Radiation in the Near Zone of a Hertzian Dipole

Transcription

1 Radiation in the Nea Zone of a Hetzian Dipole 1 Poblem Kik T. McDonald Joseph Heny Laboatoies, Pinceton Univesity, Pinceton, NJ (Apil 22, 2004) Calculate the Poynting vecto of the fields of a Hetzian oscillating dipole at all points in space. Show that the time-aveaged Poynting vecto has the same fom in the nea zone as it does in the fa zone, which confims that adiation exists both close to and fa fom the souce. 1 2 Solution 2.1 Hetzian Electic Dipole The electic and magnetic fields of an ideal, point Hetzian electic dipole p can be witten (in Gaussian units) as whose eal pats ae E = k 2 p(ˆ ˆp) ˆ ei(kωt) + p3(ˆp ˆ)ˆ ˆp ik ) e i(kωt), (1) 3 2 B = k 2 p(ˆ ˆp) 1 ) e i(kωt), (2) ik 2 E = k 2 cos(k ωt) cos(k ωt) p(ˆ ˆp) ˆ + p3(ˆp ˆ)ˆ ˆp + 3 cos(k ωt) B = k 2 p(ˆ ˆp) k sin(k ωt), (3) 2 sin(k ωt), (4) k 2 whee ˆ = / is the unit vecto fom the cente of the dipole to the obseve, p = p cos ωt ˆp is the electic dipole moment vecto, ω =2πf is the angula fequency, k = ω/c =2π/λ is the wave numbe and c is the speed of light 1, 2. We say that the adiation pat of these fields ae the tems that vay as 1/: E ad = k 2 cos(k ωt) p(ˆ ˆp) ˆ, (5) 1 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 1

2 B ad = k 2 cos(k ωt) p(ˆ ˆp). (6) In the nea zone of the dipole, whee k < 1, the adiation fields ae smalle that the othe components of E and B. 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. 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 suface of the antenna into the suounding space. The usual electodynamic measue of enegy flow is Poynting s vecto 3 (in a medium with unit elative pemeability), S = c E B. (7) 2

3 When we use the fields (3)-(4) to calculate the Poynting vecto we find six tems, some of which do not point along the adial vecto ˆ: S = c { cos k 4 p 2 (k ωt) cos(k ωt)sin(k ωt) (ˆ ˆp) ˆ (ˆ ˆp) 2 k 3 cos +k 2 p 2 (k ωt) sin 2 (k ωt) 3(ˆp ˆ)ˆ ˆp (ˆ ˆp) +cos(k ωt)sin(k ωt) = c { cos k 4 p 2 sin 2 2 (k ωt) θ ˆ 2 +k 2 p 2 (3 cos 2 θ 1) ˆ 2cosθ ˆp +cos(k ωt)sin(k ωt) ( k 3 1 k 5 4 )} cos(k ωt)sin(k ωt) k 3 cos 2 (k ωt) sin 2 (k ωt) 4 ( k 1 )}, (8) 3 k 5 whee θ is the angle between vectos and p. As well as the expected adial flow of enegy, thee is a flow in the diection of the dipole moment p. Since the poduct cos(kωt)sin(k ωt) can be both positive and negative, pat of the enegy flow is inwads at times, athe than outwads as expected fo pue adiation. Howeve, we obtain a simple esult if we conside only the time-aveaged Poynting vecto, S. Notingthat cos 2 (k ωt) = sin 2 (k ωt) =1/2 and cos(k ωt)sin(k ωt) = (1/2) sin 2(k ωt) = 0, eq (8) leads to S = ck4 p 2 sin 2 θ ˆ. (9) 8π 2 The time-aveage Poynting vecto is puely adially outwads, and falls off as 1/ 2 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 = p2 ω 4 sin 2 θ, (10) 8π 8πc 3 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. The question sometimes aises as to whethe the fields of an antenna could be pue adiation, with no nonpopagating nea fields (that take enegy to maintain). It is clea that a small oscillating dipole will geneate neaby, quasistatic dipole fields. See sec. 6 of 4 fo a kind of invese agument, that any dipole adiation fields in the fa zone imply, via a diffaction integal, that nonadiation fields ae pesent in the nea zone. 3

