Flow of Energy and Momentum inthenearzoneofahertziandipole

Transcription

1 1 Poblem Flow of Enegy and Momentum intheneazoneofahetziandipole Kik T. McDonald Joseph Heny Laboatoies, Pinceton Univesity, Pinceton, NJ Apil 11, 2007; updated May 13, 2014) Discuss the density and flow of enegy and momentum in the electomagnetic fields of an idealized Hetzian point) oscillating electic dipole. 1 2 Solution The electic and magnetic fields of an ideal, point Hetzian electic dipole of moment m cosωt can be witten in Gaussian units) as E = k 2 cosk ωt) cosk ωt) mˆ ˆm) ˆ + m3 ˆm ˆ)ˆ ˆm + cosk ωt) H = k 2 mˆ ˆm) k sink ωt), 1) sink ωt), 2) k whee ˆ = / is the unit vecto fom the cente of the dipole to the obseve, m = m ˆm is the peak electic dipole moment vecto, 2 ω =2πf is the angula fequency, k = ω/c =2π/λ is the wave numbe and c is the speed of light 1, 2. In a spheical coodinate system, θ, φ) fowhichthez-axis is along the diection of the dipole m and θ is the angle between m and, the fields ae E = k 2 m sin θ ˆθ cosk ωt) + m2 cos θ ˆ +sinθ ˆθ) cosk ωt) + k sink ωt), 3) H = k 2 m sin θ ˆφ cosk ωt) sink ωt), 4) k In the fa zone, λ, only the pats of these fields that vay as 1/ ae significant: E fa = k 2 m sin θ ˆθ cosk ωt), H fa = k 2 m sin θ ˆφ cosk ωt). 5) 1 This poblem is an extension of Some additional details, including discussion of the scala and vecto potentials of a Hetzian dipole in both the Coulomb and Loenz gauges, ae given in pob. 2 of Some consideation of Hetz vectos and scalas is given in the Appendix of 2 We eseve the symbol p fo the momentum density of the electomagnetic field. 1

2 These fields ae often called the adiation fields. Note that even in the nea zone, < λ, tems of the fom 5) ae pesent, and we say that the adiation fields exist in the nea zone as well as in the fa zone. Of couse, in the nea zone of the dipole the adiation fields ae smalle that the othe components of E and H. The most pominent featue of the fields in the nea zone is that the electic field looks a lot like that of an electostatic dipole, as shown in the figue below. Because field pattens that look like adiation ae discenable only fo > λ,theemaybe an impession that the adiation is ceated at some distance fom an antenna, athe than at the antenna itself. 2

3 2.1 Enegy Density Accoding to Maxwell the density of enegy stoed in the electomagnetic field in vacuum) is u = E2 + H 2. 6) 8π Fo the fields 3)-4), we find { u = m2 k 4 sin 2 θ + 3cos2 θ k 3 sin 2 θ k3 cos2 θ +1) 2 5 cos 2 k ωt) k2 sin 2k ωt)+ k2 cos 2 θ cos 2k ωt) 4. 7) The fist tem of eq. 7) is the enegy associated with the adiation fields, while the second tem is the electostatic enegy of the dipole m multiplied by the wave function cos 2 k ωt). The othe fou tems ae due to intefeence between the adiation fields and the quasistatic field of the oscillating dipole. Rewiting eq. 7) in tems of its Fouie components, we have { u = m2 k 4 sin 2 θ + 2k2 cos 2 θ + 3cos2 θ +1 8π 2 6 k 4 sin 2 θ + 2k2 + 3cos2 θ +1 cos 2k ωt) k 3 sin 2 θ k3 cos2 θ +1) sin 2k ωt). 8) 5 It is of inteest to ecod the total enegy density, 1 Ut) = u,t) dvol = 2π d d cos θu,t). 9) 0 1 The pat of the enegy density 8) associated with adiation vaies as 1/, so the integal of this enegy density is infinite since the foms 1)-2) tacitly assumed that the antenna has been adiating foeve ). Hence, we obtain a finite esult only if we estict ou attention to the enegy U nea in the nea zone of the antenna, which coesponds to the tems in eq. 8) that vay moe quickly than 1/. Futhemoe, we obtain a finite esult only if we conside the enegy in the egion outside some small, but finite, adius a λ. Thus, U nea = m2 { 1 2k 2 cos 2 θ d d cos θ + 3cos2 θ +1 4 a k2 + 3cos2 θ +1 cos 2k cos 2ωt +sin2k sin 2ωt) k 3 sin 2 θ k3 cos2 θ +1) sin 2k cos 2ωt cos 2k sin 2ωt) 5 = m2 { 4k 2 d 4 a k cos 2k cos 2ωt +sin2k sin 2ωt) 2 3

