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Electromagnetic Theory: Maxwell's Equations & Wave Propagation
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Chapter 1 Basic Electromagnetic Theory • Experimental laws in electromagnetism • Maxwell’s hypothesis of displacement current • Basic Maxwell’s equations in integral form • Basic Maxwell’s Maxwell s equations in differential form • Maxwell’s equations in material media • Complex Maxwell’s equations • Boundary conditions • Wave equations and Helmholtz’s equations • Poynting’s theorem Basic Experimental Laws for Electromagnetic Phenomena Charles Augustin de Coulomb (1736 - 1806) Basic Experimental Laws for Electromagnetic Phenomena Columb's law (1785), the beginning of quantitative research in electricity and magnetism Faraday's concept of field (1831) : electric field strength, g , an instantaneous vector functions of position x and time t • Linear superposition • The distance dependence: inverse square law If the distance dependence was During the age of Maxwell: Recent experiment: Gauss's law for electric fields Electric fields can arise from electric charges Basic Experimental Laws for Electromagnetic Phenomena Columb's Co u b s law a ((1785) 85) Faraday's a aday s co concept cept o of field e d ((1831) 83 ) : magneic field strength, instantaneous vector function Gauss's law for magnetic fields Later, people knew that in Columb Later Columb’ss experiment experiment, magnetic poles of magnet are the effects of molecular current loops, which can be recognized as charge pairs or magnetic dipoles, but not the effects of single magnetic charges. Dirac postulated theoretically in 1931 that magnetic fields can also l arise i ffrom magnetic i charges h called ll d magnetic i monopoles. But such particles have not been observed experimentally to date date, so that Gauss's law for magnetic fields becomes Basic Experimental Laws for Electromagnetic Phenomena Hans Christian Oersted (1777 - 1851) Basic Experimental Laws for Electromagnetic Phenomena André Marie Ampère 1775 - 1836 Basic Experimental Laws for Electromagnetic Phenomena Oersted’s observation (1819), Ampere’s law (1820), the magnetic ti effects ff t off electric l t i currents t Ampere force Lorentz force Biot-Savart law (1820) : magnetic induction • Linear superposition • The distance dependence: inverse square law Ampere's circuital law Gauss's law for magnetic induction or continuity Law of magnetic i flux fl Magnetic monopole/magnetic charge do not exist, magnetic fields can arise from electric currents (magnetic dipoles/current loops exist) Basic Experimental Laws for Electromagnetic Phenomena Basic Experimental Laws for Electromagnetic Phenomena before Faraday Electric field magnetic current do not exist Static electric field is conservative Columb's law Gauss's law Magnetic g field Ampere’s law and Biot-Savart law magnetic charge do not exist continuity Law of magnetic flux Ampere's circuital law Basic Experimental Laws for Electromagnetic Phenomena F d ' law Faraday's l off induction i d ti (1831) 法拉第1831年向英国皇家学会报告了他的电磁感应实验结果 法拉第1831年向英国皇家学会报告了他的电磁感应实验结果： “把一根203英尺的铜丝缠绕在一个大木块上，再把一根203英 尺的铜丝缠绕在前 线圈每转的中间 两线间用绝缘线隔开 尺的铜丝缠绕在前一线圈每转的中间。两线间用绝缘线隔开， 不让金属有一点接触。一根螺旋线上接有一个电流计，另一根 螺旋线则连接在一套电池组上。 螺旋线则连接在 套电池组上。… 当电路刚接通时，电流击上 发生突然的极微小的效应；当电路忽然断开时，也发生同样的 微弱的效应。但当电流不断地通过一根螺旋线时，电流计上没 有什么表现。” Basic Experimental Laws for Electromagnetic Phenomena F d ' law Faraday's l off induction i d ti (1831) Michael Faraday 1791-1867 Basic Experimental Laws for Electromagnetic Phenomena Faraday's law of induction (1831), the beginning of the investigation g on time-dependent p fields Electric fields can arise from electric charges, it can also arise from time-varying magnetic fields Basic Experimental Laws for Electromagnetic Phenomena James Clerk Maxwell 1831-1879 Basic Experimental Laws for Electromagnetic Phenomena Electromagnetic equations before Maxwell Electric fields can arise from time-varying magnetic fields magnetic fields can arise from electric currents Electric fields can arise from electric charges Magnetic monopoles/magnetic charges do not exist Maxwell’s Hypothesis of Displacement Current Ampere's circuital law For the surface S1 surrounded by the closed counter l For the surface S2 surrounded by the same closed counter l ?? Equation