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xv8.5Hamilton’s Principle and Noether’sTheorem954A. Action Integral954B. Hamilton’s Principle and MaxwellEquations955C. Noether’s Theorem and EnergyMomentum Tensors956Answers962REFERENCES969INDEX991

1FUNDAMENTALS1.1 Maxwell’s TheoryA. Maxwell’s EquationsB. Vector Analysis1.2 Electromagnetic WavesA. Wave Equation and Wave SolutionB. Unit for Spatial Frequency kC. Polarization1.3 Force, Power, and EnergyA. Lorentz Force LawB. Lenz’ Law and Electromotive Force (EMF)C. Poynting’s Theorem and Poynting Vector1.4 Hertzian WavesA. Hertzian DipoleB. Electric and Magnetic FieldsC. Electric Field Pattern1.5 Constitutive RelationsA. Isotropic MediaB. Anisotropic MediaC. Bianisotropic MediaD. Biisotropic MediaE. Constitutive Matrices–1–

21. Fundamentals1.6 Boundary ConditionsA. Continuity of Electric and Magnetic Field ComponentsB. Surface Charge and Current DensitiesC. Boundary Conditions1.7 Reflection and GuidanceA. Wave Vector kB. Reflection and Transmission of TE WavesC. Reflection and Transmission of TM WavesD. Brewster Angle and Zero ReflectionE. Guidance by Conducting Parallel PlatesAnswers

1.1 Maxwell’s Theory1.13Maxwell’s TheoryA. Maxwell’s EquationsThe laws of electricity and magnetism were established in 1873 byJames Clerk Maxwell (1831–1879). In three-dimensional vector notation, the Maxwell equations are D J t E B t H (1.1.1)(1.1.2) ·D ρ(1.1.3) ·B 0(1.1.4)where E, B, H, D, J, and ρ are real functions of position and time.E electric field strength(volts/m)B magnetic flux density(webers/m2 )H magnetic field strength(amperes/m)(coulombs/m2 )D electric displacementJ electric current density(amperes/m2 )ρ electric charge density(coulombs/m3 )Equation (1.1.1) is Ampère’s law or the generalized Ampère circuit law.Equation (1.1.2) is Faraday’s law or Faraday’s magnetic induction law.Equation (1.1.3) is Coulomb’s law or Gauss’ law for electric fields.Equation (1.1.4) is Gauss’ law or Gauss’ law for magnetic fields.We generally refer to E and D as electric fields, and H and Bas magnetic fields.Maxwell’s contribution to the laws of electricity and magnetism isthe term D/ t , which is called the displacement current. The addition of the displacement current to the electric current density J (r, t)in the original Ampère’s law has at least three major consequences.First, in a capacitor which is an open circuit for direct current, thedisplacement current insures the continuity of alternating currents inelectric circuits. Secondly, the continuity law ·J ρ t(1.1.5)

41. Fundamentalsfollows from (1.1.1) and (1.1.3) by making use of the vector identity · ( H) 0 . It is the displacement term that guarantees theconservation of electric current and charge densities. Eq. (1.1.5) statesthat the electric current and charge densities are conserved at all time.Thirdly, Faraday’s law in (1.1.2) states that surrounding a time-varyingmagnetic field, electric fields are produced, and are also time-varying.With the displacement term in (1.1.1), Ampère’s law states that aroundtime-varying electric fields, time-varying magnetic fields are produced.This interrelationship between the time-varying electric and magneticfields constitutes the foundation of electromagnetic wave theory andled Maxwell to the prediction of electromagnetic waves.In developing his theory for the electromagnetic fields in spaceand time, Maxwell conceived of a substance filling the whole spacecalled aether. In the aether, the electric fields D and E are relatedby a dielectric permittivity o , and the magnetic fields B and H arerelated by a magnetic permeability µo .D oEB µo Hwhereo 8.85 10 12µo 4π 10 7(1.1.6a)(1.1.6b)farad/meterhenry/meterwhere the numerical values for o and µo are expressed in MKS units.We now call (1.1.6) the constitutive relations for free space.With Equations (1.1.1)–(1.1.6), Maxwell’s theory of electromagnetic fields is completely expressed. Originally written in Cartesiancomponent form, Maxwell’s equations were cast in the current vectorform by Oliver Heaviside (1850–1925). In 1888, Heinrich Rudolf Hertz(1857–1894) demonstrated the generation of radio waves and experimentally verified Maxwell’s theory. Since then, electromagnetic theoryhas played a central role in the development of radio, television, wireless communications, radar, microwave heating, remote sensing, andnumerous other practical applications. The special theory of relativitydeveloped by Albert Einstein (1879–1955) in 1905 further asserted therigorousness and elegance of Maxwell’s theory. As a well-establishedscientific discipline, this sophisticated theoretical structure embodiesmany principles and concepts which serve as fundamental rules of nature and vital links for all scientific disciplines.