4 8k 3 3 4k sin 2k cos 2ωt cos 2k sin 2ωt) m a 1 + cos 2ωt) cos 2 ωt 3 =m2. 3a 3 10) The nea-field enegy oscillates in time, which implies thee is an oscillatoy exchange of enegy between the electomagnetic fields and the souce that dives the dipole antenna. 2.2 Enegy Flow Since the adiated powe comes fom the antenna fom the powe supply that dives the antenna), thee must be a flow of enegy out fom the antenna into the suounding space. The usual electodynamic measue of enegy flow is Poynting s vecto, S = c E H. 11) When we use the fields 1)-2) to calculate the Poynting vecto we find six tems, some of which do not point along the adial vecto ˆ: S = c { k 4 m 2 sin 2 θ ˆφ cosk ωt) cosk ωt)sink ωt) k k 2 m 2 2 cos θ ˆ +sinθ ˆθ) sin θ ˆφ cos 2 k ωt) sin 2 k ωt) = cm2 k +cosk ωt)sink ωt) 1 ) 3 k 5 { k sin 2 4 cos 2 k ωt) θ ˆ k2 cos 2k ωt) k3 sin 2k ωt) +sin2θ ˆθ k 2 cos 2k ωt) + k3 sin 2k ωt) k ) 2k 2 ). 12) Since the functions cos 2k ωt) andsin2k ωt) can be both positive and negative, pat of the enegy flow is inwads at times, athe than outwads as expected fo pue adiation. The pesence of an inwad flow of enegy einfoces the conclusion of the pevious section that thee is an oscillatoy exchange of enegy between souce and fields, athe than a simple flow of enegy fom the souce to the fields. Howeve, we obtain a simple esult if we conside only the time-aveaged Poynting vecto, S. Notingthat cos 2 k ωt) =1/2 and cos 2k ωt) = sin 2k ωt) = 0, eq 12) leads to S = ck4 m 2 sin 2 θ 8π ˆ. 13) The time-aveage Poynting vecto is puely adially outwads, and falls off as 1/ at all adii, as expected fo a flow of enegy that oiginates in the oscillating point dipole. The time-aveage angula distibution d P /dω of the adiated powe is elated to the Poynting vecto by d P dω = 2 ˆ S = ck4 p 2 sin 2 θ 8π 4 = p2 ω 4 sin 2 θ 8πc 3, 14)

5 which is the expession usually deived fo dipole adiation in the fa zone. Hee we see that this expession holds in the nea zone as well. We conclude that adiation, as measued by the time-aveaged Poynting vecto, exists in the nea zone of an antenna as well as in the fa zone. We veify that the Poynting vecto S coesponds to the flow of enegy such that the continuity equation, S + u =0, 15) t is satisfied. Fom eq. 8) we have, u t while fom eq.12) we find, = cm2 k 5 sin 2 θ 2k3 + k3 cos2 θ +1) sin 2k ωt) cm2 2k 4 sin 2 θ k2 3 cos 2 θ +1) cos 2k ωt), 16) 5 S = 1 S )+ 1 sin θ θ sin θs θ) { = cm2 sin 2 θ k5 sin 2k ωt) + 2k3 sin 2k ωt) + 2k2 cos 2k ωt) 5 2k4 cos 2k ωt) 1 1 ) + k3 sin 2k ωt) 1 3 ) 2k 2 2k 2 2k +3 cos 2 2 cos 2k ωt) θ 1) + k3 sin 2k ωt) 1 1 ) 5 k 2 = cm2 cm2 k5 sin 2 θ + 2k3 k3 cos2 θ +1) k 4 sin 2 θ k2 3 cos 2 θ +1) 5 sin 2k ωt) cos 2k ωt) = u t. 17) We note that the pat of the enegy density due only to the adiation fields, u ad = k4 m 2 sin 2 θ cos 2 k ωt), 18) and the pat of the Poynting vecto due only to the adiation fields, S ad = ck4 m 2 sin 2 θ cos 2 k ωt) ˆ, 19) obey the continuity equation S ad + u ad =0. 20) t We can identify pat of the enegy density as being due to the quasistatic electic dipole, u dipole = m2 3 cos 2 θ +1) 8π 6 cos 2 k ωt). 21) 5

