Why Doesn t a Steady Current Loop Radiate?

Transcription

1 Why Doesn t a Steady Current Loop Radiate? Probem Kirk T. McDonad Joseph Henry Laboratories, Princeton University, Princeton, NJ 8544 December, 2; updated March 22, 26 A steady current in a circuar oop presumaby invoves a arge number of eectrons in uniform circuar motion. Each eectron undergoes acceerated motion, and individua acceerated charges emit radiation. Yet, the current density J is independent of time in the imit of a continuous current distribution, and therefore does not radiate. How can we reconcie these two views?. Comments and Hints If the steady current were a continuous DC current, a of its mutipoe moments woud be constant, and we woud expect no radiation. For a current consisting of eectrons, it must be that the radiation is canceed by destructive interference between the radiation fieds of the arge number N of eectrons that make up the steady current. A singe eectron in uniform circuar motion emits eectric dipoe radiation, whose power is proportiona to the square of the acceeration a = v 2 /r, and hence to v/c 4. But, the eectric dipoe moment vanishes for two eectrons in uniform circuar motion at opposite ends of a common diameter; quadrupoe radiation is the highest mutipoe in this case, with power proportiona to v/c 6. It is suggestive that in case of 3 eectrons 2 apart in uniform circuar motion the time-dependent quadrupoe moment vanishes, and the highest mutipoe radiation is octupoe. For N eectrons eveny spaced around a ring, the highest mutipoe that radiates in the Nth, and the power of this radiation is proportiona to v/c 2N+2. Then, for steady motion with v/c, the radiated power of a ring of N eectrons is very sma. Verify this argument with a detaied cacuation. Reca the basic expression for the vector potentia of the radiation fieds, Ar,t= [J] c r dvo [J] dvo = Jr,t = t r/c dvo, where R is the arge distance from the observer to the center of the ring of radius a. For uniform circuar motion of N eectrons with anguar frequency ω, the current density J is a periodic function with period T =/ω, so a Fourier anaysis can be made where Jr,t = = J r e iωt, 2 This resut is impicit in the first-ever computation of eectromagnetic radiation, by FitzGerad 883 []. We can aso consider the decomposition of the eectromagnetic fieds into pane eectromagnetic waves photons, and note that a DC current has ony zero-frequency-wave components ony virtua photons, and hence no radiation of rea photons. See sec. 2. of [2].

2 with Then, J r = T Jr,t e iωt dt. 3 Ar,t= A re iωt, 4 etc. The radiated power foows from the Poynting vector [3], dp dω = c 4π R2 B 2 = c 4π R2 A 2. 5 One must be carefu in going from a Fourier anaysis of an ampitude, such as B, toafourier anaysis of an intensity that depends on the square of the ampitude. A Fourier anaysis of the average power radiated during one period T can be given as d P dω = T = 2 4π = 2 dp dω 2 dt = B 2 dt = 2 4πT 4πT T B B e iωt dt = 2 T 4π B 2 = = = B B B B e iωt dt dp dω. 6 That is, the Fourier components of the time-averaged radiated power can be written dp dω = 2 B 2 = 2 A 2 = 2 ikˆn A 2, 7 where k = ω/c and ˆn points from the center of the ring to the observer. Evauate the Fourier components of the vector potentia and of the radiated power first for a singe eectron, and then for N eectrons eveny spaced around the ring. It wi come as no surprise that a 3-dimensiona probem with charges distributed on a ring eads to Besse functions, and we must be aware of the integra representation J s = i e iφ is cos φ dφ. 8 Use the asymptotic expansion for arge index and sma argument, to verify the suppression of the radiation for arge N. J x ex/2, x, 9 This probem was first posed and soved via series expansions without expicit mention of Besse functions by J.J. Thomson [4]. He knew that atoms in what we now ca their ground state don t radiate, and used this cacuation to support his mode that the eectric 2

