Appendix A. The Dirac Delta Distribution

The Dirac delta is an element of the set of mathematical objects called distri• butions. These are a generalization of functions. Some of them are particularly suitable for describing physical entities in an approximated way. For instance, the Dirac delta can be used to represent highly concentrated distributions of mass or charge. The theory of distributions was developed in the first half of the twentieth century by the French mathematician L. Schwartz, and its name is due precisely to its connection with mass and charge distributions. Under appropriate conditions, the Dirac delta distribution 6(x) can be treated, cum grano salis, as a function of the variable x E (-00,00), defined by the conditions 6(x) = 0 for x -I- 0, { (A.I) J~oo 6(x) dx = 1.

This is an even distribution, 6(x) = 6( -x), and it is infinitely concentrated at x = O. The Dirac delta distribution can be approximated by functions, called approximant kernels of 6 (x), in appropriate limits. Some of these approximant kernels are: f(x) = (27ra)-1/2 exp( _x2 /2a) for a -+ 0 (Gaussian),

for a -+ 0 (Lorentzian), (A.2) f(x) = '\exp( -2'\lxl) for ,\ -+ 00,

f(x) = sin(ax)/7rx for a -+ 00. The main property of the Dirac delta of interest for applications is that for any function f (x) b f(xo) = 1 f(x) 6(x - Xo) dx, (A.3) where a < Xo < b, if f(x) is continuous at x = Xo. Integrating by parts, the "derivatives" of 6(x) can be introduced. They are also distributions, defined by the property 274 A. The Dirac Delta Distribution

(A A) where the labels (n) indicate nth-order differentiation. When the argument of the delta distribution is a function ¢( x), a variable change makes it possible to prove that

(A.5) where Xi are the zeros of ¢(x) in the interval (a, b). In particular,

lb f(x) 8[Q(x - xo)] dx = IQI- 1 f(xo), (A.6) or, formally, 8(Qx) = 8(x)/IQI. The product of Dirac deltas allows us to define the multidimensional version of this distribution. For instance, in three dimensions, 8(r) = 8(x)8(y)8(z), (A.7) where (x, y, z) are the Cartesian components of r. In spherical coordinates (r, (), '1') we find 1 1 8(r) = 2--=-() 8(r)8(())8(cp) = 2 8(r)8(cos())8(cp). (A.8) r sm r The multidimensional delta satisfies i f(r) 8(r - ro) d3r = f(ro), (A.9) if ro is in the integration volume V. Other important properties of the Dirac delta are the following: d 8(x) = dx ()(x), (A.lO) where ()(x) is the Heaviside function

()(x) = {O ~fx < 0, (A.ll) 1 If x> 0, also called the step function. Moreover,

8(r) = _~\72~, (A.12) 411" r which, according to Gauss's law, implies that 8(r) is the charge density to be associated with a unit point charge at the origin. Finally,

00 8(x) = -1 1 exp(ikx) dk, (A.13) 211" -00 which relates the Dirac delta to the Fourier tranform of a constant function. B. Legendre Polynomials and Spherical Harmonics

From Laplace's equation,

\7 2 4> = 0, (B.1) the separation of variables in spherical coordinates, 4>( r, (), 'P) = R( r) Y ((), 'P), produces for Y ((), 'P) the differential equation

sin() ae(sin() aeY) + a!ipY = -AY, (B.2) where A is a separation constant. Under conditions of azimuthal symmetry, the solution of Laplace's equa• tion does not depend on 'P. Requiring regularity conditions at () = °and () = Jr, the solutions of the angular part of the equation are the Leg• endre polynomials P1(x), 1= 0,1,2, ... , evaluated in terms of x = cos(): Y((),'P) = Y(()) = P1(cos()). The lth-degree Legendre polynomial P1(x) is defined by

1 d1 2 I P1(x) = ii' -dI (x - 1) . (B.3) 2. x Legendre polynomials satisfy the orthogonality relation

l 2 J Pdx) P1(x) dx = -1- 511', (B.4) -1 2 + 1 and form a set of orthogonal functions in the interval (-1, 1). Any well• behaved function can thus be expanded in a series of Legendre polynomials within that interval, and such an expansion is unique. The generating function of PI (x),