6 Howeve, thee is no magnetic field, and hence no Poynting vecto, associated with the quasistatic electic dipole. Hence, consevation of enegy in the field of the quasistatic electic dipole can be accounted fo only when we conside the intefeence tems of the Poynting vecto that involve the quasistatic dipole fields and the so-called intemediate-zone fields of eqs. 3)-4) that vay as 1/. 2.3 Momentum Density As noted by Abaham 4, 3 the Poynting vecto plays the dual ole of descibing the enegy flow in the electomagnetic field as well as the density p of momentum stoed in the field, p = E H c = S c 2, 22) whee we estict ou attention to waves in vacuum. 4 Fom eq. 12) we obtain the momentum density of the fields of the oscillating dipole, { p = m2 k sin 2 4 cos 2 k ωt) θ ˆ c +sin2θ ˆθ k 2 cos 2k ωt) k2 cos 2k ωt) k3 sin 2k ωt) + k3 sin 2k ωt) k 2 The pat of the momentum density associated only with the adiation fields is 1 1 ) 2k 2 ). 23) p ad = S ad c 2 = u ad c ˆ = k4 m 2 sin 2 θ c cos 2 k ωt) ˆ. 24) 3 Momentum Flux If we know the velocity with which the momentum is flowing, we can identify a momentum flux tenso) Π as the poduct of the momentum density and its velocity. The adiation fields popagate adially with velocity c, so we identify Π ad, = cp ad, = u ad = k4 m 2 sin 2 θ cos 2 k ωt). 25) Fo a moe geneal consideation of momentum flux, we note that the time ate of change of momentum density, p/ t, has dimensions of a foce density. We ecall that Maxwell descibed the foces associated with electomagnetic fields in tems of the stess tenso T, whose components ae fo fields in vacuum) T ij = E ie j + H i H j uδ ij, 26) 3 Vaiants of this elation wee discussed ealie by Thomson 5, 6 and by Poincaé 7. 4 Fo waves in a dielectic medium one must also conside the momentum of the oscillating dipoles of that medium, which equies some cae 8, 9. 6

7 whee u is the enegy density 6). The volume foce f density associated with the fields is the divegence of the stess tenso, f = T, 27) and the equation of continuity fo electomagnetic field momentum is This leads to the intepetation of p t T =0. 28) Π = T = uδ ij E ie j + H i H j 29) as the momentum flux tenso. The six distinct momentum flux components fo the fields of the Hetzian dipole ae { Π = m2 k 4 sin 2 θ + 1 5cos2 θ cos 2 k ωt)+ k2 cos 2θ cos 2k ωt) 2 6 k 3 sin 2 θ k1 5cos2 θ) sin 2k ωt) k2 cos 2 θ, 30) 2 5 Π θ = Π θ = m2 sin 2θ k 2 sin 2k ωt) + cos2 k ωt), 31) 5 6 Π φ = Π φ = 0, Π θθ = m2 8π Π θφ = Π φθ = 0, Π φφ = m2 8π { 5cos 2 θ 1 cos 2 k ωt)+ k2 1 3cos 2 θ) 6 + k5 cos2 θ 1) sin 2k ωt)+ k2 1 + cos 2 θ) 5 { 3cos 2 θ +1 cos 2 k ωt) k2 1 + cos 2 θ) 6 + k3 cos2 θ +1) sin 2k ωt)+ k2 3 cos 2 θ 1) 5 cos 2k ωt) 32), 33) cos 2k ωt) 34). 35) The momentum flux due only to the adiation fields is the fist tem of Π, as anticipated in eq. 25). The emaining tems of the tenso Π descibe the somewhat nontivial flow of momentum in the nea zone of the antenna. Fo compaison, we ecod the momentum flux associated with a static electic dipole m. Then, the magnetic field vanishes and the electic field is simply 3 ˆm ˆ)ˆ ˆm E static = m The field enegy density is 36) 2cosθˆ +sinθ ˆθ = m. 37) u static = m 2 3cos2 θ +1 8π 6, 38) 7

8 and the components of the momentum-flux tenso ae 4 Wave Velocities 4.1 Phase/Wavefont Velocity Π static, = m 2 1 5cos2 θ, 8π 6 39) 2 sin 2θ Π static,θ = Π static,θ = m, 6 40) Π static,φ = Π static,φ = 0, 41) Π static,θθ = m 2 5cos2 θ 1, 8π 6 42) Π static,θφ = Π static,φθ = 0, 43) Π static,φφ = m 2 3cos2 θ +1. 8π 6 44) 45) Since the electomagnetic fields of a Hetzian dipole ae not plane waves we cannot simply speak of a phase velocity. But, we can conside the velocity of a wavefont, which is the essence of the phase velocity. Since the fields of an oscillating electic dipole ae tansvese magnetic, it is cleaest to conside wavefonts of the magnetic field, eq. 4), which vanish on spheical sufaces given by tank ωt) =k, i.e., t = c 1 ω tan 1 k + nπ ω. 46) We designate the adial velocity of these sufaces as the phase velocity, v p = d dt = 1 dt/d = c 1+ 1 ) >c. 47) k 2 In the nea zone this phase velocity exceeds the speed of light in vacuum, while it appoaches the speed of light in the fa zone. 4.2 Enegy/Goup Velocity As discussed in eq. 17) of 10, we take the goup velocity to be equal to the time-aveaged) enegy velocity, v g = S u = c <c, 48) 1+ cot2 θ + 3cos2 θ+1 k 2 2k 4 sin 2 θ ecalling eqs. 8) and 13). The goup velocity is less than the speed of light at all angles and all adii, and is extemely small along the diection of the dipole moment, whee the stength of the adiation is vey weak. It is also extemely small fo k 1, although one might question the meaning of the goup velocity on this tiny length scale. Note that v g v p c 2. 8

