160 19 Maxwell’s Equations I: Faraday’s and Maxwell’s Laws where i 2 = −1, we thus have U G (t)=Re(U G e iωt ) and I(t)=Re(Je iωt ) , where U≡U (0) G and J≡(I (0) e −iα ) . By analogy with Ohm’s law we then define the complex quantity R,where U G = R·I. The quantity R is the complex a.c. resistance or simply impedance. The total impedance of a circuit is calculated from an appropriate com- bination of three types of standard elements in series or parallel, etc. 1. Ohmic resistances (positive and real) are represented by the well- known rectangular symbol and the letter R. The corresponding com- plex resistance is R R = R. 2. Capacitive resistances (negatively imaginary) correspond to a pair of capacitor plates, together with the letter C. The corresponding impedance is given by R C = 1 iωC . (A short justification: U C (t)= Q(t) C , i.e., ˙ U C (t)= I(t) C .Thuswiththe ansatz U C (t) ∝ e iωt one obtains ˙ U C (t) ≡ iωU C (t)). 3. Inductive resistances (positively imaginary) are represented by a solenoid symbol, together with the letter L. The corresponding impedance is R L =iωL . (The induced voltage drop in the load results from building-up the magnetic field, according to the relation U L (t)=L· dI(t) dt , i.e., U L (t)= L · ˙ I(t). But with the ansatz I(t) ∝ e iωt we obtain ˙ I(t) ≡ iωI(t).) One can use the same methods for mutual inductances (i.e., transformers; see exercises) 5 . 5 The input (load) voltage of the transformer is given by the relation U (1) Tr = iωL 1,2 ·J 2 , while the output (generator) voltage is given by U (2) Tr = −iωL 2,1 ·J 1 . 19.4 Applications: Complex Resistances etc. 161 d) An a.c. resonance circuit The following is well-known as example of resonance phenomena.Forase- ries RLC circuit connected as a load to an alternating-voltage generator U G (t)=U (0) G ·cos(ωt) , one has J U = 1 R = R +i ωL − 1 ωC −1 . Thus we obtain I(t)=I (0) · cos(ωt − α) , with I (0) U (0) G = 1 R = 1 R 2 +(ωL − 1 ωC ) 2 and tan α = ωL − 1 ωC R . For sufficiently small R (see below) this yields a sharp resonance at the resonance frequency ω 0 := 1 √ L · C . For this frequency the current and the voltage are exactly in phase, whereas for higher frequencies the current is delayed with respect to the voltage (inductive behavior) while for lower frequencies the voltage is de- layed with respect to the current (capacitive behavior). At the resonance frequency ω 0 the current has a very sharp maximum of height U (0) G /R, and for weak damping (i.e., for sufficiently small values of R) it decays very quickly as a function of ω, for very small deviations from ω = ω 0 ; i.e., for ω ± := ω 0 ± ε, where ε = R 2L is |ω 0 | , the current has already decreased to 70% of the maximum (more precisely: from 1×I (0) down to 1 √ 2 × I (0) ). The ratio Q := ω 0 R/L is called the quality factor of the resonance; it characterizes the sharpness of the phenomenon. In fact, Q often reaches values of the order of 10 3 or more. (Here the reader could try solving exercises 11 and 12 (file 6) from the summer term of 2002, which can be found on the internet, [2]. This can simultaneously serve as an introduction to MAPLE. In fact, it may be helpful to illustrate resonance phenomena using mathematical computer tools such as MAPLE or MATHEMATICA. See for example [12] as a rec- ommendable presentation.) 162 19 Maxwell’s Equations I: Faraday’s and Maxwell’s Laws Now consider the power loss in an a.c. circuit. By forming the derivative of the energy, we may write (dE/dt)(t)=U G (t) · I(t)=U (0) G · I (0) cos ωt · cos(ωt − α) ≡ U (0) G ·I (0) · cos α · (cos ωt) 2 +sinα ·(cos ωt ·sin ωt) . Averaged over a complete cycle the first term gives U (0) G · I (0) · 1 2 ·cos α. This is the resistive part, and represents the energy dissipated. The second term, however, vanishes when averaged over a complete cycle, and is called the reactive part. Using complex quantities one must explicitly take into account the factor 1 2 . The resistive part may be written { (dE/dt)(t)} =Re 1 2 U G J ∗ , while the reactive part (which vanishes on average) is given by Im 1 2 U G J ∗ . Further details on alternating-current theory can be found in many standard textbooks on applied electromagnetism. 