(Xl I Ir - r /I -1 = ""'~ ---r:t=lP1(cos/,),r < (B.5) 1=0 r> where /' is the angle determined by rand r/, and r < = min( r, r/), r> = max(r, r/), turns out to be particularly important in applications to electro• statics (see Chap. 4). 276 B. Legendre Polynomials and Spherical Harmonics

Some relevant properties of the Legendre polynomials are the following:

P2n+1 (0) = 0, (B.6)

(x2 -l)d~Pl(X) = lxP1(x) -lP1- 1(X). The first few Legendre polynomials are

Po(x) = 1,

Pdx) = x, (B.7) P2(x) = ~(3x2 - 1),

P3(x) = ~(5x3 - 3x). The solutions of the angular part of Laplace's equation when there is no azimuthal symmetry are the spherical harmonic functions of order (l, m), Yim(O,

2l + 1 (l - m)! m --(l )' PI (cosO) exp(im

(B.IO)

00 I z= z= yt;" (0',

00 1 1 1 r < * (' ') ( ) 1r - r '1-1 = 471" '"~ '"~ 2l + 1 rl +1 Ylm () ,'I' Yim (), 'I' , (B.13) 1=0 m=-I > where (0, '1') and (()', '1") are the angular coordinates of the vectors rand r', respectively, and r> and r < are defined as in (B.5). The first few spherical harmonics are Yoo = (471")-1/2,

YlO = (3/471")1/2 cos(),

Y1,±1 = =r=(3/871") 1/2 sinO exp(±icp), (B.14) Y20 = (5/1671")1/2(3cos2 () -1),

Y2,±1 = =r=(15/871") 1/2 sinO cos () exp(±icp) ,

Y2,±2 = =r=(15/3271")1/2sen2() exp(±2icp). C. Covariant Notation and Tensor Calculus

Tensor calculus is the formalism used in special and general relativity the• ory for operating in the four-dimensional space with spatial coordinates (x, y, z) and time coordinate ct. Generally, four-dimensional tensor calculus involves mathematical objects defined in a space with generalized coordinates (xO, xl, x2, x3). Such objects, called tensors, are characterized by transform• ing in a well-defined way under a coordinate change to the new variables (xlO , xll, x,2, x,3), which are related to the original coordinates through cer• tain functions x lO = x,O(xO, Xl , x2 , x3) ,

(C.l)

X,3 = x'3(xO, xl, x2, x3). A zeroth-rank tensor or scalar, s, is a quantity whose value does not change under the coordinate transformation, s' = s. (C.2) A first-rank contravariant tensor or, contravariant four-vector, is a set of four quantities (vO,vl,v2 ,V3 ), whose elements are generically denoted by vI-' and which transform according to

3 a 'I-' a 'I-' v-'I-'-L~a v-ao_~ v.0 (C.3) X O X O 0=0 We have introduced here the Einstein convention of summation over repeated indices. According to this convention, when an index appears twice in an expression it is implicitly assumed that the expression is summed over the repeated index, from 0 to 3. The summation over repeated indices is called contraction with respect to those indices. A first-rank covariant tensor, or covariant four-vector, Ul-" is a set of four quantities (UO,Ul,U2,U3) which transform according to

(C.4) 280 C. Covariant Notation and Tensor Calculus

Four-vector physical quantities have, in general, both a contravariant and a covariant representation. Higher rth-rank tensors are sets of 4T quantities labeled by r indices, which transform as products of r four-vector components. Each index can be of contravariant or covariant character, according to the corresponding transformation rule. For instance, any of the sixteen components of a twice• contravariant second-rank tensor tJ.LV transforms according to 8 'J.L 8 'v 'J.LV _ X x 0.(3 (C ) t - 8xo. 8x(3 t . .5 In general, we have