3 charge in an atom must be smoothy distributed. This was a cassica precursor to the view of a continuous probabiity distribution for the eectron s position in an atom. Thomson s work was foowed shorty by an extensive treatise by Schott [5], that incuded anayses in term of Besse functions correct for any vaue of v/c. 2 These pioneering works were argey forgotten during the foowing era of nonreativistic quantum mechanics, and were reinvented around 945 when interest emerged in reativistic partice acceerators. See Arzimovitch and Pomeranchuk [7], and Schwinger [8]. 2 Soution The soution given here foows the succinct treatment by Landau, sec. 74 of [9]. For charges in steady motion at anguar frequency ω in a ring of radius a, the current density J is periodic with period T =/ω, so the Fourier anaysis 2 at the retarded time t can be evauated via the usua approximation that r R r ˆn, wherer is the distance from the center of the ring to the observer, r points from the center of the ring to the eectron, and ˆn is the unit vector pointing from the center of the ring to the observer. Then, [J] = Jr,t = t r/c = J r e iωt R/c+r ˆn/c = e ikr ωt J k r e ikωr ˆn/c, where k = ω/c. The ring ies in the pane z =, centered on the origin. We use rectanguar coordinates x, y, z, cyindrica coordinates ρ, φ, z, and spherica coordinates r, θ, φ with ange θ measured with respect to the +z axis. Then, r = ρcos φ ˆx +sinφ ŷ, ˆn =sinθ ˆx +cosθ ẑ, and ˆφ = sin φ ˆx +cosφ ŷ. We first consider a singe eectron, whose azimuth varies as φ = ωt + φ,andwhose veocity is, of course, v = aω. The current density of a point eectron of charge q can be written in a cyindrica coordinate system with voume eement ρdρ dφ dz using Dirac deta functions as J = ρ charge v ˆφ = qvδρ aδzδφ ωt φ ˆφ. 2 The Fourier components J are given by J = T Jr,te it dt = qvδρ aδz eiφ φ ρωt ˆφ. 3 Using eqs. and 3 in and noting that ωt =, we find [J] = qv e ikr ωt e iφ φ ωρ sin θ cos φ/c δρ aδzˆφ. 4 ρ 2 See aso [6]. 3

4 Inserting this in eq., we have A [J]ρ dρdφdz= = A e iωt, qv e ikr ωt φ e iφ ωa sin θ cos φ/c ˆφ dφ 5 so that the Fourier components of the vector potentia are A = qv eikr φ e iφ v sin θ cos φ/c sin φ ˆx +cosφ ŷ dφ. 6 The integras yied Besse functions with the aid of the integra representation 8. The ŷ part of eq. 6 can be found by taking the derivative of this reation with respect to s: J s = i+ For the ˆx part of eq. 6 we pay the trick = = e iφ s cos φ dφ z cos φ e iφ is cos φ cos φdφ, 7 e iφ is cos φ dφ + s e iφ is cos φ sin φdφ, 8 so that e iφ is cos φ sin φdφ= e iφ is cos φ dφ = s i s J s. 9 Using eqs. 7 and 9 with s = v sin θ/c in 6 we have A = qv eikr φ i v sin θ/c J v sin θ/c ˆx i J + v sin θ/c ŷ. 2 We skip the cacuation of the eectric and magnetic fieds from the vector potentia, and proceed immediatey to the anguar distribution of the radiated power according to eq. 7, dp dω = 2 ikˆn A 2 = ck2 2 R 2 cos 2 θ A,x 2 + A,y 2 = ck2 2 R 2 = cq2 k 2 2 ˆn A 2 cot 2 θj 2 v sin θ/c+v2 c J 2 2 v sin θ/c. 2 The present interest in this resut is for v/c, but in fact it hods for any vaue of v/c. As such, it can be used for a detaied discussion of the radiation from a reativistic eectron that moves in a circe, which emits so-caed synchrotron radiation. This topic is discussed further in Lecture 2 of the Notes []. 3 Furthermore, eq. 2 hods even if the veocity v 3 If the eectron of mass m moves in a circe due to static magnetic fied B, then the anguar veocity is given by ω = kc = v/a = qb/γmc, such that k 2 =q 2 B 2 /m 2 c 4 v 2 /c 2, and eq. 2 agrees with eq of [9], noting that our θ is π/2 θ there. 4