1.1 Maxwell’s Theory5James Clerk Maxwell (13 June 1831 – 5 November 1879)James Clerk Maxwell attended University of Edinburgh (1847–1850),and studied under William Hopkins at Cambridge University (1850–1854).He was a fellow of Trinity (1855–1856), Professor of Natural Philosophy atMarischal College of the University of Aberdeen (1856–1860), and at King’sCollege (1860–1865). He was the first Cavendish Professor of ExperimentalPhysics at Cambridge University to build and direct the Cavendish Laboratory (1871–1879). He published four books and about 100 papers starting atage 14, including ‘On Faraday’s Lines of Forces’ in 1855, ‘On Physical Linesof Force’ in 1861, and ‘A Dynamical Theory of the Electromagnetic Field’ in1864. In 1865, at age 33, he retired to his country home estate to write hismonumental book A Treatise of Electricity and Magnetism (Constable andCompany, London, 1873; Dover Publications, New York, 1006 pages, 1954).Michael Faraday (22 September 1791 – 25 August 1867)Faraday became an assistant to Sir Humphry Davy at the Royal Institution on 1 March 1813. In September 1821, his experimentation demonstratedelectro-magnetic rotation, initiated the concept of electric motor. In August1831, he discovered electro-magnetic induction, and that magnetism producedelectricity through movement, the principle behind the electric transformerand generator. He became professor of chemistry in 1833. Faraday publishedmany of his results in the three-volume Experimental Researches in Electricity(1839–1855).Johann Carl Friedrich Gauss (30 April 1777 – 23 February 1855)Gauss studied mathematics at the University of Göttingen from 1795 to1798, and received his doctoral degree from the University of Helmstedt in1799. In 1807 he took the position of director of the Göttingen Observatory.In 1832 he presented a systematic use of absolute units (length, mass, time)to measure nonmechanical quantities. From 1831 to 1837 he worked closelywith Wilhelm Eduard Weber (24 October 1804 – 23 June 1891) on terrestrialmagnetism and organized a system of stations for systematic observations.André-Marie Ampère (20 January 1775 – 10 June 1836)Ampère was appointed professor at Bourg Ecole Centrale in 1802, atthe Ecole Polytechnique in 1809, and at Université de France in 1826. InSeptember 1820, Ampère showed that two parallel conductors attract eachother if they carry currents that flow in the same direction and repel if thecurrents flow in opposite directions. In 1823–1826, he completed his memoir onthe ‘Mathematical Theory of Electrodynamic Phenomena, Uniquely Deducedfrom Experience’.Charles-Augustin de Coulomb (14 June 1736 – 23 August 1806)Coulomb worked in the Corps du Génie until he retired in 1791. In 1777he invented the torsion balance, which enabled him to establish the fundamental laws of electricity by measuring the force between two small spherescharged with electricity. Between 1785 and 1791, he published seven treatiseson electricity and magnetism.