20 Maxwell’s Equations II: Electromagnetic Waves 20.1 The Electromagnetic Energy Theorem; Poynting Vector The Poynting vector whichisdefinedas S := E ×H (20.1) has the meaning of “energy current density”: S ≡ j energy . Firstly we have the mathematical identity div[E × H]=H ·curlE −E ·curlH , which can be proved using the relation div[E × H]=∂ i e i,j,k E j H k = . With the Maxwell equations curlE = − ∂B ∂t and curlH = j + ∂D ∂t one then obtains the continuity equation corresponding to the conservation of field energy: div[E × H]+E · j + E · ∂D ∂t + H · ∂B ∂t ≡ 0 , (20.2) i.e., divj energy + ∂w energy ∂t = −j · E , (20.3) where ∂w energy ∂t := E · ∂D ∂t + H · ∂B ∂t 164 20 Maxwell’s Equations II: Electromagnetic Waves is the (formal) 1 time-derivative of the energy density and −j · E describes the Joule losses, i.e., one expects j · E ≥ 0 . (The losses are essentially sources of heat production, since the Ohmic be- havior, j = σ ·E (where σ is the specific conductivity) arises due to frictional processes leading to energy dissipation and heat production due to scattering of the carriers of the current, e.g., by impurities. The case of vanishing losses is called ballistic.) For Ohmic behavior one has j = 1 κ E , where κ = 1 σ is the specific resistivity 2 , a constant property of the Ohmic material; i.e., −j · E = − E 2 κ . In the absence of electric currents the electromagnetic field energy is con- served. If the material considered shows Ohmic behavior, the field energy decreases due to Joule losses. At this point we shall just mention two further aspects: (i) the role of the Poynting vector for a battery with attached Ohmic resistance. The Poynting vector flows radially out of the battery into the vacuum and from there into the Ohmic load. Thus the wire from the battery to the Ohmic resistance is not involved at all, and (ii) Drude’s theory of electric conductivity. This theory culminates in the well-known formula j(ω)=σ(ω)E(ω) , where the alternating-current specific conductivity is given by σ(ω) ≡ σ(0)/(1 + iωτ) , with σ(0) = n V e 2 τ/m e . Here n V is the volume density of the carriers, i.e., typically electrons; m e is the electron mass, e the electron charge, and τ is a phenomenological relaxation time corresponding to scattering processes. 1 In the case of a linear relation (e.g., between E and D or B and H) the derivative is non-formal; otherwise (e.g., for the general relation D = ε 0 E +P )thisisonly formally a time-derivative. 2 The Ohmic resistance R of a wire of length l and cross-section F made from material of specific resistivity κ is thus R = κ ·( l F ). 20.2 Retarded Scalar and Vector Potentials I: D’Alembert’s Equation 165 20.2 Retarded Scalar and Vector Potentials I: D’Alembert’s Equation For given electromagnetic fields E(r,t)andB(r,t) one can satisfy the second and third Maxwell equations, i.e., Gauss’s magnetic law, divB =0,and Faraday’s law of induction, curlE ≡− ∂B ∂t , with the ansatz B(r,t) ≡ curlA(r,t)andE(r,t) ≡−gradφ(r,t) − ∂A(r,t) ∂t . (20.4) The scalar potential φ(r,t) and vector potential A(r,t)mustnowbe calculated simultaneously. However, they are not unique but can be “gauged” (i.e., changed according to a gauge transformation without any change of the fields) as follows: A(r,t) → A (r,t):=A (r,t) + gradf(r,t), φ(r,t) → φ (r,t):=φ (r,t) − ∂f(r,t) ∂t . (20.5) Here the gauge function f(r,t) in (20.5) is arbitrary (it must only be differentiable). (The proof that such gauge transformations neither change E(r,t)norB(r,t) is again based on the fact that