8 'J.L 8 'v n 8 8 ' T'J.Lv"·=~~ ... ~~ ... To.(3 .. · (C.6) pu... 8xo. 8x(3 8x'p 8x'u 8, .. · . It is worthwhile stressing that the contraction operation preserves the co• variant properties of the result. This implies that, in general, the contraction of two tensors over an arbitrary number of indices produces another tensor. For instance, given a four-vector vJ.L its modulus vJ.LvJ.L is a scalar, i.e. is in• variant under coordinate transformations. The transformed coordinates of a four-vector also form a four-vector, as they are obtained from the contraction of the Jacobian tensor associated with the transformation, JJ.Lv = 8x'J.L /8xv, and the original four-vector, v'J.L = JJ.Lvvv. When the four-dimensional space has metric properties, it is possible to define a (squared) length element corresponding to an infinitesimal displace• ment (dxO, dx1 , dx2 , dx3 ) as (C.7) where gJ.LV is a symmetric tensor, gJ.LV = gVJ.L' called the metric tensor, which characterizes the metric geometry of space. The metric tensor defines the transformation between the contravariant and the covariant components of a tensor. For instance, for a four-vector we have (C.8)

In the case of special relativity, the four-dimensional space to be con• sidered is Minkowski space, where the coordinates of a point - or event• are (C.9) In this space, the most interesting coordinate transformation is the Lorentz transformation, which relates the coordinates of an event in two inertial ref• erence frames Sand S' as (C.lO) C. Covariant Notation and Tensor Calculus 281 where the tensor aVIl is defined in (3.18). For the Lorentz transformation, which is linear, the general expressions analyzed above are considerably sim• pler. In fact, ax/v /axll = aVIl does not depend on the coordinates. Conse• quently, any set of four quantities that transform between reference frames with the same rules as the coordinates of an event, (C.U) constitutes a four-vector. As stated above, higher-rank tensors transform as products of four-vector components. For instance, for a twice-contravariant second-rank tensor, we have (C.12) In the framework of special relativity, the metric tensor defines the Minkowski metric, and is ~ ~1 ~ ~) gllv = ( 0 0 -1 0 . (C.13) o 0 0 -1 The length element associated with the Minkowski metric is then

ds2 = c2 dt2 - dx2 - dy2 - dz2 . (C.14) Finally, we point out that in Minkowski space the operation of differen• tiation with respect to the contravariant components can be associated with the product with a covariant four-vector, whose components are denoted by

(C.15)

These four derivatives are the covariant components of the four-gradient. Of course, its contravariant components all are given by the derivatives with respect to Xw The definition of the gradient four-vector makes it possible to extend some of the three-dimensional vector operations to four-dimensional space. The four-divergence of a vector VIl, for instance, is the scalar obtained from the contraction allvll . In Chaps. 3 and 9, several applications of these operations are studied. D. Vector Identities, Theorems and Operators

In the following, ¢ and 'IjJ represent scalars, and a, bye are vectors.

Three-vector products

a· (b x c) = b· (c x a) = c· (a x b)

a x (b x c) = b(a· c) - c(a· b)

Product rules

\1(¢'IjJ) = ¢\1'IjJ + 'IjJ\1¢

\1(a· b) = a x (\1 x b) + b x (\1 x a) + (a· \1)b + (b. \1)a

\1 . (¢a) = ¢\1. a + a· \1¢

\1. (a x b) = b· (\1 x a) - a· (\1 x b)

\1 x (¢a) = ¢\1 x a - a x \1¢

\1 x (a x b) = (b· \1)a - (a· \1)b + a\1· b - b\1· a

Second-order derivatives

\1 . (\1 x a) = 0

\1 x (\1¢) = 0

\1 x (\1 x a) = \1(\1. a) - \12a

Fundamental theorems

J:,2(\1¢) . dl = ¢(r2) - ¢(rI)

fv(\1 . a)d3r = JS(V) a· ds

J8(0) (\1 x a) . ds = fo a· dl 284 D. Vector Identities, Theorems and Operators

Vector operators

Cartesian coordinates

dl = dx x + dy Y + dz Z d3r = dx dy dz

Y' ¢ = ox¢ x + Oy¢ Y + oz¢ Z

Y' . a = oxax + oyay + ozaz

Y' x a = (oya z - ozay) x + (ozax - oxaz) Y + (oxay - oyax) Z Y'2¢ = o;¢ + o;¢ + o;¢

