170 20 Maxwell’s Equations II: Electromagnetic Waves 20.5 Hertz’s Oscillating Dipole (Electric Dipole Radiation, Mobile Phones) It can be readily shown that the electromagnetic field generated by a time- dependent dipole p(t)atr =0,viz B(r,t)= μ 0 4π curl ˙ p t − r c r E(r,t)= 1 4πε 0 curl curl p t − r c r , (20.14) for t ∈ (−∞, ∞), solves everywhere all four Maxwell equations. (If B(r,t) obeys Maxwell’s equations, then so does E(r,t)andvice versa, in accordance with (20.14).) A practical example of such a time-dependent electric dipole (so-called Hertz dipole) is a radio or mobile-phone antenna, driven by an alternating current of frequency ω. Since the current is given explicitly by I(t) ≡| ˙ p(t)| , the retarded vector potential has the following form, which is explicitly used in equation (20.14): A(r,t)= μ 0 4πr ˙ p t − r c . In particular, the asymptotic behavior of the fields in both the near-field and far-field range can be derived without difficulty, as follows: a) In the near-field range (for r λ,whereλ is the wavelength of light in a vacuum, corresponding to the frequency ω, i.e., λ = ω 2πc )onecan approximate p t − r c r by p(t) r . After some elementary transformation, using the identity curl curlv ≡ (grad div −∇ 2 )v and the spherical wave equation given previously, one then obtains: E(r,t) ∼ = 3(p(t) · r)r − r 2 p(t) 4πε 0 r 5 , (20.15) which is the quasi-static result E(r,t) ∝ r −3 for r → 0 with which we are already familiar, while simultaneously B(r,t) ∼ = μ 0 4πr 3 ˙p(t) ×r , i.e. , ∝ r −2 for r → 0 , and is thus less strongly divergent (i.e., asymptotically negligible w.r.t. E(r,t)) for r → 0. Here the position-dependence of the denominator dominates. 20.6 Magnetic Dipole Radiation; Synchrotron Radiation 171 b) In the far-field range (for r λ), the position-dependence of the numer- ator dominates, e.g., ∇×p t − r c ∼ = − 1 c ˆr × ˙ p t − r c r , with ˆr = r/r . Here, for r λ, one obtains asymptotically: E(r,t) ∼ = 1 4πε 0 c 2 r ˆr × ˆr × ¨ p t − r c and B(r,t) ∼ = − μ 0 4πcr ˆr × ¨ p t − r c , (20.16) i.e., S = E × H ∼ = ˆr (sin θ) 2 16π 2 ε 0 c 3 r 2 ¨ p t − r c 2 , (20.17) where θ is the angle between ¨p and ˆr. As for planar waves, the propagation vector ˆr := r/r and the vectors E and cB form a right-handed rectangular trihedron, where in addition the vector E in the far-field range lies asymptotically in the plane defined by ¨ p and r. In the far-field range the amplitudes of both E and B are ∝ ω 2 r −1 sin θ; the Poynting vector is therefore ∝ ω 4 r −2 (sin θ) 2 ˆr.Thepowerintegrated across the surface of a sphere is thus ∝ ω 4 , independent of the radius! Due to the ω 4 -dependence of the power, electromagnetic radiation beyond a limiting frequency range is biologically dangerous (e.g. X-rays), whereas low-frequency radiation is biologically harmless. The limiting frequency range (hopefully) seems to be beyond the frequency range of present-day mobile phones, which transmit in the region of 10 9 Hz. 20.6 Magnetic Dipole Radiation; Synchrotron Radiation For vanishing charges and currents Maxwell’s equations possess a symmetry which is analogous to that of the canonical equations of classical mechanics (˙q = − ∂H ∂p ;˙p = ∂H ∂q , see Part I), called symplectic invariance:thesetof equations does not change, if E is transformed into cB,andB into −c −1 E (E,cB) → 0, +1 −1, 0 E cB . To be more precise, if in equation (20.14) B is transformed into −D and E is transformed into H, while simultaneously ε 0 and μ 0 are interchanged 172 20 Maxwell’s Equations II: Electromagnetic Waves and the electric dipole moment is replaced by a magnetic one (corresponding to B = μ 0 H + J, D = ε 0 E + P ), one obtains the electromagnetic field produced by a time-dependent magnetic dipole. For the Poynting vector in the far-field range one thus obtains instead of (20.17): S = E × H ∼ = ˆr (sin θ m ) 2 16π 2 μ 0 c 3 r 2 ¨ m t − r c 2 , (20.18) where θ m is again the angle between ¨m and ˆr. We shall now show that magnetic dipole radiation for particles with non- relativistic velocities is much smaller – by a factor ∼ v 2 c 2 –thanelectric dipole radiation. Consider an electron moving with constant angular velocity ω in the xy−plane in a