FISC7006. Eletrodinâmica Clássica II. Prof. Dante H. Mosca
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1 FISC7006 Eletrodinâmica Clássica II Prof. Dante H. Mosca 2018
2 Roteiro Cap. 7 Ondas planas uniformes em meio não-condutor e suas polarizações Ondas planas não-uniformes em meio condutor e dissipação Dispersão normal e anômala, exemplos em dielétricos, condutores e plasmas Causalidade e analíticidade entre os campos D e E, relações da Kramers-Kronig Dispersão de pacotes de onda (wavepackets) e precursores do sinal Cap. 8 Condições de contorno em interfaces dielétricas e condutoras Propagação e atenuação de ondas nos modos TE e TM Guia de onda e cavidade ressonante (análises em geometria cilíndrica) Expansão dos campos em modos normais e obstáculos em guias de onda Cap. 9 Fontes localizadas oscilantes e expansão em multipolar (dipolos e quadrupolos) Expansão multipolar dos campos radiativos em ondas esféricas Fontes oscilantes dipolares e quadripolares Campos multipolares de radiação e antenas
3 Chap. 7
4 Maxwell s Eqs. in infinite, linear, homogeneous, isotropic and insulating media: Each component for E and B fields must satisfy the wave eq. : then and
5 Therefore, The requirements: gives Consequently,
6 Complex Poynting vector: Corresponding time-averaging energy density:
7 Linearly polarized wave: Circularly polarized wave:
8 uniform plane wave E - field Poincaré sphere for all polarizations Stokes parameters { I M C S } : I 2 = M2 + C2 + S2
9 For instance, see
10 Frequency dispersion characteristics
11 Taking Fourier transformation of the solutions for wave packets: If Then where rms deviations from must respect
12 Harmonic wave train
13 Dispersion relationship
14 Group and phase velocities: Leads to dispersion under refraction process:
15 Normal Anomalous Normal
16 Propagation of a pulse in a dispersive insulating medium
17 Arbitrarily, it is assume that: and Then
18 Dispersive effects on the Gaussian pulse + c.c. The width of the envelope is: Such that }
19 In a good conducting media: Free electron gas leads to the Jellium, also known as the uniform electron gas (UEG) or homogeneous electron gas (HEG), where is assumed that: Such that:
20 Whereas tr components satisfy rotational terms, the long components will satisfy: Then Meanwhile, tr components can be written as:
21 EXERCISE 1 Show that: a) b) when (good conductor)
22 Propagating damped and transversal waves in the conducting media where Non-uniform plane wave
23 EXERCICIO 2 Show that: a) b) Discuss the situation in a good conductor and shows that a penetration lenght can be defined as:
24 Examples
25 Simple model for ε(ω): electric dipole approach Consider electrons bound by a harmonic force and perturbed by an external electric field under a phenomenological damping force ( B field is disregarded).
26 Resonant absorption
27 Low frequency behavior High frequency behavior
28 EXERCISE 3 a) Explain the simple model for complex AC conductivity (see slide 24): b) Discuss the case of purely imaginary conductivity: See section 7.9 (Jackson s book, 3rd Ed.)
29 Tenuous electronic plasma of uniform density with a strong, static, uniform, magnetic induction B0 and transverse waves propagating parallel to the direction of B0. Taking A steady-state solution is: Then with
30 Calculating current density only due to electronic motion: Adding this current to the displacement current in the Maxwell eqs., one find: Now, introducing and considering the ionosphere case:
31 Amplitude of oscillations Dielectric constant
32 Magnetospheric whistlers VLF signals generated by lightning discharges travelling along Earth magnetic field lines (ducts) which produce descending pitch noise in radio receivers.
33
34
35 mech. force dens. magn. force dens. J x B
36
37 If Then
38 EXERCISE 4 a) What are Alfvén waves? b) Determine Alfvén velocities in liquid mercury and sun s photosphere. c) It is possible to obtain va >> c? Why?
39
40 Absorption coefficient of liquid water at NTP
41 Liquid water 7 decades between 4-8 x 1014 Hz 8 8 x Hz α-1 ~ 10 m about 1 % of intensity survive below 50 m. ELF communications is used in submarines.
