150 18 Magnetic Field of Steady Electric Currents b) In the dipole case one would obtain outside the (fictitious) dipole film the equivalent result B = μ 0 H , with H(r)=−grad I 4π F d 2 A n(r ) ·(r − r ) |r − r | 3 . (18.10) Proof of the equivalence of the two results proceeds analogously to Stokes’s theorem, but since it is somewhat difficult in detail, we only give an outline in a footnote 3 . An example is given in Fig. 18.1. In this context we additionally keep two useful identities in mind: A = m × r 4πr 3 Fig. 18.1. The diagram illustrates a typical section of the magnetic field lines produced by a current loop of two (infinitely) long straight wires. The wires intersect the diagram at the points (±1, 0). The plane of the loop of area A (→∞)and carrying a current I (i.e., of opposite signs in the two long wires) is perpendicular to the plane of the diagram. Exactly the same induction B(= μ 0 H) is also produced by a layer of magnetic dipoles inserted into the current loop, with the quantitative relation, dm ≡ μ 0 Ind 2 A, given in the text 3 In the following we use the antisymmetric unit-tensor e ijk and Ein- stein’s summing convention, i.e., all indices which appear twofold are summed over. With these conventions Stokes’s theorem becomes: H ∂F E j dx j = RR F e jlm ∂ l E m n j d 2 A. Now the following chain of equations is true: H ∂F e ijk dx j (x k −x k ) |r−r | 3 “ ≡ H ∂F dx j e jki ∂ k 1 |r−r | ” = RR F e jlm ∂ l e mki ∂ k 1 |r−r | n j d 2 A = − RR F e ikm e jlm ∂ lk 1 |r−r | n j d 2 A . With the basic identity e ikm e jlm = δ ij δ kl −δ il δ kj and the simple relations ∂ i 1 |r−r | = −∂ i 1 |r−r | and ∂ kk 1 |r−r | =0(forr = r) our statement of equivalence is obtained. 18.5 Gyromagnetic Ratio and Spin Magnetism 151 for the vector potential of a magnetic dipole and curl m × r r 3 = mdiv r r 3 − grad m ·r r 3 (cf. problem 5 of the exercises, summer 2002 [2]). Because of the above-mentioned equivalence it would be natural to suggest that all magnetic dipole moments are generated in this way by Amp`erian current loops. However, this suggestion would be wrong: There are magnetic moments which cannot be generated in this “classical” way, but which are related to the non-classical concept of “electron spin” (see Part III: Quantum Mechanics). The following section deals with the difference. 18.5 Gyromagnetic Ratio and Spin Magnetism An atomic electron orbiting the nucleus on a circular path of radius R with velocity v = ωR time period T = 2π ω has an angular momentum of magnitude L = R ·m e v = m e ωR 2 . AccordingtotheAmp`erian “current loop” picture it would be equivalent to a magnetic dipole moment m = μ 0 e T πR 2 = μ 0 eωR 2 2 , wherewehaveusedI = e T (m e is the electron mass). For a current loop, therefore, the gyromagnetic ratio γ := m L is given by γ ≡ μ 0 e 2m e . However, in the nineteen-twenties due to an experiment by Einstein and de Haas it was shown that for the usual magnetic materials, e.g., alloys of Fe, Co and Ni, the gyromagnetic ratio is twice as large as the above ratio. For these materials the magnetism is due almost entirely to pure spin magnetism. For the angular momentum of these alloys the “classical” orbital contribution (see Part I) is almost negligible; the (dominant!) contribution is essentially “non-classical”, i.e., due to spin magnetism, which is only understandable in a quantum mechanical context. (In fact, a profound analysis is not even 152 18 Magnetic Field of Steady Electric Currents possible in non-relativistic quantum mechanics, but only in Dirac’s relativistic version.) In elementary texts one often reads that the spin angular momentum of a (charged) particle is some kind of “proper angular momentum”, this being acceptable if one does not consider the particle to be rotating like a “spinning top”, since a spinning charge would have the classical value, γ = μ 0 2m e , for the gyromagnetic ratio and not twice this value. One has to admit that these relations are complicated and not understandable at an elementary level. 