Electromagnetic Field Theory 1 (fundamental relations and definitions)

Transcription

1 (fundamental relations and definitions) Lukas Jelinek Department of Electromagnetic Field Czech Technical University in Prague Czech Republic Ver. 216/12/14

2 Fundamental Question of Classical Electrodynamics A specified distribution of elementary charges is in state of arbitrary (but known) motion. At certain time we pick one of them and ask what is the force acting on it. Rather difficult question will not be fully answered GENERAL INTRO 2/ XXX

3 Elementary Charge Coulomb e = C smallest known amount of charge 11 significant digits!!! As far as we known, all charges in nature have values ± NeN, Z GENERAL INTRO 3/ XXX

4 Charge conservation Amount of charge is conserved in every frame (even non-inertial). Neutrality of atoms has been verified to 2 digits GENERAL INTRO 4/ XXX

5 Continuous approximation of charge distribution Volumetric density of charge C m 3 Surface density of charge C m 2 Line density of charge C m 1 ρ( r) σ( r) τ( r) Q = dv = ds = dl V S l Net charge C Continuous approximation allows for using powerful mathematics GENERAL INTRO 5/ XXX

6 Fundamental Question of Electrostatics There exist a specified distribution of static elementary charges. We pick one of them and ask what is the force acting on it. This will be answered in full details ELECTROSTATICS 6/ XXX

7 Coulomb( s) Law Measuring charge C Source charge C Radius vector of the measuring charge m Radius vector of the source charge m ( ) F r = 4πε ( r r ) qq r r 3 Permittivity of vacuum Farad Force on measuring charge N Speed of light Permeability of vacuum ε c 1 12 = = µ c 1 = m s 7 1 = 4 1 H m µ π Henry F m 1 ELECTROSTATICS 7/ XXX

8 Coulomb( s) Law + Superposition Principle ( ) F r = q 4πε ( ) qn n r r 3 n r r n Entire electrostatics can be deduced from this formula ELECTROSTATICS 8/ XXX

9 Electric Field F( r) =qe( r) ( ) E r = 1 4πε ( ) qn n r r 3 n r r n Intensity of electric field V m 1 Force is represented by field entity generated by charges and permeating the space ELECTROSTATICS 9/ XXX

10 Continuous Distribution of Charge ( ) E r = 1 4πε ( ) qn n r r 3 n r r n ( ) E r = 1 4 V πε ρ ( r )( r r ) r r 3 dv Continuous description of charge allows for using powerful mathematics ELECTROSTATICS 1/ XXX

11 Continuous Description of a Point Charge Dirac s delta function Defining property of Dirac s delta function ρ ( r) = qδ n ( r r n) F( r ) = n F( r) δ( r rn )d n V V ELECTROSTATICS 11/ XXX

12 Gauss( ) Law Mind the orientation of the surface Total charge enclosed by the surface C ρ( ) 1 E( r ) E d S ρ( r ) = r ε S Q = dv = ε ε V Differential law (local) Integral law (global) ELECTROSTATICS 12/ XXX

13 Rotation of Electric Field E = E d l = l Differential law (local) Integral law (global) ELECTROSTATICS 13/ XXX

14 Various Views on Electrostatics Integral laws of electrostatics Differential laws of electrostatics Coulomb s law Q E d S = ε S E d l = l E = ( ) ρ E( r) = r ε ( ) E r = 1 πε 4 V ρ ( r )( r r ) r r 3 dv The physics content is the same, the formalism is different. ELECTROSTATICS 14/ XXX

15 Electric potential Electric potential Defined up to arbitrary constant E = ( ) = ϕ( r) E r ϕ ( r) 4 V ( r ) 1 ρ = dv K πε + r r Scalar description of electrostatic field ELECTROSTATICS 15/ XXX

16 Voltage Potential difference is a unique number Voltage V Work necessary to take charge q from point A to point B B A ( ) ϕ( ) E d l = ϕ B A = U W B = F d l = qu A Voltage represents connection of abstract field theory with experiments ELECTROSTATICS 16/ XXX

17 Electrostatic Energy Energy is carried by charges W 1 = ρr ϕr 2 V ( ) ( ) dv Energy is carried by charges and fields W W = = 1 8πε 1 8πε ij, j i V V i qq i j r r j ( r) ( r ) ρ ρ r r dv dv W 1 = ε E r 2 V ( ) 2 dv Be careful with point charges (self-energy) Energy is carried by fields ELECTROSTATICS 17/ XXX