differentiations can be permuted, e.g., ∂ ∂t ∂f ∂x = ∂ ∂x ∂f ∂t .) In the following we use this “gauge freedom” by choosing the so-called Lorentz gauge : divA + 1 c 2 ∂φ ∂t ≡ 0 . (20.6) After a short calculation, see below, one obtains from the two remaining Maxwell equations (I and IV), divD = and curlH = j + ∂D ∂t , the so-called d’Alembert-Poisson equations: − ∇ 2 − ∂ 2 c 2 ∂t 2 φ(r,t)= E (r,t)/ε 0 and − ∇ 2 − ∂ 2 c 2 ∂t 2 A(r,t)=μ 0 j B (r,t) . (20.7) c is the velocity of light in vacuo, while E and j B are the effective charge and current density, respectively. These deviate from the true charge and true current density by polarization contributions: E (r,t):=(r,t) − divP (rt) (20.8) and j B (r,t):=j(r,t)+ J(r,t) μ 0 + ∂P ∂t . (20.9) 166 20 Maxwell’s Equations II: Electromagnetic Waves A derivation of the two d’Alembert-Poisson equations now follows. a) divD = ,→ ε 0 divE = − divP =: E ; →−∇ 2 φ − div ∂A ∂t = E /ε 0 . With div ∂A ∂t = ∂ ∂t divA and with the Lorentz gauge (20.6) we obtain the first d’Alembert-Poisson equation. b) curlH = j + ∂D ∂t → curlB −curlJ = μ 0 · j + ε 0 ∂E ∂t + ∂P ∂t ; → curl curlA = μ 0 j +curl J μ 0 + ∂P ∂t + μ 0 ε 0 ∂E ∂t . One now inserts curl curlA ≡ grad(divA) −∇ 2 A and E = −gradφ − ∂A ∂t and obtains with ε 0 μ 0 = 1 c 2 the gradient of an expression that vanishes in the Lorentz gauge. The remaining terms yield the second d’Alembert-Poisson equation. In the next section (as for the harmonic oscillator in Part I) we discuss “free” and “fundamental” solutions of the d’Alembert equations, i.e., with vanishing r.h.s. of the equation (and ∝ δ(r)). Of special importance among these solutions are planar electromagnetic waves and spherical waves. 20.3 Planar Electromagnetic Waves; Spherical Waves The operator in the d’Alembert-Poisson equations (20.7) := ∇ 2 − ∂ 2 c 2 ∂t 2 (20.10) is called the d’Alembert operator or “quabla” operator. Amongst the general solutions of the free d’Alembert equation φ(r,t) ≡ 0 20.3 Planar Electromagnetic Waves; Spherical Waves 167 (also known simply as the wave equation) are right-moving planar waves of the kind φ + (r,t) ≡ g(x −ct) . Here g(x) is a general function, defined on the whole x-interval, which must be continuously differentiable twice. g(x) describes the profile of the traveling wave, which moves to the right here (positive x-direction) with a velocity c. A wave traveling to the left is described by φ − (r,t) ≡ g(x + ct) . Choosing x as the direction of propagation is of course arbitrary; in general we could replace x by k ·r,where ˆ k := k/|k| is the direction of propagation of the planar wave. All these relations can be evaluated directly from Maxwell’s equations. In particular, it is necessary to look for the polarization direction, especially for the so-called transversality. The first two Maxwell equations, divE ≡ 0 and divB ≡ 0 , imply (if the fields depend only on x and t) that the x-components E x and B x must be constant (i.e., ! = 0, without lack of generality). Thus, only the equations curlE = − ∂B ∂t and curlB ≡ μ 0 ε 0 ∂E ∂t remain (i.e., ≡ 1 c 2 ∂E ∂t ), which can be satisfied by E ≡ g(x − ct)e y and B ≡ c −1 g(x − ct)e z , with one and the same arbitrary profile function g(x). (For an electromagnetic wave traveling to the left one obtains analogously E ≡ g(x + ct)e y and B ≡−c −1 g(x + ct)e z .) For electromagnetic waves traveling to the right (or left, respectively) the propagation direction ˆ k and the vectors E and cB thus form a right-handed rectangular trihedron (in both cases!), analogous to the three vectors ±e x , e y and ±e z . In particular, the amplitude functions of E and cB (in the cgs system: those of E and B )arealwaysidentical. The densities of the electromagnetic field energy are also identical: w E = ε 0 2 E 2 ≡ w B := B 2 2μ 0 . The Poynting vector S := [E ×H] 168 20 Maxwell’s Equations II: Electromagnetic Waves