130 17 Electrostatics and Magnetostatics Proof of the above statements is again based on Gauss’s integral theorem, according to which (with E n := E · n) Q 1 + Q 2 = ∂V 1 +∂V 2 ε 0 E n d 2 A. (17.28) However ∂S = −∂V 1 − ∂V 2 , where the change of sign reflects that, e.g., the outer normal of V 1 is an inner normal of S. But according to Gauss’s integral theorem: ∂S E n d 2 A = S divEdV. (17.29) Since divE =0, we have Q 1 + Q 2 =0, as stated. The solution of the corresponding Neumann problem (i.e., not U, but Q is given) is also essentially unique for the capacitor. Here the proof of uniqueness is somewhat more subtle (N.B. we have used the term essentially above). For a given Q j two different solutions φ k (k =1, 2) must both satisfy −ε 0 · ∂V j (∇φ k · n)d 2 A = Q j (for j =1, 2) ; thus their difference w := φ 2 − φ 1 satisfies ∂V j ∇w · nd 2 A =0, and since w is constant on ∂V j , one even has ∂(Is) (w∇w) ·nd 2 A =0. Moreover, according to Gauss’s integral theorem it follows that Is ∇·(w∇w)dV =0, i.e., Is (∇w) 2 + w∇ 2 w dV =0. Therefore, since ∇ 2 w =0 wehave Is (∇w) 2 dV =0; 17.1 Electrostatic Fields in Vacuo 131 as a consequence ∇w ≡ 0 everywhere (apart from a set of zero measure). Therefore, it is not now the potential that is unique but its essence, the gradient, i.e., the electric field itself. The capacity of a condenser is defined as the ratio of charge to voltage across the plates, or C := Q U . Elementary calculations contained in most textbooks give the following results: For a plate capacitor (with plate area F and plate separation d), a spherical capacitor (with inner radius R and outer radius R + ΔR), and for a cylindrical capacitor (with length L and inner radius R ⊥ ) one obtains, respectively: C = ε 0 F d ,C=4πε 0 R ·(R + ΔR) ΔR ,C=2πε 0 L ln R ⊥ +ΔR ⊥ R ⊥ . (17.30) In the limit ΔR R (ΔR ⊥ R ⊥ ) these results are all identical. 17.1.7 Numerical Calculation of Electric Fields The uniqueness theorem for the Dirichlet problem is useful, amongst other reasons, because it allows one immediately to accept asolution(found or guessed by any means available) as the only solution. One should not under- estimate the practical importance of this possibility! If all analytical methods fail, numerical methods always remain. These methods are not necessarily as complicated as one might think, and they can oftenbeusedinthecontextofschoolphysics. If one considers, for example, a simple-cubic grid of edge length a between lattice points, then one has (up to a small discretization error ∝ a 2 ): ∇ 2 φ n ≡ 6 j=1 (φ n+Δ j − φ n )/a 2 . (17.31) Here the six vectors n + Δ j denote the nearest neighbours (right – left, backwards – forwards, up – down) of the lattice point considered, and φ n := φ(r n ) . To obtain a harmonic function, it is thus only necessary to iterate (to the desired accuracy) until at any lattice point one has obtained φ n ≡ 1 6 6 j=1 φ n+Δ j , i.e., the l.h.s. must agree with the r.h.s. 132 17 Electrostatics and Magnetostatics 17.2 Electrostatic and Magnetostatic Fields in Polarizable Matter 17.2.1 Dielectric Behavior For a polarizable medium, such as water, the permittivity of free space ε 0 must be replaced by the product ε 0 · ε,whereε is the relative dielectric constant of the medium, which is assumed to be isotropic here (this applies to gases, liquids, polycrystalline solids and crystals with cubic symmetry 8 ; ε is a dimensionless constant of the order of magnitude O(ε) = 10 to 100 (e.g., for water ε ∼ = 81) 9 . When a dielectric (or polarizable material) is placed between the plates of a condenser, the capacity is increased by a factor ε. For a parallel plate condenser, for example, the capacity becomes: C = εε 0 F d . This enhancement is often considerable, and in practice a dielectric material, such as an insulating plastic, is inserted between condenser plates in order to increase the amount of charge that can be stored for a given U. In addition, the electric displacement vector D (see later) 10 is defined for such materials as D := ε 0 εE. Then, from Gauss’s law we obtain the first of Maxwell’s equations (i) in integral form ∂V D · nd 2 A = Q(V )(MaxwellI), (17.32) and (ii) in differential form divD = . (17.33) What are the atomistic reasons for dielectric behavior? The answer to this short question requires several sections (see below). 