9 4.3 Instantaneous Enegy Flow Velocity An instantaneous velocity v of the flow of enegy in the electomagnetic field is sometimes taken to be the atio of the Poynting vecto to the enegy density, 5 Refeing to eqs. 18)-19) we see that fo lage, v = S u. 49) v = c ˆ k 1), 50) which seems quite satisfactoy. In contast, fo k 1, eqs. 8) and 12) imply and Then, u = m 2 16π 3 cos2 θ +1)cos 2 k ωt) k 1), 51) S = cm2 k 8π sin2 θ ˆ sin 2θ θ)sin2k ωt), k 1). 52) 4ck v = 3cos 2 θ +1 sin2 θ ˆ sin 2θ θ)tank ωt), v = v = S u = 4ck sin θ tank ωt) k 1). 53) 3cos2 θ +1 Fo times nea the end of fist and thid quates of evey cycle, v exceeds c close to the oscillating dipole. The physcial significance of the classical enegy flow velocity 49) is unclea except at lage. In a quantum view, the electomagnetic fields ae in geneal associated with vitual photons whose velocity can exceed c, 6 although photons can be said to have a definite velocity only once they have been obseved; a vaiable velocity v, θ, t) such as eq. 53) coesponds to the distibution of possible velocities in an ensemble of vitual photons. Refeences 1 H. Hetz, Die Käfte electische Schwingungen, behandelt nach de Maxwell schen Theoie, Ann. d. Phys. 36, ), 5 J.J. Thomson developed the notion of field momentum density 22) essentially accoding to p = S/c 2 = uv/c 2 5, 6. See also eq. 19), p. 79 of 11, and p. 6 of 12. The idea that an enegy flux vecto is the poduct of enegy density and enegy flow velocity seems to be due to Umov 13, based on Eule s continuity equation 14 fo mass flow, ρv) = ρ/ t. 6 Fo an example of a static electomagnetic field with enegy flow velocity 49) lage than c, see15. Fo a epesentation of electomagnetic fields in tems of electomagnetic plane waves with wavespeeds that can be othe than c, see16. 9

10 The Foces of Electical Oscillations Teated Accoding to Maxwell s Theoy, Natue 39, ), epinted in chap. 9 of H. Hetz, Electic Waves Macmillan, 1900), 2 Sec. 9.2 of J.D. Jackson, Classical Electodynamics, 3d ed. Wiley, New Yok, 1999). 3 J.H. Poynting, On the Tansfe of Enegy in the Electomagnetic Field, Phil. Tans. Roy. Soc. London 175, ), 4 M. Abaham, Pinzipien de Dynamik des Elektons, Ann. Phys. 10, ), 5 J.J. Thomson, On the Illustation of the Popeties of the Electic Field by Means of Tubes of Electostatic Induction, Phil. Mag. 31, ), 6 J.J. Thomson, Recent Reseaches in Electicity and Magnetism Claendon Pess, 1893), 7 H. Poincaé, Théoie de Loentz et le Pincipe de la Réaction, Ach. Neél.5, ), 8 R. Peiels, The momentum of light in a efacting medium, Poc. Roy. Soc. London A 347, ), 9 D.F. Nelson, Momentum, pseudomomentum and wave momentum: Towad esolving the Minkowski-Abaham contovesy, Phys. Rev. A 44, ), 10 K.T. McDonald, Radiation fom Hetzian Dipoles in a Uniaxial Anisotopic Medium Nov. 16, 2007), 11 O. Heaviside, Electomagnetic Theoy, Vol. 1 Electician Pinting Co., 1893), 12 H. Bateman, The Mathematical Analysis of Electical and Optical Wave Motion, Cambidge U. Pess, 1915), 13 N. Umow, Ableitung de Bewegungsgleichungen de Enegie in continuilichen Köpen, Zeit. Math. Phys. 19, ), L. Eule, Pincipes généaux du mouvement des fluides, Acad. Roy. Sci. Belles-Lett. Belin 1755), 10

11 15 K.T. McDonald, Momentum in a DC Cicuit May 26, 2006), 16 K.T. McDonald, Decomposition of Electomagnetic Fields into Electomagnetic Plane Waves July 11, 2010), 11