Lehner G. Electromagnetic Field Theory for Engineers and Physicists
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5.3 Magnetization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 286
5.4 Forces on Dipoles in Magnetic Fields . . . . . . . . . . . . . . . . . . 291
5.5 B and H in Magnetizable Media . . . . . . . . . . . . . . . . . . . . . 293
5.6 Ferromagnetism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 299
5.7 Boundary Conditions for B and H, and the Refraction of Magnetic
Force Lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 304
5.8 Plate, Sphere, Hollow Sphere in a Uniform Magnetic Field . . . . . . . 307
5.8.1 The Planar Plate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 307
5.8.2 The Sphere . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 309
5.8.3 The Hollow Sphere . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 311
5.9 Imaging at a Plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313
5.10 Planar Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 321
5.11 Cylindrical Boundary Value Problems . . . . . . . . . . . . . . . . . . 322
5.11.1 Separation of Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322
5.11.2 Structure of Rotationally Symmetric Magnetic Fields . . . . . . . . . 324
5.11.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 326
5.11.3.1 Cylinder with an Azimuthal Surface Current . . . . . . . . . . . . . . 326
5.11.3.2 Azimuthal Surface Currents in the x-y-Plane . . . . . . . . . . . . . . 330
5.11.3.3 Current Loop and Magnetizable Cylinder . . . . . . . . . . . . . . . . 332
5.12 Magnetic Energy, Magnetic Flux and Inductance Coefficients . . . . . 335
5.12.1 Magnetic Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 335
5.12.2 Magnetic Flux . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 339
6 Time Dependent Problems I (Quasi Stationary Approximation) . . . 343
6.1 Faraday’s Law of Magnetic Induction . . . . . . . . . . . . . . . . . . 343
6.1.1 Induction by a Temporal Change of B . . . . . . . . . . . . . . . . . . . . . 343
6.1.2 Induction through the Motion of the Conductor . . . . . . . . . . . . . . 344
6.1.3 Induction by simultaneous temporal Change of B and Motion
of the Conductor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 347
6.1.4 Unipolar Machine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 349
6.1.5 Hering’s experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 351
6.2 Diffusion of Electromagnetic Fields . . . . . . . . . . . . . . . . . . . 352
6.2.1 Equations for E, g, B, A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 352
6.2.2 The Physics of these Equations . . . . . . . . . . . . . . . . . . . . . . . . . 354
6.2.3 Approximations and Similarity Theorems . . . . . . . . . . . . . . . . . . 357
6.3 Laplace Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . 360
6.4 Field Diffusion in the Two-sided Infinite Space . . . . . . . . . . . . . 364
6.5 Diffusion of a Field in a Half-Space . . . . . . . . . . . . . . . . . . . 369
6.5.1 General solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 369
6.5.2 Field Diffusion from the Surface into a Half-space (Impact of
the Boundary Conditions) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 370
6.5.3 Diffusion of the Initial Field in the Half-Space (Impact of
Initial Values) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 374
6.5.4 Periodic Field and Skin Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . 376
6.6 Field Diffusion in a Plane Plate . . . . . . . . . . . . . . . . . . . . . 381
Table of Contentsx
6.6.1 General Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 381
6.6.2 Diffusion of the Initial Field (Impact of Initial Condition) . . . . . . . 382
6.6.3 Impact of Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . 385
6.7 The Cylindrical Diffusion Problem . . . . . . . . . . . . . . . . . . . . 389
6.7.1 The Basic Formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 389
6.7.2 The longitudinal Field Bz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 391
6.7.3 The azimuthal Field B
ϕ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 395
6.7.4 Skin Effect in a Cylindrical Wire . . . . . . . . . . . . . . . . . . . . . . . . . 399
6.8 Limits of the Quasi Stationary Theory . . . . . . . . . . . . . . . . . . 402
7 Time Dependent Problems II (Electromagnetic Waves) . . . . . . . . 405
7.1 Wave Equations and their simplest Solutions . . . . . . . . . . . . . . 405
7.1.1 The Wave Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 405
7.1.2 The simplest Case: Plane Wave in an Insulator . . . . . . . . . . . . . 406
7.1.3 Harmonic Plane Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 411
7.1.4 Elliptic Polarization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 415
7.1.5 Standing Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 416
7.1.6 TE- and TM Wave . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 418
7.1.7 Energy Density in and Energy Transfer by Waves . . . . . . . . . . . 422
7.2 Plane Waves in a Conductive Medium . . . . . . . . . . . . . . . . . 423
7.2.1 Wave Equations and Dispersion Relation . . . . . . . . . . . . . . . . . . 423