circular orbit of radius R. The related electric dipole mo- ment is p(t)=eR · (cos(ωt)sin(ωt), 0) . On average the amplitude of this electric dipole moment is p 0 = eR.The corresponding magnetic dipole moment is (on average) m 0 = μ 0 πR 2 eω 2π , where we have again used – as with the calculation of the gyromagnetic ratio –therelation I = eω 2π . With ω = v R we thus obtain m 0 = 1 2 evR . Therefore, as long as p(t)andm(t) oscillate in their respective amplitudes p 0 and m 0 with identical frequency ω as in cos ωt, we would get on average the following ratio of the amplitudes of the respective Poynting vectors: 4 |S m | |S e | = m 2 0 ε 0 p 2 0 μ 0 = v 2 4c 2 . (20.19) Charged relativistic particles in a circular orbit are sources of intense, po- larized radiation over a vast frequency range of the electromagnetic spectrum, e.g., from the infrared region up to soft X-rays. In a synchrotron, electrons travelling at almost the speed of light are forced by magnets to move in 4 The factor 4 (= 2 2 ) in the denominator of this equation results essentially from the fact that the above formula p(T )=eR ·(cos(ωt), sin(ωt)) can be interpreted as follows: there are in effect two electric dipoles (but only one magnetic dipole) involved. 20.7 General Multipole Radiation 173 a circular orbit. The continual acceleration of these charged particles in their circular orbit causes high energy radiation to be emitted tangentially to the path. To enhance the effectivity the electrons usually travel through special structures embedded in the orbit such as wigglers or undulators. Synchrotron radiation is utilized for all kinds of physical and biophysical research at various dedicated sites throughout the world. 20.7 General Multipole Radiation The results of this section follow directly from (20.14) for electric dipole ra- diation (Hertz dipole) and the corresponding equations for magnetic dipole radiation (see Sect. 20.6). We recall that a quadrupole is obtained by a limit- ing procedure involving the difference between two exactly opposite dipoles, one of which is shifted with respect to the other by a vector b(= b 2 );anoc- tupole is obtained by a similar shift (with b(= b 3 )) from two exactly opposite quadrupoles, etc. As a consequence, the electromagnetic field of electric octupole radiation, for example, is obtained by application of the differential operator (b 3 ·∇)(b 2 ·∇) on the electromagnetic field of a Hertz dipole: a) In the near-field range, i.e., for r λ, one thus obtains the following quasi-static result for the E-field of an electric 2 l -pole: E ∼ = −gradφ(r,t) , with φ(r,t) ∼ = 1 4πε 0 r l+1 l m=−l c l,m (t)Y l,m (θ, φ) , (20.20) where Y l,m are spherical harmonics. The coefficients of this expansion depend on the vectors b 1 , b 2 , ,b l , andontime. b) In the far-field range, i.e., for r λ, one can approximate the expressions (b i ·∇) f t − r c r by 5 1 r (b l · ˆr) −∂ c∂t f t − r c and one obtains for the electric 2 l -pole radiation asymptotically: 5 It should be noted that the vector f (t − r c )isalwaysperpendiculartoˆr,cf. (20.16). 174 20 Maxwell’s Equations II: Electromagnetic Waves 1) E ∝ 1 r · l ν=2 (b ν · ˆ r) · −∂ c∂t l ˆr × p t − r c , (20.21) 2) cB ∼ = ˆr × E and 3) S ≡ E × H , i.e., for p(t)=p 0 cos ωt on average w.r.t. time: S(r) ∝ ω 2(l+1) r 2 l+1 ν=2 (b ν · ˆr) 2 c 2 |ˆr × p 0 | 2 ˆr . (20.22) Here we have l =1, 2, 3 for dipole, quadrupole and octupole radiation. The dependence on the distance r is thus universal, i.e., S ∝ r −2 for all l; the dependence on the frequency is simple (S ∝ ω 2(l+1) ); only the angular dependence of electric multipole radiation is complicated. However, in this case too, the three vectors ˆr, E and cB form a right-handed rectangular trihedron, similar to e x , e y , e z . Details can be found in (20.21). 6 For magnetic multipole radiation the results are similar. 