18 Electrostatic Energy vs Force Energy of a system of point charges Coulomb s law W = 1 8πε qq i j ij, r i r j j i ( ξ) F r = W = ξ q ξ 4πε ( ξ j) q r r j j r r j ξ ξ j 3 Electrostatic forces are always acting so to minimize energy of the system ELECTROSTATICS 18/ XXX

19 Electric Stress Tensor Total electric force acting in a volume Stress tensor ( ) ( ) dv ε F = ρ r E r = T d S V S 1 T = EE IE 2 2 All the information on the volumetric Coulomb s force is contained at the boundary ELECTROSTATICS 19/ XXX

20 Ideal Conductor Ideal conductor contains unlimited amount of free charges which under action of external electric field rearrange so as to annihilate electric field inside the conductor. In 3D, the free charge always resides on the external bounding surface of the conductor. In 1D and 2D it is not so Generally free charges in conductors move so as to minimize the energy ELECTROSTATICS 2/ XXX

21 Boundary Conditions on Ideal Conductor Inside conductor E( r) ϕ( r) = = const. Just outside conductor ( ) ( ) ϕ( r) ( ) ( ) σ ϕ( r) n r E r = = const. σ n r E r = = ε n ε Potential is continuous across the boundary Surface charge residing on the outer surface of the conductor Outward normal to the conductor Normal derivative ELECTROSTATICS 21/ XXX

22 Capacitance of a System of N conductors Self and mutual capacitances F Electrostatic energy Capacitances depend solely on geometry and position of conductors Q i = j Cϕ ij j W 1 = Cϕϕ 2 ij, ij j i Electrostatic system is fully characterized by capacitances (we know the energy) ELECTROSTATICS 22/ XXX

23 Capacitance of a System of two conductors Capacitance Potential difference between conductors Q = CU 1 W = CU 2 2 Charge on positively charged conductor ELECTROSTATICS 23/ XXX

24 Poisson( s) equation ϕ ( r) ( ) ρ = r ε The solution to Poisson s equation is unique in a given volume once the potential is known on its bounding surface and the charge density is known through out the volume. ELECTROSTATICS 24/ XXX

25 Laplace( s) equation ϕ( r) = The solution to Laplace s equation is unique in a given volume once the potential is known on its bounding surface. ELECTROSTATICS 25/ XXX

26 Mean Value Theorem Center of the sphere Only for spheres containing no charge ϕ 1 = 4πR ( rcenter ) 2 ϕ( r) sphere ds Radius of the sphere The solution to Laplace s equation posses neither maxima nor minima inside the solved volume. ELECTROSTATICS 26/ XXX

27 Earnshaw( s) Theorem Consequence of mean value theorem A charged particle cannot be held in stable equilibrium by electrostatic forces alone. Mind that the solution to Laplace s equation posses neither maxima nor minima inside the solved volume. This means that charged particle will always travel towards the boundary. ELECTROSTATICS 27/ XXX

28 Image Method When solving field generated by charges in the presence of conductors, it is sometimes possible to remove the conductor and mimic its boundary conditions by adding extra charges to the exterior of the solution volume. The unicity theorem claims that this is a correct solution. Image method always works with planes and spheres. ELECTROSTATICS 28/ XXX

29 Separation of Variables Constants determined by boundary conditions ϕ( r ) = ϕ ( r ) =X ( xy ) ( y) Z ( z) ϕ( r) C ϕ ( r) ijk i j k = ijk ijk ijk Semi-analytical method for canonical problems ELECTROSTATICS 29/ XXX

30 Finite Differences ( x + hyz) ( 1) ϕ,, ϕ i + jk ϕ 2ϕ + ϕ ϕ 2ϕ + ϕ ϕ 2ϕ + ϕ ϕ( r) h h h ( i+ 1) jk ijk ( i 1) jk i( j+ 1) k ijk i( j 1) k ijk ( + 1) ijk ijk ( 1) ϕ( r) = ϕ ijk = ϕ + ϕ + ϕ + ϕ + ϕ + ϕ ( i+ 1) jk ( i 1) jk i( j+ 1) k i( j 1) k ijk ( + 1) ijk ( 1) 6 Approximation by a system of linear algebraic equations Mind the mean value theorem Powerful numerical method for closed problems ELECTROSTATICS 3/ XXX