61. FundamentalsB. Vector AnalysisA vector A has a magnitude and a direction, which can be representedgraphically by a straight-line element of length proportional to themagnitude of A and with an arrow pointing in the direction of A . Ina Cartesian coordinate system (also called the rectangular coordinatesystem), we write in terms of the three Cartesian components Ax , Ay ,and Az [Fig. 1.1.1].zAzAẑŷAyx̂yAxxFigure 1.1.1 Projection of A in rectangular coordinate system.A x̂Ax ŷAy ẑAzwhere Ax , Ay , Az are the projections of A onto the x, y, z axes. Wedenote the directions of the x, y, z axes with x̂, ŷ, ẑ each of themhas unit magnitude with the scalar product x̂ · x̂ ŷ · ŷ ẑ · ẑ 1 .They are called the unit vectors. Furthermore x̂ · ŷ ŷ · ẑ ẑ · x̂ 0 .We use a hat instead of an overbar to represent the vector with unitmagnitude.Rene Descartes (31 March 1596 – 11 February 1650)Rene Descartes originated the Cartesian coordinates and founded analytic geometry. His philosophy is called Cartesianism (from Cartesius, theLatin form of his name), with the famous statement ‘I think, therefore I am.’He preached universal doubt; only one thing cannot be doubted: doubt itself.

1.1 Maxwell’s Theory7Vector Addition and SubtractionTwo vectors A and B , when they are not in the same direction or inopposite directions, determine a plane. In Cartesian components, we writeA x̂Ax ŷAy ẑAzB x̂Bx ŷBy ẑBzIt follows thatA B x̂(Ax Bx ) ŷ(Ay By ) ẑ(Az Bz )Scalar Dot ProductThe scalar or dot product of two vectors A and B , denoted by A · B ,is a scalar number,A · B Ax Bx Ay By Az BzVector Cross ProductThe vector or cross product of two vectors A and B , denoted by A B ,is a vector. In terms of their Cartesian components,A B x̂(Ay Bz Az By ) ŷ(Az Bx Ax Bz ) ẑ(Ax By Ay Bx ) x̂ŷẑ Ax A y A z B B B xyzFor the three orthogonal unit vectors x̂, ŷ, and ẑ it is seen that x̂ ŷ ẑ, ŷ ẑ x̂, ẑ x̂ ŷ.The direction of A B follows the right-hand rule, i.e., when the fingersof the right hand rotate from A to B , the thumb of the right hand points inthe direction of A B . Thus the vector A B is perpendicular to both Aand B and the plane containing A and B . It is seen that for A x̂Ax ŷAyand B x̂Bx ŷBy both in the xy -plane, A B ẑ(Ax By Ay Bx ) is inthe ẑ direction perpendicular to both A and B .Division by a vector is not defined; thus B/A and 1/A are meaninglessexpressions. If none of the operations of addition, subtraction, dot product,or cross product is imposed on A and B , the entity A B is called a dyad.In the language of tensor analysis, a dyad is a tensor of second rank, whileall vectors are tensors of first rank.

81. FundamentalsOperation of Three VectorsFor three vectors A , B , and C , we haveC · (A B) A · (B C) B · (C A)(1.1.7) Cx Cy Cz Ax Ay Az Bx By Bz Ax Ay Az Bx By Bz Cx Cy Cz B B B C C C A A A xyzxyzxyzC (A B) x̂ [Cy (Ax By Ay Bx ) Cz (Az Bx Ax Bz )] ŷ [Cz (Ay Bz Az By ) Cx (Ax By Az By )] ẑ [Cx (Az Bx Ax Bz ) Cy (Ax By Ay Bx )] (x̂Ax ŷAy ẑAz )(Cx Bx Cy By Cz Bz ) (Cx Ax Cy Ay Cz Az )(x̂Bx ŷBy ẑBz ) A(C · B) (C · A)B(1.1.8)Notice that the vector C (A B) is perpendicular to C and lies in theplane determined by A and B .Operation with the del OperatorThe del operator is a vector differential operator written as x̂ ŷ ẑ x y zThe following can be proved in Cartesian coordinates or in vector form: · (E H) H · ( E) E · ( H)(1.1.9) · ( A) 0(1.1.10) ( Φ) 0(1.1.11) ( E) ( · E) 2 E(1.1.12)where 2 · 2 2 2 x2 y 2 z 2(1.1.13)is the Laplacian operator in the rectangular coordinate system.Pierre-Simon Laplace (28 March 1749 – 5 March 1827)Pierre-Simon Laplace was appointed to a chair of mathematics at theÉcole Militaire in Paris at the age of 19. During the French Revolution hehelped to establish the metric system. The Laplace equation 2 · Φ 0 waspublished in 1813.