20.8 Relativistic Invariance of Electrodynamics We have already seen in Part I that classical mechanics had to be amended as a result of Einstein’s theory of special relativity. In contrast, Maxwell’s theory is already relativistically invariant per se and requires no modification (see below). It does no harm to repeat here (see also Section 9.1) that prior to Ein- stein’s theory of relativity (1905) it was believed that a special inertial frame existed, the so-called aether or world aether, in which Maxwell’s equations had their usual form, and, in particular, where the velocity of light in vacuo had the value c ≡ 1 √ ε 0 μ 0 , whereas in other inertial frames, according to the Newtonian (or Galilean) additive behavior of velocities, the value would be different (e.g., c → c+v). In their well known experiments, Michelson and Morley attempted to measure the motion of the earth relative to the aether and thus tried to verify this behavior. Instead, they found (with great precision): c → c. 6 Gravitational waves obey the same theory with l = 2. Specifically, in the distri- bution of gravitational charges there are no dipoles, but only quadrupoles etc. 20.8 Relativistic Invariance of Electrodynamics 175 In fact, Hendryk A. Lorentz from Leiden had already established before Einstein that Maxwell’s equations, which are not invariant under a Galilean transformation, are invariant w.r.t. a Lorentz transformation –asitwaslater called – in which space and time coordinates are “mixed” (see Section 9.1). Furthermore, the result of Michelson and Morley’s experiments follows nat- urally from the Lorentz transformations. However, Lorentz interpreted his results only as a strange mathematical property of Maxwell’s equations and (in contrast to Einstein) not as a scientific revolution with respect to our basic assumptions about spacetime underlying all physical events. The relativistic invariance of Maxwell’s equations can be demonstrated most clearly in terms of the Minkowski four-vectors introduced in Part I. a) The essential point is that in addition to ˜x := (x, y, z, ict) , the following two quadruplets, 1) ˜ A := A, i Φ c and , 2) ˜ j := (j, ic) are Minkowski four-vectors, whereas other quantities, such as the d’Alem- bert operator, are Minkowski scalars (which are invariant), and the fields themselves, E plus B (six components) correspond to a skew symmetric tensor F μ,ν (= −F ν,μ ) generated from ˜ A, viz F μ,ν := ∂A ν ∂x μ − ∂A μ ∂x ν (e.g., F 1,2 = −F 2,1 = B 3 ), with x 1 := x, x 2 := y, x 3 := z,andx 4 := ict. b) One defines the “Minkowski nabla” ˜ ∇ := ∂ ∂x , ∂ ∂y , ∂ ∂z , ∂ ∂ict = ∇, ∂ ∂ict . Similarly to Euclidian space R 3 , where the Laplace operator ∇ 2 (≡ Δ) is invariant (i.e., does not change its form) under rotations,inMinkowski space M 4 the d’Alembert operator ˜ ∇ 2 (≡ ) is invariant under pseudo- rotations. In addition, similar to the fact that the divergence of a vector field has a coordinate-invariant meaning with respect to rotations in R 3 , analogous results also apply for the Minkowski divergence, i.e. one has an invariant meaning of ˜ ∇·˜v with respect to pseudo-rotations in M 4 . 176 20 Maxwell’s Equations II: Electromagnetic Waves c) For example, the continuity equation, divj + ∂ ∂t =0, has a simple invariant relativistic form (which we shall use below): ˜ ∇· ˜ j := 4 ν=1 ∂j ν ∂x ν =0. (20.23) d) Analogously, gauge transformations of the kind A → A +gradg(r,t),Φ→ Φ − ∂g(r,t) ∂t can be combined to ˜ A(˜x) → ˜ A(˜x)+ ˜ ∇g(˜x) . We are now prepared for the explicit Minkowski formulation of Maxwell’s equations. As mentioned, the homogeneous equations II and III, divB =0; curlE = − ∂B ∂t are automatically satisfied by introducing the above skew symmetric field tensor F μν := ∂ μ A ν − ∂ ν A ν , with μ, ν =1, ,4; which is analogous to the representation of E and B by a scalar potential plus a vector potential. The remaining inhomogeneous Maxwell equations I and IV, divE ≡ /ε 0 and curlB ≡ μ 0 j , simply yield the following result, with Einstein’s summation convention 7 : ∂ μ F νμ = ∂ μ ∂ ν A μ − ∂ μ ∂ μ A ν ≡ μ 0 j ν . With the Lorentz gauge, divA + 1 c 2 ∂φ ∂t = ∂ μ A μ =0, 8 the first term on the l.h.s. vanishes, i.e., we again obtain the d’Alembert- Poisson equation − ˜ ∇ 2 ˜ A ≡ μ 0 ˜ j. 7 Using the Einstein convention one avoids clumsy summation symbols: If an index appears twice, it is summed over. 8 Here we again use the permutability of partial derivatives. 