Spherical coordinates

dl = dr r + r dB 8 + r sin £1 drp cp

d3r = r2 sin £1 dr dB drp

Y'¢ = or¢ r + r-1oo¢ 8 + (rsinB)-locp¢ cp

Y'. a = r-20r(r2ar) + (r sin O)-loo(sinO ao) + (r sin O)-locpacp

Y' x a = (r sinO)-l [00 (sin 0 acp) - ocpao] r+r-1[(sin O)-locpar -Or(racp)] 8 +r-l[or(rao) - ooar] cp

Y'2¢ = r-20r(r20r¢) + (r2 sin B)-100(sin Ooo¢) + (r2 sin2 O)-10~¢

Cylindrical coordinates

dl = dr r + r drp cp + dz Z

d3r = r dr drp dz

Y'¢ = or¢ r + r-10cp¢ cp + oz¢ Z

Y'. a = r-10r(rar) + r-1o",acp + ozaz Y' x a = [r- 10cpaz - ozacp] r + [ozar - oraz] cp + r-1[Or(ra",) - o",ar] Z

Y'2¢ = r-10r(ror¢) + r-2o~¢ + o;¢ E. Operation of PhysicSolver

PhysicSolver is a computer program designed to solve two-dimensional prob• lems in electrostatics and magnetostatics using the finite element method described in Chap. 5. The problems may involve three-dimensional geome• tries with either planar or axial symmetry. A problem with planar symmetry is solved in the (x, y) plane. A problem with axial or cylindrical symmetry is solved in one half of the (r, z) plane. To use the software it is recommended that a simple sketch of the prob• lem is first made on paper. Regions of the plane where materials with differ• ent physical properties are present should be identified. Mirror symmetries should be determined and the plane must be divided accordingly. The solu• tion domain must be defined. If the domain extends to infinity, an artificial boundary must be fixed at some distance from the region of interest. Finally, the boundary conditions at all parts of the boundary must be defined.

Units of length (mm, cm, m, inch or feet) must be selected to obtain the solution. Though they can be specified or changed at any time before the solution is calculated, it is recommended that the units of length be specified as early as possible. All other units are fixed and cannot be changed. The program uses a mixed unit system. Results are displayed in ampere, volt, coulomb, gauss and oersted. The unit of force is the newton and the unit of energy is the joule. To proceed with the solution, points, lines, and arcs have to be created to define the geometry of the problem. These steps lead to the definition of regions. In each one of these regions the material is assumed to be homoge• neous. It is also possible to divide a region where the material is homogeneous into two or more subregions where different behaviors of the solution are ex• pected, for instance, in the vicinity of a sharp point or of a region where material properties change.

The symmetry of the problem - planar or cylindrical - must be selected. One must also of course tell the program whether the problem to be solved is one of electrostatics or magnetostatics. 286 E. Operation of PhysicSolver

To apply the finite element method, the software creates a mesh. This mesh can be refined by regions before creating it. This is done to adapt the size of the mesh to the expected variations of the solution in each region. In a region where a smooth variation is expected the mesh can be coarser than in a region where the solution is likely to have more abrupt variations.

To completely define the problem, boundary conditions must be fixed once the regions have been created. They must be applied at every line or arc on the outside boundary, but cannot be applied at any internal line or arc. Only one type of boundary condition can be applied at each line or arc. Boundary conditions cannot be changed after a solution has been obtained.

To apply a boundary condition to a line or arc, select one of the Boundary Condition Tools. The cursor becomes an arrow with a legend, indicating which type of boundary condition tool has been selected. Click on the lines or arc where that boundary condition applies. The lines or arcs change color, indicating that the boundary condition has been accepted.

The available boundary conditions are the following.