19 Maxwell’s Equations I: Faraday’s Law of Induction; the Continuity Equation; Maxwell’s Displacement Current Maxwell’s first and second equations, divD = and divB = 0 (i.e., Gauss’s law for the electric and the (non-existent) magnetic charges, respectively) also apply without change for time-dependent electrodynamic fields. This is different with respect to the third and fourth Maxwell equations: a) Faraday’s law of induction (Faraday 1832) curlE = − ∂B ∂t , (19.1) and b) Amp`ere’s law including Maxwell’s displacement current: curlH = j + ∂D ∂t . (19.2) These two equations, (19.1) and (19.2), will be discussed in the following subsections. To aid our understanding of the last term in (19.2), known as Maxwell’s displacement current, we shall include a subsection on the conti- nuity equation. This general equation contains an important conservation law within it, the conservation of total charge (see below). 19.1 Faraday’s Law of Induction and the Lorentz Force; Generator Voltage In 1832 Faraday observed that a time-dependent change of magnetic flux φ B (F )= F B · nd 2 A through a current loop Γ = ∂F gives rise to an electromotive force (i.e., a force by which electric charges of different sign are separated). This corresponds to a generator voltage which is similar to the off-load voltage between the two poles of a battery. (In the interior of a battery the current flows from the minus pole to the plus pole; only subsequently, in the external load circuit, does the current flow from plus to minus.) In Fig. 19.1 we present a sketch of the situation. 154 19 Maxwell’s Equations I: Faraday’s and M axwell’s Laws Fig. 19.1. Sketch to illustrate Faraday’s law of induction. The three arrows on the r.h.s. of the figure denote a magnetic induction B.AnorientedloopΓ is plotted, as w ell as a paved surface F , which is inserted into Γ (Γ = ∂F) but which does not need to be p lanar as in the diagram. A change in magnetic flux φ B (F ):= RR F B ·nd 2 A gives rise to an induced voltage U i (Γ ) = r 1−ε H r 1+ε E ·dr ! between two infinitesimally close points r 1+ε and r 1−ε on the loop. These tw o points – which are formally the initial and end points of the loop – can serve as the poles of a voltage generator (the initial point r 1+ε corresponds to the negative pole). The related quantitative equation is Faraday’s law: U i (t)=− dφ B (F ) dt Faraday’s law of induction states that the induced voltage (i.e., the gen- erator voltage mentioned above) U i = r 1−ε r 1+ε E(r) ·dr between (arbitrary) initial points r 1+ε and (almost) identical end points r 1−ε of an (almost) closed line 1 Γ = ∂F obeys the following law: U i (t)=− dφ B (F ) dt . (19.3) As already mentioned, the initial and end points of the (almost) closed loop Γ differ only infinitesimally. They correspond to the minus and plus poles of the generator, i.e., U i is the generator voltage. It does not matter at which position of the curve Γ the voltage is “tapped”, nor does it matter whether the change of the magnetic flux re- sults a) from a change of Γ (i.e., form or size) relative to the measuring equipment, b) from a change of the magnetic induction B(r,t), or c) by a combination of both effects. 