31 Method of Moments Assumed to be known in volume where the charge resides Distribution of charge is unknown Simple functions for which the potential integral can be easily evaluated ϕ ( r) = 1 4 V πε ρ ( r ) r r dv ρ ( r) αρ n n ( r) n V ( r) ( r ) 1 ρ ρ m n ρ ( r) ϕ( r) dv = α dv dv m n n 4 πε r r V V Known Approximation by a system of linear algebraic equations Known Powerful numerical method for open problems ELECTROSTATICS 31/ XXX

32 Dielectrics Material in which charges cannot move freely Clusters are electrically neutral Charges are forming clusters (atoms, molecules) Under influence of electric field the clusters change shape or rotate P r C m Electric field induces electric dipoles with density ( ) 2 Number of dipoles in unitary volume ELECTROSTATICS 32/ XXX

33 Electric Field of a Dipole Two opposite charges very close to each other r r r r center 1 2 ϕ ( r) ( ) 1 q 1 q 1 p r r = 4πε 4πε 4πε r r1 r r2 r r center center 3 ( ) ρ( ) p = q r r = r r 1 2 d V V Electric dipole moment C m Formula for two opposite charges General formula ELECTROSTATICS 33/ XXX

34 Field Produced by Polarized Matter ϕ ( r) P( r ) ( r r ) ( ) ( ) V P r P r S = d = d dv 4πε 4πε 4πε V r r S r r V r r Only apply at infinitely sharp boundary (unrealistic) Potential of volumetric charge density This formula holds very well outside the matter and, curiously, it also well approximates the field inside ELECTROSTATICS 34/ XXX

35 Electric Displacement Electric displacement C m 2 Dr ( ) =ρ( r) ( ) = ( ) + ( ) ε Dr E r P r ( ) D d S = ρ r dv = Q S V Only free charge (compare to divergence of electric field) ELECTROSTATICS 35/ XXX

36 Linear Isotropic Dielectrics ε Relative permittivity r ( r) = 1+ χ ( r) e P( r) =εχ ( r) E( r ) Dr ( ) = εε ( r) E( r) = ε( r) E( r) e r Electric susceptibility Permittivity F m 1 All the complicated structure of matter reduces to a simple scalar quantity ELECTROSTATICS 36/ XXX

37 Fields in Presence of Dielectrics 1/2 Analogy with electric field in vacuum can only be used when entire space is homogeneously filled with dielectric. ( ) ε( r) E( r) Dr = Inequality is due to boundaries Analogy with vacuum can only be used when space is homogeneously filled with dielectric ELECTROSTATICS 37/ XXX

38 Fields in Presence of Dielectrics 2/2 ( ) ( ) ϕ( ) E r = E r = r ( r) ( r) ρ( r) ε ϕ = ϕ ( r) ρ( ) = r ε Not a function of coordinates Poisson s equation holds only when permittivity does not depend on coordinates ELECTROSTATICS 38/ XXX

39 Dielectric Boundaries ( ) ( ) ( ) ϕ ( ) ϕ ( ) n r = = E r E r 1 2 r r 1 2 ( r) ϕ ( r) ϕ1 2 n( r) ε1 1( ) ε2 2( ) E r E r = σ ε ε = σ 1 2 n n Normal pointing to region (1) Both conditions are needed for unique solution ELECTROSTATICS 39/ XXX

40 Electrostatic Energy in Dielectrics W 1 = ε E r 2 V ( ) 2 dv W 1 = E r Dr 2 V ( ) ( ) dv ELECTROSTATICS 4/ XXX

41 Forces on Dielectrics This only holds when charge is held constant 1 2 1Q W = CU = 2 2C W 1 = E r Dr 2 V 2 ( ) ( ) dv ( ) = W ξ ξ F r ELECTROSTATICS 41/ XXX

42 Electric Current Current density A m 2 Charge C Velocity of charge m s 1 N ( r r ) v ( ) (, ) = δ ( ) J r t q t t k= 1 k k k Volumetric density represented by Dirac delta m 3 Charges in motion are represented by current density CURRENT 42/ XXX