Field Normal: The resulting field is normal to the boundary. In electrostat• ics, this is a Dirichlet boundary condition for the potential, and the boundary is an equipotential line at zero potential. In magnetostatics, this is equivalent to a Neumann boundary condition for the vector potential.

Field Confined: The resulting field is parallel to the boundary. In electro• statics, this is a Neumann boundary condition for the scalar potential. In magnetostatics, this is a Dirichlet boundary condition for the vector poten• tial. The vector potential at such a boundary vanishes. Fixed Potential: In this case a value for the potential is fixed at the bound• ary. In electrostatics, it applies to the scalar potential and the resulting field is normal to the boundary. In magnetostatics, it applies to the vector potential, and the field is parallel to that portion of the boundary.

Infinite Elements: These approximate an infinite boundary, and the net result is an improvement of the accuracy of the solution in the finite domain.

Material properties must be assigned to the different regions. In electro• statics the regions can correspond to vacuum, or to a conducting or dielectric material. For conductors one must fix the value of the potential whereas for dielectrics the value of the dielectric constant must be given.

In magnetostatics, the regions can correspond to vacuum, to coils carry• ing electric current, to linear magnetic materials or to permanent magnets. Electric currents can be given as total currents or as current densities. For E. Operation of PhysicSolver 287 linear magnetic materials one must specify the relative permeability, a di• mensionless number introduced in Chap. 1I.

A permanently magnetized material is described by assuming a fixed di• rection of magnetization in the plane. When an external magnetic field is applied to such a material, the magnetic field has two components: one is parallel to the magnetization and the other is perpendicular. The external field causes additional magnetization to develop in the material, and our con• cern is how to calculate the total magnetization at each point in the material when the total magnetic field is given at that point. To do this, we separate the problem into its two components. In the direction perpendicular to the magnetization, we simply assume B = MoH. This approximation is used only for simplicity. It is good if the perpendicular field component is small com• pared with the parallel component, a condition which holds for most common applications of permanent magnets. If this condition is not met, large errors can result.

For the parallel component, we describe the behavior by a linear interpo• lation in the (H,B) plane, passing through the points (-He, 0) and (0, Br ), where Br is the residual inductance and He is the coercive force. Manufactur• ers of permanent magnets usually supply He and Br for each material. The approximation is good in the range -He < H < 0, since the materials are strongly non-linear outside that range.

The solution consists of the set of values of the potential at the nodes of the mesh and a linear interpolation inside each finite element. Once the solution has been obtained, additional results or plots of the electrostatic equipotentials or magnetostatic lines of force can be obtained from the Re• ports menu.

Table E.1 has been included to facilitate the conversion between egs and 81 units. The coefficients 3 and 9 appearing in this table are respectively related to the value of the speed of light and its square. To evaluate the conversion factors with higher precision the coefficient 3 has to be replaced by 2.9979246. 288 E. Operation of PhysicSolver

Table E.!. SI and cgs unit conversion

Quantity SI unit Conversion factor cgs unit length meter (m) 102 centimeter (cm) mass kilogram(kg) 103 gram (g) time second (s) 1 second (s) force newton (N) 105 dyne (dyn) energy, work joule (J) 107 erg power watt (W) 107 erg/s electric charge coulomb (C) 3 x 109 statcoulomb (ues) charge density C/m3 3 x 103 statcoulomb / cm 3 electric current ampere (A) 3 x 109 statampere (statA) current density A/m2 3 x 10-2 statA/cm2 potential volt (V) 1/300 statvolt (statV) electric field Vim (1/3) x 10-4 statV /cm = gauss (G) polarization C/m2 3 x 105 G displacement C/m2 3 x 105 G magnetic field tesla (T) 104 G magnetization A/m 10-3 G H-field A-loop/m 47l' x 10-3 oersted (Oe) conductivity mho/m 9 x 109 l/s resistance ohm (!?) (1/9) x 10-11 s/cm capacity farad (F) 9 x 10 11 cm magnetic flux weber (Wb) 108 G cm2 induction henry (H) (1/9) x 10-11 stat henry (statH) Index