7.2.2 The Process is Harmonic in Space . . . . . . . . . . . . . . . . . . . . . . . 425
7.2.3 The Process is Harmonic in Time . . . . . . . . . . . . . . . . . . . . . . . . 427
7.3 Reflection and Refraction . . . . . . . . . . . . . . . . . . . . . . . . 431
7.3.1 Reflection and Refraction for Insulators . . . . . . . . . . . . . . . . . . . 431
7.3.2 Fresnel’s Equations for Insulators . . . . . . . . . . . . . . . . . . . . . . . 433
7.3.3 Nonmagnetic Media . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 436
7.3.4 Total Reflection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 439
7.3.5 Reflection at a Conducting Medium . . . . . . . . . . . . . . . . . . . . . . 441
7.4 Potentials and their Wave Equations . . . . . . . . . . . . . . . . . . 442
7.4.1 The Inhomogeneous Wave Equation for A and
ϕ . . . . . . . . . . . . 442
7.4.2 Solution of the Inhomogeneous Wave Equations (Retardation) . 445
7.4.3 The Electric Hertz Vector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 447
7.4.4 Vector Potential for D and Magnetic Hertz Vector . . . . . . . . . . . . 448
7.4.5 Hertz Vectors and Dipole Moments . . . . . . . . . . . . . . . . . . . . . . . 450
7.4.6 Potentials for Uniformly Conductive Media without Volume
Charges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 452
7.5 Hertz’s Dipole . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 454
7.5.1 Fields of Oscillating Dipoles . . . . . . . . . . . . . . . . . . . . . . . . . . . . 454
7.5.2 Far Field and Radiation Power . . . . . . . . . . . . . . . . . . . . . . . . . . 460
7.6 Frame Antenna . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 463
7.7 Waves in Cylindrical Wave Guides . . . . . . . . . . . . . . . . . . . 466
7.7.1 Basic Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 466
7.7.2 TM Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 469
7.7.3 TE Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 470
7.7.4 TEM Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 470
Table of Contents xi
7.8 Rectangular Wave Guide . . . . . . . . . . . . . . . . . . . . . . . . 475
7.8.1 Separation of Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 475
7.8.2 TM Waves in a Rectangular Wave Guide . . . . . . . . . . . . . . . . . . 476
7.8.3 TE Waves in a Rectangular Wave Guide . . . . . . . . . . . . . . . . . . 478
7.8.4 TEM waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 479
7.9 Rectangular Cavities . . . . . . . . . . . . . . . . . . . . . . . . . . . 481
7.10 Circular Wave Guide . . . . . . . . . . . . . . . . . . . . . . . . . . . 485
7.10.1 Separation of Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 485
7.10.2 TM Waves in a Circular Cylindrical Wave Guide . . . . . . . . . . . . . 487
7.10.3 TE Waves in a Circular Cylindrical Wave Guide . . . . . . . . . . . . . 489
7.10.4 The Coaxial Cable . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 490
7.10.5 Telegrapher's Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 493
7.11 The Wave Guide as a Variational Problem . . . . . . . . . . . . . . . 495
7.12 Boundary and Initial Value Problems . . . . . . . . . . . . . . . . . . 498
7.12.1 The Initial Value Problem of the Infinite, Uniform Space . . . . . . . 499
7.12.2 The Boundary Value Problem of the Half-Space . . . . . . . . . . . . 503
8 Numerical Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . 506
8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 506
8.2 Basics of Potential Theory . . . . . . . . . . . . . . . . . . . . . . . . 506
8.2.1 Boundary Value Problems and Integral Equations . . . . . . . . . . . 506
8.2.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 510
8.2.2.1 The One-dimensional Problem . . . . . . . . . . . . . . . . . . . . . . . . . . 510
8.2.2.2 Dirichlet’s Boundary Value Problem of a Sphere . . . . . . . . . . . . 512
8.2.3 Mean Value Theorems of Potential Theory . . . . . . . . . . . . . . . . . 515
8.3 Boundary Value Problems as Variational Problems . . . . . . . . . . . 517
8.3.1 Variational Integrals and Euler’s Equations . . . . . . . . . . . . . . . . . 517
8.3.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 520
8.3.2.1 Poisson’s Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 520
8.3.2.2 Helmholtz Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 523
8.4 Method of Weighted Residuals . . . . . . . . . . . . . . . . . . . . . 528
8.4.1 Collocation Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 529
8.4.2 Method of Fractional Regions . . . . . . . . . . . . . . . . . . . . . . . . . . . 531
8.4.3 Momentum Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 532
8.4.4 Method of Least Squares . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 532
8.4.5 Galerkin Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 533
8.5 Random-Walk Processes . . . . . . . . . . . . . . . . . . . . . . . . 537
8.6 Method of Finite Differences . . . . . . . . . . . . . . . . . . . . . . . 541
8.6.1 Fundamental Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 541
8.6.2 An Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 546
