 dz 0 j! 0 I 0 4 j 0I 0 4 e jkr R e jkr R cos kz 0 R e jkr dz 0 ; (.78) which agrees with Eq. (.70). In radiation zone, the scalar potential is irrelevant but in near eld region, it should be considered together with the vector potential in a self consistent manner. The radiation impedance rad can be de ned by rad j 4 0 e jkr 4 4 R + e jkr R cos(kz)dz (.79) Remember that this is for a center-fed antenna. For an antenna of arbitrary length l with a current standing wave I (z) I m sin [k (l jzj)] (.80) the impedance is modi ed as rad j l 0 Im e jkr 4 I (0) j 0 4 sin (kl) l l l R e jkr R + e jkr R + e jkr R cos (kl) e jkr R cos (kl) e jkr R sin [k (l jzj)] dz sin [k (l jzj)] dz (.8) where R ; q (z l) + a ; R p z + a (.8) Note that the current seen by the generator is I (0) I m sin (kl) :.8 Radiation from Small Sources The retarded vector potential due to a nonrelativistic ( ; ' ) charged particle A (r; t) 0 ev(t 0 ) 4 jr r p (t 0 )j (.83) ret can be generalized for a collection of moving charges as A (r; t) J r 0 ; t 0 4 jr 3 jr r 0 j c r 0 dv 0 (.84) j

24 where J (r; t) is the current density. If the current is oscillating at!; we have A (r; t) 0 4r ej(!t kr) J r 0 e jkr0 dv 0 (.85) where the following approximation is used, e j! t jr r 0 j c ' e j(!t kr)+jkr0 Note that kjr r 0 j ' kr k r 0 ; r r 0 If the radiation source is small compared with the wavelength, that is, if!r 0 c, or kr 0 Eq. (.85) may be approximated by A (r; t) 0 4r ej(!t kr) J r 0 + jk r 0 k r0 dv 0 (.86) Note that in the limit of dc (or very low frequency) current (!! 0; k! 0), Eq. (.86) does reduce to the vector potential in magnetostatics. By assumption, jk r 0 j. As we will see, each term in this series expansion can be identi ed as electric and magnetic multipole radiation elds. The lowest order radiation vector potential is A (r; t) 0 4r ej(!t kr) J r 0 dv 0 (.87) The integral of the current density can be calculated as follows. The x component of the integral is J x dv rx JdV r (xj) dv xr JdV (.88) The rst integral vanishes, I r (xj) dv xj ds 0 because the closed surface S is at in nity where all sources vanish. Then J x dv S xr dv d dt p x where p x xdv 4

25 is the x component of the electric dipole moment. In general, JdV d dt p (.89) The term of next order J (r) (k r)dv (.90) can be calculated in a similar manner. J (r) (k r)dv r (k J) dv k r JdV and r (k J) dv dv (k r) JdV k d! Q (k r) JdV dt we nd J (r) (k r)dv k m + k d dt! Q (.9) where! Q rrdv (.9) is the quadrupole moment and m r JdV (.93) is the magnetic dipole moment. Therefore, to order kr 0 ; the vector potential is given by A (r;t) 0 4r ej(!t and the radiation magnetic eld H is kr) _p jk m + j k d dt! Q (.94) H 0 r A ' j 0 k A 4cr ej(!t kr) p n+ c n (n m) c n... n Q! (.95) where n r r is the radial unit vector. 5

26 Radiation by a Nonrelativistic Charge undergoing Circular Motion: Electric dipole radiation We assume a charge q is undergoing circular motion with a radius a and angular frequency!, as shown in Fig.. The dipole moment is p (t) qa (cos (!t) e x + sin (!t) e y ) Then the radiation magnetic eld is H (r; t) 4cr ej(!t! 4cr ej(!t kr) p n kr) n p and the radiation power is given by P 0 r jhj d 4" 0 3c 3!4 jpj 4" 0 3c 3 jpj (.96) The radiation magnetic eld may be written as H (r; t)! 4cr ej(!t kr) n p H (r; t) j aq! kr) ej(!t p (e j cos e ) (.97) 4cr The eld is plane polarized at and circularly polarized at 0: The radiation magnetic led due to an electric dipole is proportional to p; that is, acceleration of the charge. Radiation by a Small Current Loop: Magnetic dipole radiation If the loop radius is a (ka ) ; the magnetic dipole moment is m z a I 0 e j!t The radiation magnetic eld is H 4c r ej(!t kr) n (n m)! 4c r ej(!t kr) n (n m) 6

27 z x ωt a q y Figure.0: Charge in circular motion. The radiation power is The radiation resistance is P r 0 jhj d 0k 4 (4) a I 0 0 I0 (ka) 4 6 R rad 0 (ka) 4 6 sin d.9 Radiation of Angular Momentum The Poynting vector S E H (.98) is the energy ux density. Since the electromagnetic energy is carried at the velocity c; the momentum ux density may be de ned by and the momentum density by Similarly, the angular momentum ux density is given by c E H (.99) c E H (.00) r (E H) (.0) c 7

28 and the angular momentum density by c r (E H) " 0r (E B) " 0 r (E (r A)) (.0) The vector E (r A) can be expanded as E r A E i ra i (E r)a (.03) and we have However, r (E B) r (E i ra i ) r [(E r)a] (.04) r [(E r)a] r i (E i r A) E A (re) (r A) r i (E i r A) E A since re 0 in source free region. Then the total angular momentum is " 0 r (E B)dV " 0 r (E i ra i ) dv + " 0 (E A)dV (.05) where use is made of I r i (E i r A) dv r A(EdS) 0 It is evident that in Eq. (.05) the rst term containing the factor r can be identi ed as the orbital angular momentum. Then the last term can be interpreted as the spin angular momentum. Consider a circularly polarized plane wave propagating in the z direction. If the eld has positive helicty, the electric elds components are E x (z; t) E 0 cos (!t kz) E y (z; t) E 0 sin (!t kz) Corresponding vector potentials are A x (z; t) E 0! A y (z; t) + E 0! 8 sin (!t kz) cos (!t kz)

29 The spin momentum density is " 0 E A! " 0E 0e z (.06) For negative helicity wave, E x (z; t) E 0 cos (!t kz) E y (z; t) E 0 sin (!t kz) A x (z; t) A y (z; t) E 0 sin (!t! kz) E 0 cos (!t! kz) the spin direction is reversed, " 0 E A! " 0E 0e z (.07) as expected. Example: Radiation of angular momentum by an electric dipole Electric multipoles radiate Transverse Maganetic (TM) modes having no radial component of magnetic eld, H r 0: Then the angular momentum ux density becomes r (E H) c (r E) H c The radiation vector potential by an electric dipole is Corresponding magnetic eld is A 0 4r ej(!t kr) _p (.08) H 0 r A ' j e j(!t 4cr kr) ej(!t kr) 4r k _p n p (.09) where jk n j! c 9

30 The electric eld can be found from the Maxwell s equation " r B r (r A) r (r A) r A r (r A) + k A r A is r A d dr j k r e j(!t kr)! n _p r r e j(!t kr) n _p Then the radial component of the electric eld is The rate of angular momentum radiation is When applied to a charge undergoing circular motion, kr) E r ej(!t n _p (.0) " 0 cr r (r E) H d c 8" 0 c 3 (n _p) (n p) d (.) p x e cos (!t) ; p y e sin (!t) we nd n _p e! sin sin (!t) and e z (n p) e! sin sin (!t) Then dl dt e! 3 6" 0 c 3 e z P! e z (.) 30