is related to the total field-energy density of the wave, w total := w E + w B , as follows: S ≡ c ˆ k · w total , as expected. Spherical waves traveling outwards, Φ + (r, t):= g(t − r c ) r , (20.11) for r>0 are also solutions of the free d’Alembert equation. This can easily be seen from the identity ∇ 2 f(r)=r −1 d 2 (r · f (r)) dr 2 . If the singular behavior of the function 1/r at r = 0 is again taken into account by exclusion of a small sphere around the singularity, one obtains from the standard definition divv := lim V →0 1 V ∂V v ·nd 2 A for sufficiently reasonable behavior of the double-derivative ¨g(t) the following identity: g(t − r c ) r ≡ ∇ 2 − 1 c 2 ∂ 2 ∂t 2 g(t − r c ) r ≡−4πδ(r)g t − r c . (20.12) This corresponds to the analogous equation in electrostatics: ∇ 2 1 r ≡−4πδ(r) . As a consequence we keep in mind that spherical waves traveling outwards, Φ + (r,t):= g t − r c r , are so-called fundamental solutions of the d’Alembert-Poisson equations. (The corresponding incoming spherical waves, Φ − (r, t):= g ( t+ r c ) r ,are also fundamental solutions, but in general they are non-physical unless one is dealing with very special initial conditions, e.g., with a pellet bombarded from all sides by intense laser irradiation, which is performed in order to initiate a thermonuclear fusion reaction.) 20.4 Retarded Scalar and Vector Potentials II 169 20.4 Retarded Scalar and Vector Potentials II: The Superposition Principle with Retardation With equation (20.12) we are now in a position to write down the explicit solutions of the d’Alembert-Poisson equations (20.7), viz, φ(r,t)= dV E r ,t− |r−r | c 4πε 0 |r −r | , A(r,t)= dV μ 0 j B r ,t− |r−r | c 4π|r −r | . (20.13) In principle these rigorous results are very clear. For example, they tell us that the fields of single charges and currents a) on the one hand, can be simply superimposed, as in the static case with Coulomb’s law, while b) on the other hand, the retardation between cause and effect has to be taken into account, i.e., instead of t = t (instantaneous reaction) one has to write t = t − |r −r | c , i.e., the reaction is retarded, since electromagnetic signals between r and r propagate with the velocity c. c) Huygens’s principle 3 of the superposition of spherical waves is also con- tained explicitly and quantitatively in equations (20.13). d) According to the rigorous result (20.13) the mutual influences propagate at the vacuum light velocity, even in polarizable matter. (This does not contradict the fact that stationary electromagnetic waves in dielectric and permeable matter propagate with a reduced velocity (c 2 → c 2 /(εμ)). As with the driven harmonic oscillator (see Part I) these stationary waves develop only after a finite transient time. The calculation of the tran- sition is one of the fundamental problems solved by Sommerfeld, who was Heisenberg’s supervisor.) The explicit material properties enter the retarded potentials (20.13) only through the deviation between the “true charges” (and “true currents”) and the corresponding “effective charges” (and “effective currents”). One should remember that in the rigorous equations (20.13) the effective quantities enter. Only in vacuo do they agree with the true quantities. 3 See section on optics . special importance among these solutions are planar electromagnetic waves and spherical waves. 20.3 Planar Electromagnetic Waves; Spherical Waves The operator in the d’Alembert-Poisson equations. can simultaneously serve as an introduction to MAPLE. In fact, it may be helpful to illustrate resonance phenomena using mathematical computer tools such as MAPLE or MATHEMATICA. See for example. divB =0,and Faraday’s law of induction, curlE ≡− ∂B ∂t , with the ansatz B(r,t) ≡ curlA(r,t)andE(r,t) ≡−gradφ(r,t) − A( r,t) ∂t . (20.4) The scalar potential φ(r,t) and vector potential A( r,t)mustnowbe calculated