1.1 Maxwell’s Theory9Gradient of a ScalarWhen the del operator operates on a scalar function Φ(x, y, z), the result is a vector Φ x̂ Φ ŷ Φ ẑ Φ x y z(1.1.14)called the gradient of Φ(x, y, z). The differential form of the gradient of Φ as definedstates that Φ Φ Φ ŷ lim ẑ lim y 0 y z 0 z x·µ¶ x1 xΦ(x , y, z) Φ(x , y, z) x̂ lim x 0 x22·µ¶ 1 y yΦ(x, y , z) Φ(x, y , z) ŷ lim y 0 y22·µ¶ 1 z zΦ(x, y, z ) Φ(x, y, z ) ẑ lim z 0 z22 Φ x̂ lim x 0(1.1.15)When Φ(x, y, z) Φ(x) is a function of x only, Φ(x) is a vector pointing in thedirection of increasing x with the magnitude equal to the slope of the function atx.EXAMPLE 1.1.1 Electric field vector as gradient of a potential function.When there is no time variation, we may write the electric field vector E asE Φ(E1.1.1.1)and call Φ a potential function. As the gradient Φ points in the direction ofincreasing potential Φ, the electric field E points from high potential towards lowpotential, similar to water flowing from a high altitude to lower ground.Giving the potential of a point charge Q isΦ Q4πrthe electric field is QΦ r4πr2Thus the electric field points from high potential to low potential.E —END OF EXAMPLE 1.1.1—

101. FundamentalsDivergence of a VectorThe divergence of a vector function is a scalar, defined as · D x̂· (x̂Dx ŷDy ẑDz ) ŷ ẑ x y z Dx Dy Dz x y z(1.1.16)z(x0 , y0 , z0 ) z x yyxFigure 1.1.2 Differential volume x y z .Consider a differential volume with sides x, y, z centered around apoint (x0 , y0 , z0 ) [Fig. 1.1.2]. The divergence as defined states that 1 x x ·D lim y z Dx (x0 , y0 , z0 ) Dx (x0 , y0 , z0 ) x 0 x y z22 y 0 z 0 y y z x Dy (x0 , y0 , z0 ) Dy (x0 , y0 , z0 )zz z z) Dz (x0 , y0 , z0 ) x y Dz (x0 , y0 , z0 22(1.1.17)

1.1 Maxwell’s Theory11Gauss Theorem or Divergence TheoremThe first term in the braces is equal to the field component Dx at thesurface at x x0 x2 multiplied by the surface area y z . We define asurface normal vector dS pointing outward of the volume such that at the xsurface at x x0 x2 , dS x̂ y z and at the surface at x x0 2 ,dS x̂ y z . Then the negative sign in the second term is due to D dotmultiplied by dS . All six terms account for the six differential areas boundingthe differential volume V x y z with a surface normal dS . We thusexpress the divergence of D as · D lim V 01 dS · D V(1.1.18)Applying (1.1.18) to a large volume V containing an infinite number of suchinfinitesimal differential volumes [Fig. 1.1.3], we note that integrating the divergence over the volume surfaces shared by adjacent differential volumes willhave no contribution because the surface normals point in opposite directionsand thus cancel. The result is the divergence theorem or Gauss theoremVSFigure 1.1.3 Derivation of divergence theorem.dV · D dS · DV(1.1.19)SThe divergence theorem states that the volume integral of the divergence ofthe vector field D is equal to the total outward flux D through the surfaceS enclosing the volume.