43 Local Charge Conservation N ρ J ( r, t) = qδ k ( r rk ( t) ) = t t k= 1 ( r, t) Charge is conserved locally at every space-time point CURRENT 43/ XXX

44 Global Charge Conservation When charge leaves a given volume, it is always accompanied by a current through the bounding envelope S (, ) J r t d S ( ) Q t = t Charge can neither be created nor destroyed. It can only be displaced. CURRENT 44/ XXX

45 Stationary Current When charge enters a volume, another must leave it without any delay J ( r) = S ( ) d = J r S There is no charge accumulation in stationary flow CURRENT 45/ XXX

46 Ohm( s) Law Conductivity S m 1 J ( r) =σ( r) E( r) This simple linear relation holds for enormous interval of electric field strengths CURRENT 46/ XXX

47 Electromotive Force Stationary flow of charges cannot be caused by electrostatic field. The motion forces are non-conservative, are called electromotive forces, and are commonly of chemical, magnetic or photoelectric origin. l ( ) d E r l ( ) d = E r l l For curves passing through sources of electromotive force For curves not crossing sources of electromotive force CURRENT 47/ XXX

48 Boundary Conditions for Stationary Current ( ) ( ) ( ) ϕ ( ) ϕ ( ) n r = = E r E r 1 2 r r 1 2 ( r) ϕ ( r) ϕ1 2 n( r) ε1 1( ) ε2 2( ) E r E r = σ ε ε = σ 1 2 n n ( r) ϕ ( r) ϕ1 2 n( r) σ1 1( ) σ2 2( ) E r E r = σ σ = 1 2 n n Charge conservation forces the continuity of current across the boundary CURRENT 48/ XXX

49 Electric Current Current A I S ( ) = J r d S Cross-section of current path m 2 Existence of high contrast in conductivity between conductors and dielectrics allows for well defined current paths. CURRENT 49/ XXX

50 Resistance (Conductance) Potential difference (voltage) V Resistance Ω Current A Conductance S U = RI I = GU 1 L R = = G σs Resistance of a cylinder homogeneous cylinder of conductive material Cross-section of current path m 2 Length along current path m CURRENT 5/ XXX

51 Resistive Circuits and Kirchhoff( s) Laws In a loop On a resistor At a junction U = U i electromotive U = RI I = i i i i i i Kirchhoff s laws are a consequence of electrostatics and law s of stationary current flow CURRENT 51/ XXX

52 Joule( s) Heat Power lost via conduction W Power lost on resistor W ( ) ( ) d σ( r) E( r) 2 P = E r J r V = dv V V 2 2 U P = UI = RI = R Electric field within conducting material produce heat CURRENT 52/ XXX

53 Fundamental Question of Magnetostatics There exist a specified distribution of stationary current. We pick a differential volume of it and ask what is the force acting on it. MAGNETOSTATICS 53/ XXX

54 Biot-Savart( s) Law Permeability of vacuum µ = 4π 1 H m 7 1 Measuring current element A Source current element A ( ) F r µ = 4π ( ) dv ( ) dv ( ) J r J r r r r r 3 Force on measuring current N Radius vector of the measuring current m Radius vector of the source current m MAGNETOSTATICS 54/ XXX

55 Biot-Savart( s) Law + Superposition Principle ( ) ( ) F ( r) = J ( r) dv 3 4 µπ J r r r V r r dv Entire magnetostaticscan be deduced from this formula MAGNETOSTATICS 55/ XXX

56 Magnetic Field ( ) = ( ) dv Br ( ) F r J r ( ) Br = ( ) ( ) µπ J r r r r r dv 3 4 V Magnetic field (Magnetic induction) T MAGNETOSTATICS 56/ XXX

57 Divergence of Magnetic Field Br ( ) = ( ) d = Br S S There are no point sources of magnetostaticfield MAGNETOSTATICS 57/ XXX

58 Curl of Magnetic Field Ampere( s) Law Br ( ) =µ J ( r) ( ) d Br l =µi l Total current captured within the curve A MAGNETOSTATICS 58/ XXX

59 Magnetic Vector Potential Magnetic vector potential Defined up to arbitrary scalar function B Br ( ) = Ar ( ) Ar ( ) = ( ) µ = dv ψ 4π J r + V r r ( r) Reduced description of magnetostaticfield MAGNETOSTATICS 59/ XXX