1 A sketch is recommended. The normal vector n of the area F should coincide with the orientation of the loop Γ = ∂F. 19.1 Faraday’s Law of Induction and the Lorentz Force 155 We thus realize that Faraday’s law of induction (19.3), which appears so simple, embodies a great deal of experimental information. In addition we may already surmise at this point that Maxwell’s theory is relativistically invariant, as we shall show later in detail. For case b), i.e., constant Γ but variable B, using Stokes’s theorem, from the integral form of the law of induction, ∂F E ·dr = − F ∂B ∂t · nd 2 A, one obtains the differential form (19.1). To derive case a) as well, we additionally use the expression for the Lorentz force on an electrically charged particle, which moves with velocity v in a magnetic field. This law was formulated by Hendryk A. Lorentz in Lei- den decades after Faraday’s discovery 2 , and it states the following: The force F L on an electrically charged particle moving in a magnetic induction B(r,t) with velocity v(r,t)isF L = qv × B (Lorentz force) . We shall now consider an infinitesimal line element dr (e.g., dle y )ofthe current loop Γ = ∂F and move this line element during a time interval δt with velocity v (e.g. in the x-direction), such that the line element defines a surface element (vδt) ×dr . The corresponding change in magnetic flux is given by δφ B = δt[v × dr] ·B = −δtdr ·[v × B] . We then calculate the integral of this force along the loop, i.e., r 1−ε r 1+ε dr · F L . But F L = q · E . Therefore, we finally obtain U i ≡ E · dr ≡− δφ B δt . Case c) follows from b) and a) by linear superposition and the product law of differentiation. Consequently, we can state that the integral version of Faraday’s law contains more than the differential version (19.1); inter alia the Lorentz force, i.e., not only b), but also a). 2 Shortly before Einstein’s special theory of relativity Lorentz also formulated his famous Lorentz transformations. 156 19 Maxwell’s Equations I: Faraday’s and Maxwell’s Laws 19.2 The Continuity Equation The conservation law for total charge mentioned above, states that Q total (t):= ∞ dV(r) (t)=constant, where we integrate over the whole space R 3 . This implies that the electric charge contained in a finite volume V , Q(V ):= V dV(r) , can only increase by an influx of charge from outside, i.e., dQ(V ) dt = − ∂V j · nd 2 A. (19.4) Now for constant volume we have: dQ(V ) dt = V ∂ ∂t dV, and by Gauss’s integral theorem: − ∂V j · nd 2 A = − V divjdV. As a consequence we have V ∂ ∂t +divj dV ≡ 0 , i.e. not divj ≡ 0, but instead the continuity equation divj + ∂ ∂t ≡ 0 . (19.5) This equation, including its derivation, should be kept in mind, since ana- logous continuity equations apply to other conservation laws. 19.3 Amp`ere’s Law with Maxwell’s Displacement Current As a result of the continuity equation, (19.5), Amp`ere’s law curlH = j 19.3 Amp`ere’s Law with Maxwell’s Displacement Current 157 cannot remain unchanged for time-dependent electromagnetic fields, since div curlH ≡ 0 , whereas according to (19.5) divj = 0 applies. Maxwell found the correct solution by adding a displacement current ∂D ∂t to the true current j,togive: curlH = j + ∂D ∂t . Due to the permutability of partial derivatives, e.g., ∂ ∂x ∂ ∂t = ∂ ∂t ∂ ∂x , one obtains div ∂D ∂t = ∂ ∂t divD . But according to the first Maxwell equation one has divD = .Thus,with the continuity equation one obtains: div j + ∂D ∂t ≡ 0 , as expected. The fourth Maxwell equation was therefore deduced to be: curlH = j + ∂D ∂t (Maxwell IV; differential version) . (19.6) (By analogy with Faraday’s law this equation could be termed Maxwell’s law 3 .) In the corresponding integral form (see below) one considers a metal ca- pacitor. A true current enters plate 1 of this capacitor from a lead 1, whereas from plate 2 a corresponding true current flows outwards through a lead 2. In the space between plate 1 and plate 2 no true current flows, but the D-field changes with time, since the curls of the magnetic field surrounding the true currents in the leads cannot stop in the interspace: curls must always continue (a diagram is recommended here!) As with Faraday’s law of induction, the integral form of the fourth Maxwell equation is again somewhat more general than the differential form (Note the time derivative in front of the integral): ∂F H · dr = F j · nd 2 A + d dt F D · nd 2 A. (19.7) 3 The fact that on the r.h.s. of (19.1) and (19.2) the time derivatives of the electro- magnetic fields B and D enter with different signs is ultimately responsible (as we shall see) for the existence of electromagnetic waves. The difference in signs also reminds us of a similar difference in the canonical equations in classical me- chanics (see Part I) and gives another pointer to the existence of matter waves (see Part III). 