of continuity for the volume ol me V enclosed b by closed surface S2 − S1 Applying Gauss Gauss's s law We have Since then Maxwell‘s hypothesis yp of displacement p current Finally Basic Maxwell Equations in Time-Domain or Instantaneous Maxwell Equations in Integral Form (1-1) (1-2) (1-3) (1 4) (1-4) Equation of continuity in integral form (1-9) Definitions (1-11) (1-12) : in C(Coulomb) : in A(Ampere) = C/s : in C/cm3 : in A/cm2 (1 13) Electric (1-13) El t i fforce : in N(Newton)/C = V(Volt)/m (1-15) Lorentz force (1-14) Ampere force : in N/Am = Wb(Weber)/m2 Vectors and Vector Functions Th position The iti vector t or coordinate di t vector t off a point, i t x : Unit vectors in the direction of coordinates x, y and z x, y and z : components of the vector x Vectors and Vector Functions Th position The iti vectors t off ttwo points, i t x and d x’ Vectors and Vector Functions Th di The distance t vector t between b t two t points, i t x and d x’ A vector function f off x and x’ Vectors and Vector Functions V t function Vector f ti off position iti x For example, the vector function of Coulomb electric field Vectors and Vector Functions is a vector function of position vector x Components of a vector function are components of the vector function They are scalar function of position vector x (x, y, z) Vectors and Vector Functions An arbitrary vector function of position (vector) x and time (scalar) t are scalar functions f off position x and time t Algebraic Formulas for Vector Functions Addition Scalar product or dot product (A dot B) Vector product or cross product (A cross B) Line Element, Surface Element and Volume Element Line element dl, a vector Surface element dS, a vector Volume element dV, a scalar Surface S and its enclosed l d counter t l Volume V and its enclosed l d surface f S The Flux and the Circulation The flux of a vector function on a surface S The flux of a vector function on a closed surface S The circulation of a vector function on a counter l Vector Analysis The gradient of a scalar function Gradient – the directional derivative , taking in the direction of th maximum the i value. l (G (Gradient di t ϕ) In rectangular coordinates The g gradient of a scalar function is a vector function. Vector Analysis The divergence of a vector function Divergence – Volume density of the outward flux through a closed surface (Divergence A) In rectangular coordinates Th divergence The di off a vector t function f ti is i a scalar l function. f ti Gauss’ theorem Volume V is enclosed by a closed surface Vector Analysis The curl of a vector function Curl (Rotation) – Surface density of the circulation along a closed counter taking in the direction of the maximum value counter, value. The n component of the curl, Surface density of the circulation, where n is the direction of the normal unit vector t off the th surface f element l t ΔS Curl A In rectangular coordinates The curl of a vector function is still a vector function. Stoke’s Stoke s theorem Surface S is enclosed by a closed counter l Vector Analysis The scalar Laplacian of a scalar function Laplacian – The divergence of the gradient of a scalar function (Laplacian ϕ) In rectangular coordinates The Laplician of a scalar function is still a scalar function. Vector Analysis Diff Differential ti l operations ti off scalar l and d vector t functions f ti Operation Symbol Object Result Gradient Scalar Vector Divergence Vector Scalar Curl Vector Vector Laplacian Scalar Scalar Maxwell equations in Differential Form The divergence equations Gauss's law for electric fields and magnetic fields Applying the definition of divergence We have and We g get divergence g equations: q Similarly Equation of continuity becomes: Maxwell equations in Differential Form The curl equations Circuital equations Applying the definition of curl We have and We get and Basic Maxwell Equations in Time-Domain or Instantaneous Maxwell Equations In integral form In derivative form (1-1) (1-5) (1-2) (1-6) (1-3) (1 7) (1-7) (1 4) (1-4) (1-8) (1 8) (1-9) (1-10) Equation of continuity Equation of continuity is not independent independent, it can be derived from the Maxwell equations (Exercise) Basic Maxwell Equations in Time-Domain or Instantaneous Maxwell Equations (1-5) (1-6) (1-7) (1-8) The most important equations