60 Poisson( s) equation ( ) J ( r) Ar = µ The solution to Poisson s equation is unique in a given volume once the potential is known on its bounding surface and the current density is known through out the volume. MAGNETOSTATICS 6/ XXX

61 Boundary Conditions Surface current on the boundary ( ) ( ) ( ) = µ ( ) n r B r B r 1 2 K r ( ) ( ) ( ) n r B r B r = 1 2 ( ) ( ) A r A r = 1 2 Normal pointing to region (1) MAGNETOSTATICS 61/ XXX

62 Magnetostatic Energy W 1 = Ar J r 2 V ( ) ( ) dv W 1 = Br 2µ V ( ) 2 dv For now it is just a formula that works it must be derived with the help of time varying fields MAGNETOSTATICS 62/ XXX

63 Magnetostatic Energy Current Circuits M = M = ( ) j j i ( i) J r J r µ dv dv 4 II r r ij ji i j π i j V V j i j i Mutual-Inductance H 1 1 W = LI + M II 2 2 N 2 i i i= 1 i j ij i j L i = Self-Inductance ( ) i i i ( i) J r J r µ dv dv 2 4 I r r π i V V i i i i H i i MAGNETOSTATICS 63/ XXX

64 Mutual Inductance Thin Current Loop Magnetic flux induced by i-th current through j-th current Wb ( ) d Φ = B r S ji i j j S j M ij = Φ I ji i MAGNETOSTATICS 64/ XXX

65 Magnetic Materials Material response is due to magnetic dipole moments Magnetic moment comes from spin or orbital motion of an electron Magnetic field tends to align magnetic moments M r A m Magnetic field induces magnetic dipoles with density ( ) 1 Number of dipoles in unitary volume MAGNETOSTATICS 65/ XXX

66 Magnetic Field of a Dipole Dipole is assumed at the origin r Ar µ = m r 4π r 3 ( ) ( ) Br ( ) µ 3 r r m m 4π r r = 5 3 m 1 = ( ) dv 2 r J r V Magnetic dipole moment A m 2 Magnetic dipole approximates infinitesimally small current loop MAGNETOSTATICS 66/ XXX

67 Field Produced by Magnetized Matter ( ) Ar M ( r ) ( r r ) ( ) d ( ) dv M r S M r 3 µ µ µ = = + dv 4π 4π 4π V r r S r r V r r Only applies at infinitely sharp boundary (unrealistic) Potential of volumetric current density This formula holds very well outside the matter and, curiously, it also well approximates the field inside MAGNETOSTATICS 67/ XXX

68 Magnetic Intensity Magnetic Intensity A m 1 H ( r) = J ( r) ( ) ( ) = H ( r) + M ( r) Br µ ( ) d H r l =I l Only free current MAGNETOSTATICS 68/ XXX

69 Linear Isotropic Magnetic Materials Relative permeability r ( r) = 1+ ( r) µ χ m M ( r) =χ m ( r) H ( r ) Br ( ) = µµ ( r) H r ( r) = µ ( r) H ( r) Magnetic susceptibility Permeability H m 1 All the complicated structure of matter reduces to a simple scalar quantity MAGNETOSTATICS 69/ XXX

70 Fields in Presence of Magnetic Material ( ) ( ) ( ) Br = Br = Ar 1 Ar = J r µ ( r) ( ) ( ) Ar ( ) = µ J ( r) ( ) Ar = Coulomb( s) gauge Not a function of coordinates Poisson s equation holds only when permittivity does not depend on coordinates MAGNETOSTATICS 7/ XXX

71 Magnetic Material Boundaries ( ) ( ) ( ) = ( ) n r H r H r 1 2 K r ( ) µ ( ) µ ( ) n r H r H r = Normal pointing to region (1) Both conditions are needed for unique solution MAGNETOSTATICS 71/ XXX

72 Magnetostatic Energy in Magnetic Material W 1 = Br 2µ V ( ) 2 dv W 1 = H r Br 2 V ( ) ( ) dv MAGNETOSTATICS 72/ XXX

73 Magnetic Materials Paramagnetic small positive susceptibility (small attraction linear) Diamagnetic small negative susceptibility (small repulsion linear) Ferromagnetic large positive susceptibility (large attraction nonlinear) MAGNETOSTATICS 73/ XXX