121. FundamentalsCurl of a VectorThe curl of a vector field H is a vector defined as H x̂ x̂ ŷ ẑ x y z Hz Hy ŷ y z H x̂ xHxŷ yHy Hx Hz ẑ z x ẑ z Hz Hy Hx x y (1.1.20)Consider a differential volume of sides x, y, z centered around a point(x0 , y0 , z0 ) . In the Cartesian coordinate system, the differential form of thecurl of H as defined states that H lim x 0 y 0 z 0 1x̂ x 1 ŷ y x xH(x0 , y0 , z0 ) H(x0 , y0 , z0 )22 y y, z0 ) H(x0 , y0 , z0 )H(x0 , y0 22 1 z z) H(x0 , y0 , z0 ) ẑ H(x0 , y0 , z0 z221 lim x 0 x y z y 0 z 0 x̂ x z x y y z z zHx (x0 , y0 , z0 ) Hx (x0 , y0 , z0 )22 x x, y0 , z0 ) Hz (x0 , y0 , z0 )Hz (x0 22 ẑ y z x z z zHy (x0 , y0 , z0 ) Hy (x0 , y0 , z0 )22 ŷ x y y yHz (x0 , y0 , z0 ) Hz (x0 , y0 , z0 )22 x xHy (x0 , y0 , z0 ) Hy (x0 , y0 , z0 )22 y yHx (x0 , y0 , z0 ) Hx (x0 , y0 , z0 )22(1.1.21)

1.1 Maxwell’s Theory13Stokes TheoremThe ẑ component of (1.1.21) is Hy Hx x y 1 x x lim y Hy (x0 , y0 , z0 ) Hy (x0 , y0 , z0 ) x 0 x y22 y 0ẑ · ( H) ( H)z y y x Hx (x0 , y0 , z0 ) Hx (x0 , y0 , z0 )22The first term in the bracket is equal to the component Hy at x x0 x2multiplied by the differential length y . We define a vector differential lengthdl [Fig. 1.1.4] such that for the side y at x x0 x2 , dl ŷdy ; for ythe side x at y0 2 , dl x̂dx ; for the side y at x x0 x2 ,dl ŷdy ; and for the side x at y y0 y,dl x̂dx.Ifweuse2the fingers of the right hand to trace the direction of dl along the loop, theright-hand thumb points in the surface normal direction ẑ . Thusẑ y(x0 , y0 )dl x̂dxdl ŷdyC xFigure 1.1.4 Derivation of ẑ -component of the curl of a vector field.ẑ · ( H) lim x 0 y 01 Sdl · H(1.1.22)Cwhere C denotes the contour circulating the area S x y . Similarresults are derivable for the x̂ and ŷ components of H . For a differentialarea S with a surface normal in the direction of ŝ , we haveŝ · ( H) lim S 01 Sdl · HC(1.1.23)

141. FundamentalsWe now apply (1.1.21) to an open surface S , subdivide into N differentialareas [Fig. 1.1.5]. For a differential area Sj bounded by a contour Cj andwith a surface normal sˆj , we have S j sˆj Sj and S j · ( H)j dl · HCjAdding the contributions of all N differential areas [Fig. 1.1.5], we findNlim Sj 0N j 1 S j · ( H)j dl · HC SCCSCFigure 1.1.5 Derivation of Stokes’ theorem.Since the common part of the contours in two adjacent elements is traversedin opposite directions by the two contours, the net contribution of all thecommon parts in the interior sums to zero and only the contribution from theexternal contour C bounding the open surface S remains in the line integralon the right-hand side. The left-hand side becomes a surface integral, and theresult is Stokes’ theorem:dS · ( H) dl · H(1.1.24)CStokes’ theorem states that the surface integral of the curl of the vector fieldH over an open surface S is equal to the closed line integral of the vectoralong the contour enclosing

The first version of the book was published in 1975 by Wiley Interscience, New York, entitled Theory of Electromagnetic Waves, which was based on my 1968 Ph.D. thesis, where the concept of bian-isotropic media was introduced. The book was expanded and published in 1986 with the present title and its second edition appeared in 1990.