17.2.2 Dipole Fields; Quadrupoles An electric dipole at a position r in vacuo can be generated by the following elementary dumbbell approach. 8 Generalizations to non-cubic crystals will be treated below in the context of crystal optics. 9 The product ε 0 · ε is sometimes called the absolute dielectric constant of the material andalsowrittenasε; however this convention is not followed below, i.e., by ε we always understand the relative dielectric constant, especially since ε 0 does not appear in the cgs system whereas the relative dielectric constant is directly taken over into that system. In addition it should be mentioned that the dielectric constant is not always constant but can be frequency dependent. 10 So-called for historical reasons. 17.2 Electrostatic and Magnetostatic Fields in Polarizable Matter 133 Firstly consider two exactly opposite point charges each of strength ±q, placed at the ends of a small dumbbell, r +(a/2) and r − (a/2). Then take the limit (i) a → 0 while (ii) q →∞,insuchawaythatqa → p(=0), whereas the limit qa 2 → 0. The result of this procedure (the so-called “dipole limit”) is the vector p, called the dipole moment of the charge array. Similarly to Dirac’s δ-function the final result is largely independent of intermediate configurations. The electrostatic potential φ (before performing the limit) is given by φ(r)= q 4πε 0 1 |r − r − a 2 | − 1 |r − r + a 2 | . (17.34) Using a Taylor expansion w.r.t. r and neglecting quadratic terms in |a|,one then obtains: φ(r) ∼ = − qa 4πε 0 · grad r 1 |r − r | = qa 4πε 0 · r − r |r − r | 3 . (17.35) The electrostatic potential φ Dp due to an electric dipole with dipole vector p at position r is thus given by φ Dp (r)= p 4πε 0 · r − r |r − r | 3 , (17.36) and the corresponding electric field, E Dp = −gradφ Dp ,is(→ exercises): E(r) Dp = (3(r −r ) ·p)(r − r ) −|r −r | 2 p 4πε 0 |r − r | 5 . (17.37) In particular one should keep in mind the characteristic kidney-shaped appear- ance of the field lines (once more a sketch by the reader is recommended). A similar calculation can be performed for an electric quadrupole.The corresponding array of charges consists of two opposite dipoles shifted by a vector b. 11 The Taylor expansion must now be performed up to second order. Monopole, dipole and quadrupole potentials thus decay ∼ r −1 , ∼ r −2 and ∼ r −3 , whereas the fields decay ∝ r −2 , ∝ r −3 and ∝ r −4 , respectively. 17.2.3 Electric Polarization In a fluid (i.e., a gas or a liquid) of dielectric molecules, an external electric field E polarizes these molecules, which means that the charge center of the (negatively charged) electron shells of the molecules shifts relative to the (positively charged) nuclei of the molecule. As a consequence, a molecular electric dipole moment is induced, given by 11 Another configuration corresponds to a sequence of equal charges of alternating sign at the four vertices of a parallelogram. 134 17 Electrostatics and Magnetostatics p = αε 0 E , where α is the so-called molecular polarizability, which is calculated by quan- tum mechanics. Now, a volume element ΔV of the fluid contains ΔN := n V ΔV molecules of the type considered. Moreover, the electric moment of this vol- ume element is simply the sum of the electric moments of the molecules contained in ΔV , i.e., Δp ≡ P ΔV . This is the definition of the so-called electric polarization: P ≡ Δp ΔV , where Δp := r i ∈ΔV p(r i ); i.e., the polarization P is the (vector) density of the electric moment (dipole density). Under the present conditions this definition leads to the result P = n V ε 0 αE . Furthermore, the displacement D, is generally defined via the dipole den- sity as D := ε 0 E + P . (17.38) For dielectric material we obtain D = ε 0 ·(1 + χ) · E where χ = n V α is the electric susceptibility, i.e., ε =1+χ. (N.B.: In a cgs system, instead of (17.38) we have: D := E +4πP , where P = n v αE , for unchanged α. Hence ε =1+4πχ =1+χ, i.e., χ =4πχ . Unfortunately the prime is usually omitted from tables of data, i.e., the authors of the table rely on the ability of the reader to recognize whether given data correspond to χ or χ . Often this can only be decided if one knows which system of units is being used.) 17.2.4 Multipole Moments and Multipole Expansion The starting point for this rather general subsection is the formula for the potential φ(r) of a charge distribution, which is concentrated in a region of 17.2 Electrostatic and Magnetostatic Fields in Polarizable Matter 135 vacuum G : φ(r)= G (r )dV 4πε 0 |r − r | . We assume that for all r ∈ G the following inequality holds: |r||r |, i.e., that the sampling points r are very far away from the sources r . A Taylor expansion w.r.t. r can then be performed, leading to 1 |r − r | ∼ = 1 r + 3 i=1 (−x i ) ∂ ∂x i 1 r + 1 2! 