4 2.2 Hetzian Magnetic Dipole IfE(,t) and B(,t) ae solutions to Maxwell sequations infee space (i.e., whee the chage density ρ and cuent density J ae zeo), then the dual fields E (,t)=b(,t), B (,t)=e(,t), (11) ae solutions thee also. The Poynting vecto is the same fo the dual fields as fo the oiginal fields, 2,3 S = c E B = c B E = S. (12) Taking the dual of fields (3)-(4) with the change of notation p m, we find the fields E = k 2 m(ˆ ˆm) 1 ) e i(kωt), (13) ik 2 B = k 2 m(ˆ ˆm) ˆ ei(kωt) whose eal pats ae cos(k ωt) E = k 2 m(ˆ ˆm) B = k 2 m(ˆ ˆm) ˆ + m3( ˆm ˆ)ˆ ˆm cos(k ωt) cos(k ωt) +m3( ˆm ˆ)ˆ ˆm + 3 ik ) e i(kωt), (14) 3 2 sin(k ωt), (15) k 2 k sin(k ωt), (16) 2 which ae also solutions to Maxwell s equations. These ae the fields of an oscillating point magnetic dipole, whose peak magnetic moment is m. In the nea zone, the magnetic field (16) looks like that of a (magnetic) dipole. While the fields of eqs. (3)-(4) ae not identical to those of eqs. (15)-(16), the Poynting vectos ae the same in the two cases. Hence, the time-aveage Poynting vecto, and also the angula distibution of the time-aveaged adiated powe ae the same in the two cases. The adiation of a point electic dipole is the same as that of a point magnetic dipole (assuming that m = p), both in the nea and in the fa zones. Measuements of only the intensity of the adiation could not distinguish the two cases. Howeve, if measuements wee made of both the electic and magnetic fields, then the nea zone fields of an oscillating electic dipole, eqs. (3)-(4), would be found to be quite diffeent fom those of a magnetic dipole, eqs. (15)-(16). This is illustated in the figue on p. 5, which plots the atio E/H = E/B of the magnitudes of the electic and magnetic fields as a function of the distance fom the cente of the dipoles. To distinguish between the cases of electic and magnetic dipole adiation, it suffices to measue only the polaization (i.e., the diection, but not the magnitude) of eithe the electic of the magnetic field vectos. 2 The enegy adiated by a Hetzian magnetic dipole was fist calculated by Fitzgeald in , pio to Hetz calculations fo electic dipoles. Appaently Fitzgeald consideed atoms to involve chages oscillating in cicles, and hence he thought that atoms would emit magnetic dipole adiation. He seems not to have noticed that such classical atoms would be unstable due to this adiation. See, fo example, 7. 3 An application of the concept of dual field to macoscopic, plana antennas is given in 6. 4

5 A Appendix: Self-Dual Antennas Heaviside intoduced the concept of dual electomagnetic fields in the 1880 s while also consideing magnetic chages and cuents, ρ e and J m, that ae conceptually the dual of electic chages and cuents, ρ e and J e. In geneal, if fields E and B ae geneated by electic chages and cuents ρ e and J e, then the dual fields E and B of eq. (11) ae geneated by the dual magnetic chages ρ m = ρ e and J m = J e. Since it appeas that thee ae no magnetic chages and cuents in Natue, it will not be possible in geneal to geneate the dual fields E and B. Howeve, if only magnetic dipoles (o highe multipoles) ae needed to geneate the dual fields, it may be possible to do so with appopiate configuations of electic cuents. Accoding to the definition (11) the dual fields cannot equal the oiginal fields. Howeve, it is possible that the dual fields ae equal to the oiginal fields to within a phase facto ±i, in which case the fields ae said to be self dual. Thatis, E = B = αe, B = E = αb, equies that α 2 = 1, (17) such that E = ±ib, B = ie, (self dual). (18) Pehaps the most familia examples of self-dual electomagnetic fields ae ciculaly polaized plane waves, such as E = E 0 (ˆx ± i ŷ) e i(kzωt), B = ˆx E = E 0 (ŷ i ˆx) e i(kzωt) = ie. (19) A set of self-dual fields (E sd, B sd ) can be fomally geneated fom any solution (E, B) to Maxwell s equations accoding to E sd = E ie = E ± ib, B sd = B ib = B ie = ie sd. (20) Of couse, in geneal it will not be possible physically geneate these fields with only electic chages and cuents. Howeve, we can make a self-dual antenna by combining a Hetzian 5

6 electic dipole antenna of moment p with a Hetzian magnetic dipole antenna of moment m = ±ip. In this case, the fa-zone electic field of waves emitted in diection has ˆ has diection E sd ˆ (ˆ ˆp) ± i ˆp =(ˆ ˆp)ˆ ˆp ± i ˆ ˆp. Thus,whenˆ is pependicula to ˆp, the electic field has diection ˆp ± i ˆ ˆp, so that the adiation emitted in this diection is ciculaly polaized (and theefoe self dual as noted in eq. (19)). Refeences 1 H. Hetz, The Foces of Electical Oscillations Teated Accoding to Maxwell s Theoy, Weidemann s Ann. 36, 1 (1889), Natue 39, 402 (1889); 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, 343 (1884), 4 K.T. McDonald Time-Revesed Diffaction (Sep. 5, 1999), 5 G.F. Fitzgeald, On the Quantity of Enegy Tansfeed to the Ethe by a Vaiable Cuent, Tans.Roy.DublinSoc.3 (1883), On the Enegy Lost by Radiation fom Altenating Electic Cuents, Bit. Assoc. Rep. 175, 343 (1883), 6 H.G. Booke, Slot Aeials and Thei Relation to Complementay Wie Aeials (Babinte s Pinciple), J. IEE 93, 620 (1946), 7 J.D. Olsen and K.T. McDonald, Classical Lifetime of a Boh Atom (Ma. 7, 2005), 6