8.7 Finite Elements Method . . . . . . . . . . . . . . . . . . . . . . . . . 549
8.8 Method of Boundary Elements . . . . . . . . . . . . . . . . . . . . . 557
8.9 Method of Image Charges . . . . . . . . . . . . . . . . . . . . . . . . 562
8.10 Monte-Carlo Method . . . . . . . . . . . . . . . . . . . . . . . . . . . 563
ii Table of Contentsx
A Appendices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 569
A.1 Electromagnetic Field Theory and Photon Rest Mass . . . . . . 569
A.1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 569
A.1.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 574
A.1.2.1 Uniformly Charged Spherical Surface . . . . . . . . . . . . . . . . . 574
A.1.2.2 The Plane Capacitor and its Capacitance . . . . . . . . . . . . . . 575
A.1.2.3 The Ideal Electric Dipole. . . . . . . . . . . . . . . . . . . . . . . . . . . . 577
A.1.2.4 The Ideal Magnetic Dipole . . . . . . . . . . . . . . . . . . . . . . . . . . 577
A.1.2.5 Plane Waves. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 579
A.1.3 Measurements and Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 582
A.1.3.1 Magnetic Fields of Earth and Jupiter . . . . . . . . . . . . . . . . . . 582
A.1.3.2 Schumann-Resonances . . . . . . . . . . . . . . . . . . . . . . . . . . . . 583
A.1.3.3 Fundamental Limit -- The Uncertainty Relation . . . . . . . . . . 584
A.2 Magnetic Monopoles and Maxwell’s Equation . . . . . . . . . . 585
A.2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 585
A.2.2 Dual Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 587
A.2.3 Properties of Magnetic Monopoles . . . . . . . . . . . . . . . . . . . . . . . . . . 590
A.2.4 The Search for Magnetic Monopoles . . . . . . . . . . . . . . . . . . . . . . . . 591
A.3 On the Significance of Electromagnetic Fields and Potentials
(Bohm-Aharonov Effects) . . . . . . . . . . . . . . . . . . . . . 593
A.3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 593
A.3.2 The Role of Fields and Potentials . . . . . . . . . . . . . . . . . . . . . . . . . . . 595
A.3.3 The Ehrenfest Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 597
A.3.4 Magnetic Field and Vector Potential in an infinitely long Coil . . . . . . 598
A.3.5 Interference of Electron Beams at Double Slit . . . . . . . . . . . . . . . . . 599
A.3.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 603
A.4 Liénard-Wiechert Potentials . . . . . . . . . . . . . . . . . . . . 603
A.5 The Helmholtz Theorem . . . . . . . . . . . . . . . . . . . . . . 608
A.5.1 Derivation and Interpretation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 608
A.5.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 611
A.5.2.1 Uniform Field inside a Sphere . . . . . . . . . . . . . . . . . . . . . . . 611
A.5.2.2 Point Charge inside a Conducting Hollow Sphere . . . . . . . . 615
A.6 Maxwell’s Equations and Relativity . . . . . . . . . . . . . . . . 616
A.6.1 Galilean and Lorentz Transformation . . . . . . . . . . . . . . . . . . . . . . . . 616
A.6.2 Lorentz Transformation as an Orthogonal Transformation . . . . . . . . 618
A.6.3 Some Consequences of the Lorentz Transformation . . . . . . . . . . . . 622
A.6.3.1 Lorentz Contraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 622
A.6.3.2 Time Dilatation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 624
A.6.3.3 Relativistic Addition of Velocities . . . . . . . . . . . . . . . . . . . . . 624
A.6.3.4 Aberration and Doppler Effect . . . . . . . . . . . . . . . . . . . . . . . 626
A.6.4 Lorentz Transformation of Maxwell’s equations . . . . . . . . . . . . . . . . 627
A.6.5 4-Vectors and 4-Tensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 629
A.6.5.1 Definitions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 629
A.6.5.2 Some Important 4-Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . 630
A.6.5.3 Field Tensor F. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 636
Table of Contents xiii
A.6.6 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 639
A.6.6.1 Surface Charges and their Fields . . . . . . . . . . . . . . . . . . . . . 639
A.6.6.2 Currents and Volume Charges . . . . . . . . . . . . . . . . . . . . . . . 641
A.6.6.3 Force of a Current on a moving Charge . . . . . . . . . . . . . . . . 643
A.6.6.4 Field of a Uniformly Moving Point Charge . . . . . . . . . . . . . . 644
A.6.7 Final Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 645
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 647
Generally Recommended Reading . . . . . . . . . . . . . . . . . . . . . . 648
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 649
Table of Contentsxiv
List of Symbols
General Symbols
~
(For example ) refers to a function f which is derived via an
integral transform (Fourier, Hankel, Laplace transform)
~ Proportionality sign (for example F ~ m)
* (For example z*, w*) refers to the complex conjugate of a
quantity (e.g. for z, w), or to a dual quantity (e.g. A* to A, ϕ* to
ϕ)
n,
A perpendicular component if used as an index
t, || A tangential component if used as an index
The circle indicates that the integral is to be taken over a closed
contour (line integral) or over a closed surface (surface integral).