20.8 Relativistic Invariance of Electrodynamics 177 From the “simple” Lorentz transformations for the x-andt-components of the Minkowski four-potential ˜ A, for a transition between different inertial frames the following “more complicated” Lorentz transformations for elec- tromagnetic fields result: The longitudinal components E x and B x remain unchanged, whereas one obtains for the transverse components: B ⊥ (r,t)= B ⊥ (r ,t )+ v c 2 × E (r ,t ) 1 − v 2 c 2 , (20.24) E ⊥ (r,t)= E ⊥ (r ,t ) − v × B (r ,t ) 1 − v 2 c 2 . (20.25) These results can be used to obtain the E-andB-fields of a moving point charge from the Coulomb E -field in the co-moving frame. 9 9 See the exercises at http://www.physik.uni-regensburg.de/forschung/krey, sum- mer 2002, file 9. 21 Applications of Electrodynamics in the Field of Optics 21.1 Introduction: Wave Equations; Group and Phase Velocity Firstly we shall remind ourselves of the relationship between the frequency ν = 2π ω and wavelength λ = 2π k (k =wavenumber) of an electromagnetic wave in vacuo: ω =2πν = c · k = c · 2π λ , or λ · ν = c. Secondly, electromagnetic waves cover an extremely wide spectral range. For example, radio waves have wavelengths from 1 km or more (long-wave) via 300 m (medium-wave) to about 50 m (short-wave). This range is followed by VHF (very high frequency), then the range of television and mobile-phone frequencies from ≈ 100MHz to 10 GHz; then we have radar, light waves, X- rays and, at very short wavelengths or high frequencies, γ-rays. It is useful to remember that the wavelengths of visible light range from λ ≈ 8000 ˚ A (or 800 nm, red )downto≈ 4000 ˚ A (400 nm, violet) 1 On the lower frequency side of the visible range come infrared and far-infrared,andonthe high-frequency side ultraviolet and soft-X-ray radiation. Thirdly, in connection with X-rays and γ-radiation, it is useful to re- member that these phenomena arise from quantum transitions, see Part III (Fermi’s “golden rules”), according to the formula ΔE ≡ E i − E f ≡ hν , i.e. by transitions from a higher initial energy E i to a lower final energy E f . The radiation may have a continuous distribution of frequencies (so-called bremsstrahlung,orbraking radiation), or it may contain a discrete set of spectral “lines”. The quantity h is Planck’s constant: h =6.625 ·10 −34 Ws 2 ≡ 4.136 · 10 −15 eVs . 1 Some readers may prefer the characteristic atomic length 1 ˚ A, whereas others use units such as 1 nm (≡ 10 ˚ A). Which one is more appropriate, depends on the problem, on the method used, and on personal preferences. 180 21 Applications of Electrodynamics in the Field of Optics X-rays have typical energies of ΔE ≈ 10 keVto ≈ 1 MeV, characteristic for the electron shell of atoms, whereas for γ-radiation one is dealing with excitations of nuclei, i.e., ΔE ≈ 1MeVup to1GeV. Fourthly, Planck’s formula for black-body radiation: The total energy of the electromagnetic field contained in a volume V at Kelvin temperature T is given by U(T ) ≡ V ∞ 0 dνu(ν, T) , with the spectral energy density u(ν, T)= 8πν 2 c 3 hν e hν k b T − 1 . (21.1) For the surface temperature of the sun, i.e., for T ≈ 6000 K, the function u(ν, T) has a pronounced maximum in the green range, i.e., for ν = c λ with λ ≈ 6000 ˚ A(= 600 nm) . See Fig. 21.1: The vacuum velocity of electromagnetic waves (e.g., light) is c 0 := (ε 0 μ 0 ) − 1 2 . In polarizable matter, the (stationary) velocity of electromagnetic waves is smaller: c m = c 0 n , Fig. 21.1. Planck’s black-body radiation formula. For the reduced frequency f (≡ hν/(k B T ) in the text) Planck’s function P (f):=f 3 /(exp f − 1) is shown as a double- logarithmic plot. It has a pronounced maxi- mum around f ≈ 2 . that Maxwell’s equations, which are not invariant under a Galilean transformation, are invariant w.r.t. a Lorentz transformation –asitwaslater called – in which space and time coordinates are. Details can be found in (20.21). 6 For magnetic multipole radiation the results are similar. 20.8 Relativistic Invariance of Electrodynamics We have already seen in Part I that classical mechanics. 10 9 Hz. 20.6 Magnetic Dipole Radiation; Synchrotron Radiation For vanishing charges and currents Maxwell’s equations possess a symmetry which is analogous to that of the canonical equations of classical