Aberration of stars 16, 32 Conductance coefficient 221 Abraham-Lorentz model 183 Conducting media 219 Absorption 242 - ionic, superionic 220 Action 44 Conductivity 219, 237 Advanced perturbation 148 - DC 246 Aepinus, Franz Maria 5 Conservation laws 138 Ampere's law 102, 125 Constitutive equations 191,198 - modified 127 Continuity equation 54, 80 Ampere, Andre-Marie 5, 101 Convection fields 158 Arago, Dominique Franc;ois 5, 101 - dipole approximation 166 Axial vector 132 Cooper, Leon 254 Copernicus, Nicolas 2 Bardeen, George 254 Cornu, Alfred 33 BCS theory 254 Cosmology 1 Biot, Jean Baptiste 5,7,101 Coulomb gauge 136,149 Biot-Savart law 104 Coulomb's law 52,56 Bohr radius 169, 191 - validity 72 Bohr, Niels 265 Coulomb, Charles Augustin 5 Bound fields 158 Covariance of physical laws 13, 18, Boundary conditions 70,81 173 - in material media 200 Critical Boyle, Robert 7 - magnetic field 253 Bradley, James 16,32 - temperature 251,255 Brahe, Tycho 2 Curie's law 215 Brewster angle 233 Current 53 Brugmans, Anton 5 - density 53 Bruno, Giordano 2 -- molecular 196 Bucherer experiment 39, 183 - superconducting 256

Capacity coefficient 68 Davy, Humphrey 6 - in dielectric media 221 Depolarization tensor 210 Cauchy, Augustin Louis 8 Depolarizing field 208 Causality 241 Descartes, Rene 6 Charge 52 Diamagnetism 216 - density 53 Dielectric constant 206, 242 -- molecular 195 - Lorentz model 245 - inversion 133 Dielectrics 205 Cisternay du Fay, Charles Franc;ois de Diffraction 153 5 Diffusion Classical mechanics 13 - of heat 80, 240 Clausius-Mossotti equation 214,247 - of matter 79,240 Collimated beam 152 Dipole 61,108 290 Index

- field, covariant form 177 -- in free space 148 - moment 61 -- in material media 227 -- density 195 Electromotive force 111,126 - radiation 163 Electron 10 Dirac delta distribution 273 - Abraham-Lorentz model 183 Dispersion 242 - charge 53 - anomalous 245 - classical radius 71 Dispersion relation 149 - electromagnetic theory 183 - in material media 229 - mass 39 Displacement 196, 202 -- electromagnetic 183 - current 127 - renormalized 185 Doppler effect 189 - self-energy 184 Doppler, Christian Johann 32 - stability 184 - superconducting 256 Einstein, Albert 9, 18,40,228 Electrostatic Electric - energy 65 - charge 52 -- of a system of conductors 68 -- conservation 53,55 - potential 45, 58 -- density 53 - shielding 85 -- quantization 53 Electrostatics 51 -- relativistic invariance 53,55 - current relevance 72 - conductivity 219,237 - symmetry with magnetostatics 116 - current 53 Elementary particles 10,142 -- density 53 Empedocles 6 Energy conservation 40 - dipole 61 Energy-momentum tensor 179 - displacement 196, 202 Environmental pollution 73 - field 56 Equivalence principle 10 -- covariant form for charges 177 Ether 7,15,141 -- of a charge in uniform motion Euclid 6 170,175 Euler-Lagrange equations 44, 69 - force 47 Evenson, Kenneth 33 - neutrality of matter 72 Extinction theorem 239 - polarization 195, 240 - potential 136, 155 Faraday effect 6 - susceptibility 205, 240 Faraday's law 125 -- Lorentz model 243 Faraday, Michael 6,53,125 Electromagnetic Fermat's principle 237 - angular momentum 188 Fermat, Pierre de 7 - conservation laws 138 Ferroelectricity 216 - energy 139 Ferromagnetism 218 -- in material media 201 Fick's law 80 - field tensor 174 Field tensor 174 - fields 129,147 Finite element method 92 -- covariant formulation 173 Fizeau's experiment 17,32 -- in material media 191 Fizeau, Armand Hippolyte 7,17,33 -- of a moving charge 156 Flux 105 mass 71,183 - quantization 264 - momentum 140 - quantum 266 -- flux 141 Fluxoid 266 - potentials 135,147 Force 14 -- covariant formulation 173 - classical definition 39 - waves 9, 127 - relativistic 39 -- in conducting media 237 Foucault, Jean Bernard 33 Index 291