are the two curl equations with time dependent dependent. The interactions between electric field and magnetic field in time-varying y g state are described by y the two curl equations. q From this set of equations, J.C.Maxwell derived wave equations, and find that the speed of electromagnetic wave is just equal to th experimental the i t l value l off th the speed d off lilight. ht J.C.Maxwell predicted that time-varying electromagnetic fields exist in the form of waves and light is electromagnetic wave in a special frequency band. But unfortunately, he did not see the experimental verification of his prediction. Electromagnetic Waves 25 years after the Maxwell’s prediction, 9 years after the death of J.C.Maxwell, in 1888, H.R.Hertz, did the experiment of electromagnetic wave wave. and Maxwell’s Maxwell s hypothesis is verified verified. Heinrich Rudolph Hertz 1857 - 1894 Application of Electromagnetic Waves G. Marconi sent and received his first radio signal in Italy in 1895. Guglielmo Marconi (1874 – 1937) with his early wireless apparatus in 1896. Application of Electromagnetic Waves A.S.Popov did the experiment of telecommunication by electromagnetic wave in Russia in May, 1895, independently. Aleksandr Stepanovic Popov 1859 - 1906 7 May 1895 250 mètres Maxwell’s Equations in Material Media • Classification of charges g and currents • Polarization and magnetization vectors • Electric induction and magnetic field strength • Maxwell’s Equations in material media • Simple medium • Nondispersive and dispersive media • Linear Li and d nonlinear li media di • Isotropic and anisotropic media • Biisotropic and bianisotropic media Maxwell’s Equations in Material Media Charges and Currents In basic Maxwell equations, all kinds of charges and currents are included Divergence equation for Free charge density (charges in space and on conductor) Polarization (bounded) charge density, electric dipole Curl equation for Free current density, density movement of free charges, charges including conduction current, convection current, and independent source Polarization current density, movement of bounded charges Magnetization (molecular) current density density, molecular current loop (orbital and spin), magnetic dipole Maxwell’s Equations in Material Media Maxwell equations q become (1-5) (1-17) (1-6) (1-18) ( ) (1-7) (1-8) (1 19) (1-19) (1-20) : Polarization vector: the volume density of the electric dipole moment : Magnetization vector: the volume density of the magnetic dipole moments Maxwell’s Equations in Material Media (1-17) ( (1-18) ) (1-19) (1-20) (1-22) (1-23) (1-25) We have (1-26) (1 27) (1-27) (1-28) (1-29) Maxwell’s Equations in Material Media ((1-26) 6) (1-27) (1-28) (1-29) Definition of electric induction Definition of magnetic g field strength g (1-30) (1-31) Maxwell equations become (1-32) (1-33) (1-34) (1 34) (1-35) Maxwell’s Equations in Material Media Maxwell equations taking account of the effect of polarization and magnetization In integral form Here, In differential form are free charges and free currents Maxwell’s Equations in Material Media Constitutional equations: Relations among and 1. Simple media Stable nondispersive, Stable, nondispersive linear, linear isotropic media is linearly proportional to , and (1 36) (1-36) : electric susceptibility is linearly proportional to (1 37) (1-37) : magnetic susceptibility : permittivity : permeability Constitutional equations for simple media (1-38) (1-39) and are content scalars with respect to Time, frequency and field strength Maxwell’s Equations in Material Media Constitutional equations for simple media (1-39) (1-38) M Maxwell ll equations ti for f simple i l media di In derivative form (1-42) (1-43) (1 44) (1-44) ((1-45)) : Conduction current, Ohm’s law : Free current, except conduction current Maxwell’s Equations in Material Media N di Nondispersive i and d dispersive di i media di Nondispersive media: the responses of polarization and magnetization are instant instant. The electric and magnetic inductions depend upon only the present value of electric and magnetic g field strengths g Dispersive media: the responses of polarization and magnetization are not instant. The electric and magnetic inductions depend upon not only the present value of electric and magnetic field