3 i,k=1 (−x i )(−x k ) ∂ 2 ∂x i ∂x k 1 r + (17.39) Substitution of (17.39) into the formula for φ(r) finally results in φ(r)= 1 4πε 0 ⎧ ⎨ ⎩ Q r + p ·r r 3 + 1 2 3 i,k=1 q i,k 3x i x k − r 2 δ i,k r 5 + ⎫ ⎬ ⎭ , (17.40) with a) the total charge of the charge distribution, Q := G (r )dV ; b) the dipole moment of the charge distribution, a vector with the three components p i := G (r )x i dV ; and c) the quadrupole moment of the distribution, a symmetric second-order tensor, with the components q i,k = G (r )x i x k dV . Higher multipole moments q i 1 , ,i l are calculated analogously, i.e., in terms of order l (l =0, 1, 2, )oneobtains φ(r) ∼ = Q 4πε 0 r + p ·r 4πε 0 r 3 + ∞ l=2 3 i 1 , ,i l =1 (−1) l l! 1 4πε 0 q i 1 , ,i l · ∂ l ∂x i 1 ∂x i l 1 r . (17.41) However, only a minor fraction of the many 2 l -pole moments q i 1 , ,i l ac- tually influence the result; in every order l, there are only 2l + 1 linear in- dependent terms, and one can easily convince oneself that the quadrupole 136 17 Electrostatics and Magnetostatics tensor can be changed by the addition of an arbitrary diagonal tensor, q i,k → q i,k + a · δ i,k , without any change in the potential. As a result, for l = 2 there are not six, but only 2l + 1 = 5 linearly independent quadrupole moments q i,k = q k,i . Analogous results also apply for l>2, involving so-called spherical harmonics Y lm (θ, ϕ), which are listed in (almost) all relevant books or collections of formulae, especially those on quantum mechanics. One can write with suitable complex expansion coeffi- cients c l,m : φ(r)= Q 4πε 0 r + p ·r 4πε 0 r 3 + ∞ l=2 1 4πε 0 r l+1 · +l m=−l c l,m Y l,m (θ, ϕ) . (17.42) Due to the orthogonality properties of the spherical harmonics, which are described elsewhere (see Part III), the so-called spherical multipole moments c l,m , which appear in (17.42), can be calculated using the following integral involving the complex-conjugates Y ∗ l,m of the spherical harmonics: c l,m = 4π 2l +1 ∞ 0 drr 2 π 0 dθ sin θ 2π 0 dϕ(r)r l Y ∗ l,m (θ, ϕ) . (17.43) Dielectric, Paraelectric and Ferroelectric Systems; True and Effective Charges a) In dielectric systems, the molecular dipole moment is only induced, p molec. = ε 0 αE. b) In contrast, for paraelectric systems one has a permanent molecular dipole moment, which, however, for E = 0 vanishes on average, i.e., by perform- ing an average w.r.t. to space and/or time. (This is the case, e.g., for a dilute gas of HCl molecules.) c) Finally, for ferroelectric systems, e.g., BaTiO 3 crystals, below a so-called critical temperature T c a spontaneous long-range order of the electric po- larization P exists, i.e., even in the case of infinitesimally small external fields (e.g. E → 0 + ) everywhere within the crystal a finite expectation value of the vector P exists, i.e., for T<T c one has P =0. In each case where P = 0 Gauss’s law does not state that ∂V ε 0 E · nd 2 A = Q(V ) , 17.2 Electrostatic and Magnetostatic Fields in Polarizable Matter 137 but, instead (as already mentioned, with D = ε 0 E + P ): ∂V D · nd 2 A = Q(V ) . (17.44) Alternatively, in differential form: divD = i.e., precisely: div(ε 0 E + P )=, (17.45) and not simply divε 0 E = . For a parallel plate condenser filled with a dielectric, for given charge Q, the electric field E between the plates is smaller than in vacuo,sincepartof the charge is compensated by the induced polarization. The expression E := divε 0 E represents only the remaining non-compensated (i.e. effective)chargedensity, E = true − divP , (see below). In the following, will be systematically called the true charge density ( ≡ true ). In contrast, the expression E := ε 0 divE will be called the effective charge density. These names are semantically some- what arbitrary. In each case the following equation applies: E = −divP . Calculating the Electric Field in the “Polarization Representation” and the “Effective Charge Representation” a) We shall begin with the representation in terms of polarization, i.e., with the existence of true electric charges plus true electric dipoles,andthen apply the superposition principle. In the case of a continous charge and dipole distribution, one has the following sum: φ(r)= (r )dV 4πε 0 |r − r | + dV P (r ) ·(r − r ) 4πε 0 |r − r | 3 . (17.46) This dipole representation is simply the superposition of the Coulomb potentials of the true charges plus the contribution of the dipole potentials ofthetruedipoles. 