Four-vector 42 Kohlrausch, Rudolph 9 Fourier, Jean Baptiste 81 Franklin, Benjamin 4, 52 Lagrangian 44 Fresnel prism 236 - density 186 Fresnel, Augustin Jean 6,7 Landau, Lev Davidovich 254 Landau-Ginzburg theory 254, 268 Galilean transformations 14, 15 Laplace's equation 59,79 Galilei, Galileo 2 - in other branches of physics 79 Galileo diagram 31,35 Laplace, Pierre Simon de 7 Gauge Larmor formula 160 - choice 104, 136 Least action principle 44 - theory 186 Legendre polynomials 275 - transformation 135, 186 Length contraction 23 Gauss's law 58, 125 Lienard-Wiechert potentials 156 Gauss's theorem 56 Light cone 29 Gilbert, William 4 Linear response 240 Ginzburg, Vitali 254 Lines of force 62 Goos-Hiinchen effect 237 Local field 211 Gradient transformation 186 London Gray, Stephen 4 - first equation 256 Green's function - gauge 260 - for Poisson's equation 81 - penetration depth 258 - for the wave equation 154 - second equation 258 Green's theorem 82 - theory 256 Green, George 83 Lorentz - field 212 H-field 197 - force 47, 101 - in a superconductor 260 -- covariant form 179 Hamilton principle 44 - gauge 138,174 Hamiltonian 47 - length contraction 23 Hauksbee, Francis 4 - model for electric susceptibility 243 Heat conduction 80 213 Helmholtz equation 257 - relation - sphere 212 Henry, Joseph 125 Hertz, Heinrich 9, 127 - transformations 20,24,41, 130, 175 Hooke, Robert 7 -- inverse 25 Hysteresis 202 -- properties 26 Lorentz, Hendrik Antoon 19, 23, 38, Image charge 84, 98 243 Induction coefficient 68, 113, 115 Lorentz-Lorenz relation 247 Industrial painting 73 Inertial reference system 13 Macroscopic Interface 200, 230 - fields 192 Intermediary field 185 -- sources 198 Invariance - wave function 257 - of electric charge 53 Magnetic - of proper quantities 20 - dipole 108 - of the speed of light 18 - energy 110 Invariant interval 27 -- of a superconductor 266 Isotope separation 118 - field 101 -- critical 253 Kamerlingh Onnes, Heike 251 -- of Earth 117 Kepler, Johannes 2 -- time-dependent 125 Kinematic laws 231 - flux 105, 115 292 Index