strengths but also the time derivatives of all orders of them. (1-52) (1-53) Maxwell’s Equations in Material Media Linear and nonlinear media Linear media: the relation between electric and magnetic inductions and electric and magnetic field strengths are linear. This means that the constitutive parameters μ and ε are constants and independent upon the field strengths. Nonlinear media: the relation between electric and magnetic inductions and electric and magnetic field strengths are nonlinear functions. This means that the constitutive parameters μ and ε are not constants and depend upon the field strengths. Maxwell’s Equations in Material Media Isotropic and Anisotropic media Isotropic media: The electric field vector and the polarization vector are in the same direction, and the susceptibilities in all directions are equal, so are the magnetic field and the magnetization vectors vectors. In isotropic media media, electric and magnetic susceptibilities are scalars. Anisotropic media: The electric field induces polarization in a direction other than that of the electric field, and the susceptibilities are different in different directions directions, the medium is known as an electric anisotropic medium. Similar behavior in magnetization is called magnetic anisotropic medium. In anisotropic media, the electric and magnetic susceptibilities are no longer scalars but tensors of rank 2 or matrices Maxwell’s Equations in Material Media Anisotropic media Electric anisotropic medium: The component of polarization d depends d nott only l on th the same componentt off electric l t i fifield ld b butt also the other components of the field. In the tensor form: Maxwell’s Equations in Material Media Anisotropic media Electric anisotropic medium, electric susceptibility tensor (1-55) (1-54) Magnetic anisotropic medium, magnetic susceptibility tensor (1-57) (1-56) Electric anisotropic medium (1-58) (1 63) (1-63) (1 61) (1-61) Magnetic anisotropic medium (1 59) (1-59) (1-64) (1-62) Maxwell’s Equations in Material Media Anisotropic media Electric anisotropic medium (1-63) (1-61) Magnetic anisotropic medium (1-64) (1-62) (1 62) Unit tensor: (1-60) (1 60) Permitivity tensor: (1-65) Permeability tensor: (1 66) (1-66) Maxwell’s Equations in Material Media Bi i t Bianisotropic i and d bianisotropic bi i t i media di In biisotropic and bianisotropic media, cross coupling between the electric and magnetic fields exists. When placed in an electric or magnetic field, a biisotropic p or bianisotropic p medium becomes both polarized and magnetized. Bianisotropic media Bianisotropic media Micro-helix model of bianisotropic and bianisotropic media Electric dipole Magnetic dipole Micro-helix Maxwell Equations in Frequency-Domain p Maxwell Equations q or Complex Time harmonic (sinusoidal) scalar functions Time harmonic (sinusoidal) vector functions Complex vector functions Instantaneous vector functions and complex vector functions Derivative of the sinusoidal function Maxwell Equations in Frequency-Domain p Maxwell Equations q or Complex Maxwell Equations I t t Instantaneous (ti (time-domain) d i ) C Complex l (frequency-domain) (f d i ) Equation of continuity Maxwell Equations in Frequency-Domain p Maxwell Equations q or Complex Complex Maxwell Equations In integral form In derivative form E Equation ti off continuity ti it Maxwell Equations in Frequency-Domain p Maxwell Equations q or Complex Constitutional equations for simple media Complex Maxwell equations for uniform simple media Complex Maxwell equations in source-free region region, ρ =0, =0 J =0, =0 (equations for wave propagation) Maxwell Equations in Frequency-Domain p Maxwell Equations q or Complex Complex permittivity and permeability Maxwell Equations in Frequency-Domain p Maxwell Equations q or Complex Polarization loss Magnetization loss Electric and magnetic loss angles Loss tangents Constitutional equations Maxwell equations Maxwell Equations in Frequency-Domain p Maxwell Equations q or Complex Loss angle for conducting media Loss tangent for polarization and conduction L Loss tangent t t for f conduction d ti Equivalent Magnetic Charge and Magnetic Current Maxwell Equations q in Duality y Form Instantaneous (time-domain) Complex (frequency-domain)
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