74 Ferromagnetic Materials Spins are ordered within domains Magnetization is a non-linear function of field intensity Magnetization curve Hysteresis, Remanence Susceptibility can only be defined as local approximation Above Curie( s) temperature ferromagnetism disappears Exact calculations are very difficult use simplified models (soft material, permanent magnet) MAGNETOSTATICS 74/ XXX

75 Faraday( s) Law Minus sign is called Lenz( s) law Φ t Time variation of magnetic flux E ( r, t) d l = (, t) d t Br S l S Br E( r, t) = t (, t) Time variation in magnetic field produces electric field that tries to counter the change in magnetic flux (electromotive force) QUASISTATICS 75/ XXX

76 Lenz( s) Law The current created by time variation of magnetic flux is directed so as to oppose the flux creating it. QUASISTATICS 76/ XXX

77 Time Varying RL Circuits In a loop At a junction U ( t i ) = U electromotive ( t) I ( t ) = i i i i ( ) = ( ) U t RI t i i I t U ( t) = L + M t ( ) I ( t) t On a resistor On an inductor Circuit laws are valid as long as the variations are not too fast QUASISTATICS 77/ XXX

78 Time Varying Potentials Potential calibration (, t) ( r, t) Ar = σµϕ (, t) = Ar (, t) Br (, t) ϕ( r, t) E r Ar = t (, t) In time varying fields scalar potential becomes redundant QUASISTATICS 78/ XXX

79 Source and Induced Currents Those are fixed, not reacting to fields (, t) (, t) (, t) (, t) (, t) H r = J r + J r = J r +σe r source induced source QUASISTATICS 79/ XXX

80 Diffusion Equation (, t) Ar Ar (, t) σµ = µ J r, source t (, t) ( t) ( t) H r H ( r, t) σµ = J r, source t ( t) ( t) E r, 1 J r, source E( r, t) σµ = ρ source ( r, t) + µ t ε t Material parameters are assumed independent of coordinates QUASISTATICS 8/ XXX

81 Maxwell( s)-lorentz( s) Equations (, t) (, t) Br (, t) (, t) t (, t) (, t) ρ( r, t) (, t) Dr H r = J r + t E r = Br = Dr = Interaction with materials (, t) = ρ( r, t) E( r, t) + J ( r, t) Br (, t) f r Equations of motion for fields Equation of motion for particles (, t) = εe( r, t ) + P( r, t) (, t) = µ H ( r, t) + M ( r, t) Dr Br ( ) Absolute majority of things happening around you is described by these equations ELECTRODYNAMICS 81/ XXX

82 Boundary Conditions ( ) ( t) ( t) n r,, E r E r = 1 2 ( ) (, t) (, t) = (, t) n r H r H r 1 2 K r Normal pointing to region (1) ( ) ( t) ( t) n r,, B r B r = 1 2 ( ) (, t) (, t) = σ( r, t) n r D r D r 1 2 ELECTRODYNAMICS 82/ XXX

83 Electromagnetic Potentials (, t) = Ar (, t) Br (, t) ϕ( r, t) E r Ar = t Ar (, t) = σµϕ( r, t) εµ ϕ t (, t) Lorentz( s) calibration ( r, t) ELECTRODYNAMICS 83/ XXX

84 Wave Equation 2 (, t) Ar (, t) (, t) σµ Ar Ar, 2 source t εµ = t µ J r ( t) Material parameters are assumed independent of coordinates ELECTRODYNAMICS 84/ XXX

85 Poynting( s)-umov( s) Theorem Power passing the bounding envelope Energy storage E J dv = source ( E H) ds + σe dv + ε + µ dv 2 t E H V S V V Power supplied by sources Heat losses Energy balance in an electromagnetic system ELECTRODYNAMICS 85/ XXX

86 Linear Momentum Carried by Fields Volume integration considerably change the meaning of Poynting( s) vector p 1 = c 2 V ( E H) dv This formula is only valid in vacuum. In material media things are more tricky. ELECTRODYNAMICS 86/ XXX

87 Angular Momentum Carried by Fields L 1 = r E H c 2 V ( ) dv This formula is only valid in vacuum. In material media things are more tricky. ELECTRODYNAMICS 87/ XXX