- force 47 Mutual inductance 110 - induction 126 - media, linear 216 Natural width of spectral lines 244 - moment 108 Navier, Claude Louis 8 -- density 197 Newton's laws 13 -- in a superconductor 261 Newton, Isaac 3 - monopole 102, 119 Nuclear reactor 48 - scalar potential 103 - susceptibility 216 Ochsenfeld, Rudolf 252 Magnetization 197 Oersted, Hans Christian 5, 101 - in a superconductor 260 Ohm's law 219, 237 - spontaneous 218 Optics 3 Magnetostatics 101 Oscillator strengths 245 - symmetry with electrostatics 116 Magnets 4,101,218 Paramagnetism 216 - superconducting 112 Particle acceleration 118 Mass Permeability 217 - as a function of velocity 38 Photocopying 73 - conservation 38, 80 Photon 10, 183, 185 - equivalence with energy 40 - mass 72 Plane wave 149 Material media 191 - linear 205 Planetary model of the atom 168 Plasma frequency 245 Maxwell stress tensor 140 Plemelj relations 241 Maxwell's equations 51,125,129 Poisson's equation 59, 79 - covariant formulation 176 - in other branches of phys\cs 79 - in material media 197 - solution 81 - symmetries 129 Poisson, Simeon Denis 5 Maxwell, James Clerk 8,51,66,127 Polar liquids 215 Meissner effect 252 Polar vector 132 Meissner, Walter 252 Polarizability 211, 244 Method of images 84, 98 - electronic 214 Metric tensor 41 - ionic 214 Michelson, Albert Abraham 15, 33 Polarization 195, 240 Michelson-Morley experiment 15, 32 - as a function of temperature 215 Microscopic fields 191 - as source of the wave fields 238 Millikan, Robert Andrews 53 - of spheres and ellipsoids 206 Minkowski - spontaneous 216 - diagram 28,35 - waves 229 - force 43 Potential - velocity 42 - coefficient 68 Minkowski, Hermann 31 - momentum 46 Moment tensor 178 Poynting vector 140 Morley, Edward William 15 - in material media 201 Motors 119,120 - of waves 151 Miiller formula 112 -- in material media 229,238 Multipole 62 Proper - expansion - length 20 -- electrostatic 59 - mass 38 -- magnetostatic 106 - time 22 - moment Pseudovector 132 -- Cartesian 63 Ptolemy 1 -- spherical 64 Musschenbroeck, Pieter 4 Quadrupole moment 63 Index 293

- density 195 Separation of variables Quark 10, 53, 71 - Cartesian coordinates 86 - spherical coordinates 89 Radiation 147 Simultaneity 19 - absorption 242 Sky light 167 - dipole approximation 163 Snell laws 231 - energy 159 Snell, Willebord 7 - fields 158 Sommerfeld, Arnold 265 -- covariant properties 180 Space-time continuum 41 -- dipole approximation 166 Speed of light 32 -- of a moving charge 158 Spherical - gauge 138,149 - harmonics 275 - power 160 - wave 152 Rapidity 34 Superconductivity Reflection coefficient 232 - electromagnetic theory 251 Reflectivity 232 - present relevance 269 Refractive index 228, 242, 245 - semiclassical theory 265 Relativistic Superposition principle 52, 56 - action 44 Susceptance 261 - analytical dynamics 44 Susceptibility - dynamics 37 - electric 205, 240 - energy 39 - in a superconductor 261 - force 39 - Lorentz model 243 - invariants 177 - magnetic 216 - kinematics 13 Symmetry - momentum 37 - charge inversion 133 -- canonical 44 - in physics 142 -- conservation 38 - rotation 130 - work 39 - space reflection 131 - time inversion 134 Relativity - general theory 37 Tensor calculus 279 - in classical mechanics 3, 13 Thomson dispersion 72 - principles 18 Time dilation 22 - special theory 18 Total internal reflection 233 Resistance coefficient 221 Transmission coefficient 232 Rest Transmissivity 232 - energy 40 Transverse gauge 138 - mass 38 Tunnel effect 97, 236 Retarded - observation 157 Variational principle - perturbation 148 - for conducting media 222 - time 156 - for electrostatics 69 Romer, Ole 7,32 - for magnetostatics 113 Vector potential 45, 104, 135, 155 Salam, Abdus 10 Velocity addition law 26, 32 Savart, Felix 5, 101 Volta, Alessandro 5 Scalar 42 Vorticity 257 Scalar potential 136,155 Scanning tunnel microscopy 97 Wave Schrodinger equation 81 - dispersion 242 Schrieffer, Robert 254 - equation 137,147 Self-energy 67,184 -- covariant form 174 Self-induction 113 -- Green's function 154 294 Index

-- in linear media 227 - transverse 150 - evanescent 235 - vector 148 - frequency 148 Wavelength 166 - monochromatic 149 Weber, Wilhelm 9 - number 149 Weinberg, Steven 10 - phase 149 Wilke, Johan 5 - polarization 150 World line 28 - propagation 148 -- at an interface 230 -- covariant properties 180 Young, Thomas 7 Springer and the environment

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