,
Nabla symbol “del” and “del dot” operator
(del in Cartesian coordinates: )
Divergence of the vector a
Circulation of the vector a (curl)
Laplace operator
^
Quantum mechanical operators, e.g. .
Used to indicate multiplication of scalar quantities
Scalar multiplication of vectors, scalar product, dot product
Vector product of two vector quantities
Dyadic product. It is an operator whose result is a tensor matrix
with elements:
f
˜
⊥
"
∇ ∇•
x∂
∂
y∂
∂
z∂
∂
,,〈〉
a∇•
a∇×
∇
2
∆≡
H
ˆ
⋅
•
×
ab
ab()
ik
a
i
b
k
()=
Latin Letters
A vector and its cartesian components
a, a (Surface) area, magnitude and vector (where the are caould be
confused with the vector potential)
A, A (Surface) area magnitude and vector
Magnetic vector potential
Electric vector potential (in analogy to the magnetic vector
potential).
arg(z) Argument of a complex number (phase angle)
ber(), bei() Kelvin’s function
Magnetic flux density, magnetic induction, B-field
Amplitude of the magnetic field of an electromagnetic wave
Normal component of
Tangential component of
c Speed of light in vacuum
c
G
Group velocity of light
c
Ph
Phase velocity of light
Influence coefficients
C
ik
Capacitance coefficient
C Capacitance
C’ Capacitance per unit length
cos ( ),
cosh( )
Cosine and hyperbolic cosine function
Displacement field (electric)
Normal component of
a a
x
a
y
a
z
,,,
a
x
a
y
a
z
,,
A
A*
B
B
0
B
n
B
B
t
B
c
ik
D
D
n
D
xvi List of Symbols Latin Letters
List of Symbols Latin Letters xvii
Tangential component of
Differential of a surface element (vector quantity)
Magnitude of the vector of the surface element
Partial derivative with respect to t, x, …
Normal component of the gradient of the function
dt Differential of time
Differential of a line element (vector)
The magnitude of the differential of a line element
Differential of a volume element
Differential of a solid angle
Differential of an angle
Distance, thickness of a layer, Skin depth
Divergence of the vector
Electric Field vector, E-field
Magnitude of or the complex field
Normal component of
Tangential component of
Amplitude of the electric field of an electromagnetic wave
Amplitude of the incident, reflected, transmitted wave
Incident electromagnetic wave
Driving electric field
D
t
D
dA da,
dA da,
t∂
∂
x∂
∂
…,,
n∂
∂φ
φ
ds
ds
dτ
dΩ
dα
d
a∇• a
E
E E EE
x
iE
y
+=
E
n
E
E
t
E
E
0
E
i0
E
r0
E
t0
,,
E
i
E
e
Total elliptic integral of the 2nd kind
Unity vector in the direction of the u coordinate
Dyadic product
Electron charge
The number e (=2.718…), the basis of the natural logarithm
exp(x)
=
Exponential function
erf() Error function
erfc() Complement to the Error function [1 - erf()]
f() A function
f Frequency
Force vector
Magnitude of the force
F
Potential, besides and
G Conductance
G’ Conductance per unit length
Green’s function
Green’s function for Dirichlet’s boundary value problem
Green’s function for Neumann’s boundary value problem
Electric current density, volume current density
Magnetic current density (duality to the electric current density)
Magnetization current density (volume bound)
Gradient of the function f (grad f )
E
π
2
---
k,
e
u
e
u
e
u
e
e
e
x
F
F
ϕφ
G rr
0
,()
G
D
rr
0
,()
G
N
rr
0
,()
gg
e
,
g
m
g
mag
∇f
xviii List of Symbols Latin Letters
h Planck’s Constant
h bar, Planck’s constant divided by ( )
Magnetic field, H-field
Amplitude of the H-field of an electromagnetic wave
Normal component of
Tangential component of
H(x-x
0
)
Heaviside’s step function (generalized function)
H Hamiltonian
Hamilton operator
I Current
Modified Bessel function of the 1st kind of order m
i
Imaginary number symbol
Bessel function of order m
Modified Bessel function of the 2nd kind of order m
Total elliptical integral of the 1st kind
Surface current density
Surface bound current density
Wave vector = vector of wave number
Magnitude of the wave vector (wave number)
L, l Length
l Propagation number, wave number
Coefficient of inductance
Ñ 2π h 2π⁄
H
H
0
H
n
H
H
t
H
H
ˆ
I
m
()
i 1–=
J
m
()
K
m
()
K
π
2
---
k,
k
k
mag
k
k
L
ik
List of Symbols Latin Letters xix