88 Frequency Domain F( r,t) R ˆ ( ) F r,ω C 1 = ˆ 2π (, t) F( r, ) e j F r d ωt ω ω j ( ω) F( r ) ˆ ωt F r, =, t e dt F r (, t) t ( ) j ωfˆ r, ω Spatial derivatives are untouched Time derivatives reduce to algebraic multiplication F r (, t) Fˆ ( r, ω) r ξ r ξ Frequency domain helps us to remove explicit time derivatives ELECTRODYNAMICS 88/ XXX

89 Phasors (, ω) = F ( r, ω) ˆ ˆ* F r 1 (, ) Re ˆ(, ) e j ω F rt = ω t dω π F r Reduced frequency domain representation ELECTRODYNAMICS 89/ XXX

90 Maxwell( s) Equations Frequency Domain (, ω) (, ω) j ωεe( r, ω) Hˆ r = Jˆ r + ˆ (, ω) j ωµ Hˆ ( r, ω) Eˆ r = ( ω) Hˆ r, = ( ω) Eˆ r, = ( r ) ρˆ, ω ε We assume linearity of material relations ELECTRODYNAMICS 9/ XXX

91 Wave Equation Frequency Domain (, ω) jωµσ ( j ωε) (, ω ) µ (, ω) Ar ˆ + Ar ˆ = J ˆ r source Helmholtz( s) equation ELECTRODYNAMICS 91/ XXX

92 Heat Balance in Time-Harmonic Steady State Valid for general periodic steady state Cycle mean E J dv = E H ds + σe dv source V S V Re ˆ ˆ dv Re ˆ ˆ d σ ˆ dv 2 E J = + 2 E H S 2 E * * source V S V 2 Valid for time-harmonic steady state ELECTRODYNAMICS 92/ XXX

93 Plane Wave Electric and magnetic fields are orthogonal to propagation direction ( ω) = ( ω) Eˆ r, E e jknr ˆ k H ( r, ω) = ( ω) e ωµ n E jknr Unitary vector representing the direction of propagation Electric and magnetic fields are mutually orthogonal ( ω) n E = ( ω) n H = k 2 ( j ) = jωµσ + ωε Wave-number The simplest wave solution of Maxwell( s) equations ELECTRODYNAMICS 93/ XXX

94 Plane Wave Characteristics k ( j ) = jωµσ + ωε Re k ; Im k > < 2π λ = Re k Vacuum ω k = c Re k ; Im k > = λ = c f v f Z = = ω Re k ωµ k General isotropic material v = c f Z c µ = µ = 377 Ω ε δ 1 = Im k δ ELECTRODYNAMICS 94/ XXX

95 Cycle Mean Power Density of a Plane Wave Power propagation coincides with phase propagation 1 Re k ( ) ( ) ( ) 2 2 Im k, t, t nr E r H r = E ω e n 2 ωµ ELECTRODYNAMICS 95/ XXX

96 Guided TEM Wave Wave propagation identical to a planewave k 2 ( ) = j ωµσ+jωε Geometry of a planewave ( ω) = ( xyω) Eˆ r, E,, e ( ω) = ( xyω) Hˆ r, H,, e jkz jkz Hˆ = k ωµ ( z Eˆ ) Transversal field pattern is static-like E = H = n E = Generalization of a planewave Boundary condition on the conductor ELECTRODYNAMICS 96/ XXX

97 Circuit Parameters of the TEM Wave Enclosing conductor ( ω) = ˆ ( ω) Uˆ z, U e jkz Iˆ ( ω) k Q = H dl = ωµ ε l pul ( ω) = Iˆ ( ω) Iˆ z, e jkz Uˆ ( ω) B ωµ = E dl = k A Φ pul µ Z v TRL phase ( ) ( ω) Uˆ ω ωµ ε ωµ L L = = = = Iˆ k C k µ C pul pul = 1 = 1 C L εµ pul pul pul pul Per unit length Between conductor Velocity of phase propagation ELECTRODYNAMICS 97/ XXX

98 The Telegraph Equations (, ) I ( zt, ) U zt z = L pul (, ) U ( zt, ) I zt z = C pul t t Circuit analog of Maxwell s equations ELECTRODYNAMICS 98/ XXX

99 Lukas Jelinek Department of Electromagnetic Field Czech Technical University in Prague Czech Republic Ver. 216/12/14