5 exceeds the speed of ight c/n in a medium of index of refraction n, inwhichcaseakind of synchroton-čerenkov radiation is emitted []. Since each ampitude A varies as /R at arge distance R from the source, the tota radiated power varies as /R 2. 4 We now turn to the case of N eectrons uniformy spaced around the ring. The initia azimuth of the nth eectron can be written φ n = n N. 22 The th Fourier component of the tota vector potentia is simpy the sum of components 2 inserting φ n in pace of φ : A = N qv n= eikr φ n i v sin θ/c J v sin θ/cˆx i J + v sin θ/cŷ N = qveikr i m v sin θ/c J v sin θ/cˆx i J v sin θ/cŷ + e in/n. n= This sum vanishes uness is a mutipe of N, in which case the sum is just N. Theowest nonvanishing Fourier component has order N, and the radiation is at frequency Nω. We recognize this as Nth-order mutipoe radiation, whose radiated power foows from eq. 2 as dp N dω = cq2 k 2 N 2 cot 2 θjn 2 Nvsin θ/c+v2 c J 2 2 N Nvsin θ/c. 24 For arge N but v/c we can use the asymptotic expansion 9, and its derivative, to write eq. 24 as 23 J x ex/2 x,x, 25 dp N dω cq2 k 2 N e v 2N 4π 2 sin 2 θ 2 c sin θ + cos 2 θ N dp E dω N,v/c. 26 In eqs. 25 and 26 the symbo e inside the parentheses is not the charge but rather the base of natura ogarithms, Forcurrentsin,say,aoopofcopperwire,v cm/s, so v/c, whie N 23. The radiated power predicted by eq. 26 is extraordinariy sma! Note, however, that this neary compete destructive interference depends on the eectrons being uniformy distributed around the ring. Suppose instead that they were distributed with random azimuths φ n. Then the square of the magnetic fied at order m has the form B m 2 N 2 e imφ n n= = N + n e imφ φ n = N This is in contrast to caims [2] that the power varies as /R for Čerenkov radiation emitted by a partice in uniform circuar motion. 5

6 Thus, for random azimuths the power radiated by N eectrons at any order is just N times that radiated by one eectron. If the charge carriers in a wire were ocaized to distances much smaer than their separation, radiation of steady currents coud occur. However, in the quantum view of metaic conduction, such ocaization does not occur. The random-phase approximation is reevant for eectrons in a so-caed storage ring, for which the radiated power is a major oss of energy or source of desirabe photon beams of synchrotron radiation, depending on one s point of view. We do not expound here on the interesting topic of the formation ength for radiation by reativistic eectrons, which ength sets the scae for interference of mutipe eectrons. See, for exampe, [3]. 2. Comment An interesting comment by Lai [4] is that the radiation terms in the Lienard-Wiechert expressions for the eectric and magnetic fieds of an acceerated charge sec. 63 of [9] for the case of uniform circuar motion incudes a piece corresponding the fieds of a ring of charge in steady circuar motion. That is, the interference of the radiation fieds of a steady oop of current is not competey destructive, even though no radiation survives. 5 References [] K.T. McDonad, FitzGerad s Cacuation of the Radiation of an Osciating Magnetic Dipoe June 2, 2, [2] K.T. McDonad, Decomposition of Eectromagnetic Fieds into Eectromagnetic Pane Wave Juy, 2, [3] J.H. Poynting, On the Transfer of Energy in the Eectromagnetic Fied, Phi. Trans. Roy. Soc. London 75, , [4] J.J. Thomson, The Magnetic Properties of Systems of Corpusces describing Circuar Orbits, Phi. Mag. 36, , [5] G.A. Schott, Eectromagnetic Radiation Cambridge U.P., 92, [6] R. Rivera and D. Viarroe, An exact soution for severa charges in cassica eectrodynamics, J. Math. Phys. 4, , 5 The Poynting vector of a steady oop of moving charge if not eectricay neutra forms azimutha oops with the same sense as the current. In the view that any nonzero Poynting vector shoud be termed radiation [5], a steady oop of current does support radiation athough this does not carry energy away from the current oop. 6

7 [7] L. Arzimovitch and I. Pomeranchuk, The Radiation of Fast Eectrons in the Magnetic Fied, J. Phys. USSR 9, , [8] J. Schwinger, On Radiation by Eectrons in a Betatron 945, On the Cassica Radiation of Acceerated Eectrons, Phys. Rev. 75, , [9] L.D. Landau and E.M. Lifshitz, Cassica Theory of Fieds, 4th ed. Butterworth- Heinemann, Oxford, 987. [] K.T. McDonad, Reativistic Radiation Effects, Lecture 2, Physics 26/5, [] T.M. Rynne, G.B. Baumgartner, and T. Erber, The anguar distribution of synchrotron- Čerenkov radiation, J. App. Phys. 49, , [2] A. Ardavan et a., Experimenta observation of nonsphericay-decaying radiation from a rotating superumina source, J. App. Phys. 96, , [3] M.S. Zootorev and K.T. McDonad, Cassica Radiation Processes in the Weizsäcker- Wiiams Approximation, Aug. 25, 999, [4] H.M. Lai, Static contribution from the radiation fied, Am. J. Phys. 46, , [5] K.T. McDonad, On the Definition of Radiation by a System of Charges, Sept. 6, 2, 7