158 19 Maxwell’s Equations I: Faraday’s and Maxwell’s Laws 19.4 Applications: Self-inductance and Mutual Inductances; Transformers; Complex Resistances; Alternating-current Resonance Circuit a) Inductance Consider a solenoid of length l and circular cross-section (radius r 0 ) filled with a material of relative permeability μ (typically a soft-iron core). Let l be the length of the solenoid, while r 0 is the radius of the circular cross-section and the number of windings is N 1 . The total magnetic flux through all the windings of the solenoid is thus φ B = N 1 · μμ 0 H · πr 2 0 . Applying Amp`ere’s law, ∂F H · dr = I(F ) , to a long closed loop circulating around the core with all windings, implies for the magnetic field H in the interior of the solenoid: H = N 1 l ·I, where I is a quasi-static, i.e., slowly (periodically) changing current in the coil. Hence we have φ B = μμ 0 N 2 1 l πr 2 0 ·I, i.e. ,φ B ≡ L ·I, where L is the (self-)inductance of the solenoid, given by L(=: L 1,1 )=μμ 0 N 2 1 l πr 2 0 . (19.8) The field energy of the solenoid can be written E mag = μμ 0 2 H 2 · πr 2 0 l. With H = N 1 l ·I one obtains the equivalent result E mag = L 2 I 2 , i.e. , dE mag dt = L ·I dI dt = L dI dt I = −U i · I, as expected. 19.4 Applications: Complex Resistances etc. 159 b) The transformer The index 1 above refers to the number of windings for a single solenoid. Now consider a second solenoid of equal length wrapped around the first solenoid and possessing only a marginally larger cross-sectional radius r 0 , but a significantly different number of windings N 2 . (This arrangement is merely designed so that the same flux of magnetic field lines passes through both coils.) The terminals of solenoid 2 are assumed to remain open, i.e., no current flows through solenoid 2, in contrast to solenoid 1, through which a current I 1 flows. The magnetic field H 2 in the interior of solenoid 2 due to the current I 1 in 1 is thus H 2 = N 1 l I 1 . The magnetic flux through solenoid 2 is then (φ B ) 2 = L 2,1 ·I 1 , where L 2,1 is the mutual inductance, L 2,1 = L 1,1 · N 2 N 1 . It follows that (in the case of a low-frequency 4 alternating current) the in- duced voltages measured at the ends of the closed solenoid 1 and the open solenoid 2 behave as the ratio of the corresponding number of windings, (U i ) 2 :(U i ) 1 = N 2 : N 1 . This is the principle of the transformer. c) Complex alternating-current resistances Consider a circuit in which an alternating current (a.c.) I(t)=I (0) · cos(ωt − α) is generated by a voltage U G (t)=U (0) G ·cos(ωt) of angular frequency ω. The phase of the current I(t)isthusshiftedwith respect to that of the voltage U G (t) (for positive α: the maxima of the current are delayed w.r.t. the maxima of the generator voltage). Making use of the Euler-Moivre relation: e iωt =cos(ωt)+isin(ωt) , (19.9) 4 Only later will we see quantitatively what “low frequency” means in this context. . the gyromagnetic ratio and not twice this value. One has to admit that these relations are complicated and not understandable at an elementary level. 19 Maxwell’s Equations I: Faraday’s Law of Induction;. Einstein and de Haas it was shown that for the usual magnetic materials, e.g., alloys of Fe, Co and Ni, the gyromagnetic ratio is twice as large as the above ratio. For these materials the magnetism. thus realize that Faraday’s law of induction (19.3), which appears so simple, embodies a great deal of experimental information. In addition we may already surmise at this point that Maxwell’s