138 17 Electrostatics and Magnetostatics b) Integrating by parts (see below), equation (17.46) can be directly trans- formed into the equivalent effective-charge representation, i.e., with E := −divP one obtains: φ(r)= E (r )dV 4πε 0 |r − r | . (17.47) The electric polarization does not appear in equation (17.47), but instead of the density of true charges one now has the density E = − divP of effective (i.e., not compensated) charges (see above). For simplicity we have assumed that at the boundary ∂K of the integration volume K the charge density and the polarization do not jump discontinously to zero but that, instead, the transition is smooth. Otherwise one would have to add to equation (17.47) the Coulomb potential of effective surface charges σ E d 2 A := (σ + P · n)d 2 A, i.e., one would obtain φ(r)= K E (r )dV 4πε 0 |r − r | + ∂K σ E (r )d 2 A 4πε 0 |r − r | . (17.48) Later we shall deal separately and in more detail with such boundary divergences and similar boundary rotations. Essentially they are related to the (more or less elementary) fact that the formal derivative of the unit step function (Heaviside function) Θ(x)(= 1 for x>0; =0 for x<0; = 1 2 for x =0) is Dirac’s δ-function: dΘ(x) dx = δ(x) . Proof of the equivalence of (17.47) and (17.46) would be simple: the equiv- alence follows by partial integration of the relation r − r |r − r | 3 =+∇ r 1 |r − r | . One shifts the differentiation to the left and obtains from the second term in (17.46): − ∇ r ·P (r ) 4πε 0 |r − r | dV . Together with the first term this yields the required result. 17.2 Electrostatic and Magnetostatic Fields in Polarizable Matter 139 17.2.5 Magnetostatics Even the ancient Chinese were acquainted with magnetic fields such as that due to the earth, and magnetic dipoles (e.g., magnet needles)wereusedby mariners as compasses for navigation purposes. In particular it was known that magnetic dipoles exert forces and torques on each other, which are analo- gous to those of electric dipoles. For example, the torque D, which a magnetic field H exerts on a magnetic dipole, is given by D = m ×H , and the force F on the same dipole, if the magnetic field is inhomogeneous, is also analogous to the electric case, i.e., F =(m ·grad)H (see below). On the other hand a magnetic dipole at r itself generates a magnetic field H, according to 12 H(r)=−gradφ m , with φ m = m ·(r − r ) 4πμ 0 |r − r | 3 . All this is completely analogous to the electric case, i.e., according to this convention one only has to replace ε 0 by μ 0 , E by H and p by m. However, apparently there are no individual magnetic charges, or mag- netic monopoles, although one has searched diligently for them. Thus, if one introduces analogously to the electric polarization P a so-called magnetic po- larization J 13 , which is given analogously to the electric case by the relation Δm = JΔV , then one can define a quantity B := μ 0 H + J , the so-called “magnetic induction”, which is analogous to the “dielectric dis- placement” D := ε 0 E + P ; but instead of divD = 12 Unfortunately there are different, although equivalent, conventions: many au- thors write D = m B ×B,wherein vacuo B = μ 0 H, and define m B as magnetic moment, i.e., m B = m/μ 0 , where (unfortunately) the index B is omitted. 13 All these definitions, including those for P and J, are prescribed by international committees and should not be changed. . For a plate capacitor (with plate area F and plate separation d), a spherical capacitor (with inner radius R and outer radius R + ΔR), and for a cylindrical capacitor (with length L and inner radius. by a factor ε. For a parallel plate condenser, for example, the capacity becomes: C = εε 0 F d . This enhancement is often considerable, and in practice a dielectric material, such as an insulating. practical importance of this possibility! If all analytical methods fail, numerical methods always remain. These methods are not necessarily as complicated as one might think, and they can oftenbeusedinthecontextofschoolphysics.