42
43
44
45
46
47
48
49
50 Photon tunneling leads to a number of effects in which light propagates over small distances faster than the speed of light in vacuum. If the concepts of information and noise are defined properly, it can be shown that the principle of causality is applied.
51 Cap. 8
52 (D,H) and (E,B) fields at the interface of an ideal conductor (c) Possible boundary condition on the H field: Perfect or ideal conductor
53 Inside thin layer or skin depth length inward into the non-ideal conductor : where is assumed an harmonic variation along the the outward normal n as: Then, Leading to:
54 Non-ideal conductor
55 At the interface: Dissipation by ohmic loss, leads to: Then, the power loss by effective surface current is:
56 Cylindrical cavity and waveguides Assuming sinusoidal time dependent fields inside the hollow metallic cylinder: Equivalently, Or yet,
57 By defining: Then General solutions:
58 TEM Wave = = =
59 EXERCISE 5 Discuss the conditions of E waves and H waves.
60 TE & TM modes
61 Waveguide Propagation of waves inside the hollow with uniform cross section for both TE and TM modes Wave impedances
62 General solutions for TM (TE) wavess Cut-off frequencies Propagation only if wave number is real for each (finite number) mode: Otherwise one find cut-off or evanescent modes.
63 Phase velocity becomes infinite exactly at cut-off
64 = Total power flow along the axial component of S Field energy per unit length of waveguide Such that:
65 zero Then and eles way
66 TE and TM fields in a cylindrical cavity resonator.
67
68
69
70
71
72 Power loss at non-ideal surfaces
73 EXERCISE 6 What are Schumann resonances? See Section 8.9, Jackson s book, 3rd Ed.
74
75
76
77 EXERCICIO 7 Analise a formação de momentos dipolares efetivos de pequenas aberturas em campos externos coeficientes de amplitudes de propagação dos campos =
78 Cap. 9
79 Fontes localizadas oscilantes
80 Multipolar Expansion Radiation or far region (k r >> 1): Rapid fall off...
81 Near and transition regions (k r <<1) :
82 Contribuição monopolar elétrica A contribuição é necessariamente estática!
83 Contribuição dipolar elétrica
84 Exercício 8 Considerar uma fonte dipolar elétrica. (a) Mostrar que os campos são: (b) Mostrar que os campos de radiação são: (c) Mostrar que a potência irradiada por unidade de ângulo sólido é:
85 Contribuições simétrica e antissimétrica de uma fonte dipolar magnética simétrica antissimétrica momento de dipolo magnético
86 Exercício 9 (a) Discutir as consequências de existirem componentes transversais elétrica e magnética na equação do potencial vetor: (b) Mostrar que os campos de uma fonte dipolar magnética são: (c) Analizar a simetria de intercâmbio dos campos dipolares elétrico e magnético. (d) Comparar a distribuição angular da potência irradiada e a polarização dos campos de radiação de dipolos de natureza elétrica e magnética.
87 Confguração espaço-temporal do campo dipolar
88 Parte simétrica do potencial vetor de uma fonte dipolar magnética Campos de radiação quadripolar de natureza elétrica
89 Análise do campo quadripolar dens. de momento de quadripolo Obs.:
90 Confgurações
91 Exercício 10 O potencial de um quadupolo elétrico escrito como: (a) Mostre que:
92 (b) Mostre que caso de uma expansão multipolar em coordenadas esférico-polares tem-se um inter-relacionamento híbrido com o momentos multipolares tal que:
93 Exercício 11 Considere uma fonte de carga q e seu potencial elétrico num dado instante t, tal que: P Admitindo que: mono dip quad
94 (a) Mostre que : p 0.6 q d zˆ (b) Mostre que : (c) Mostre que V : mono (r) = 0,20000 Vo Vdip (r) = 0,02400 Vo Vquad (r) = 0,00032 Vo (d) Compare com o valor exato do potencial V. Obs.:
95 Distribuição angular da potência irradiada de natureza quadripolar elétrica zero
96 Análise do espalhamento Linear polarized unpolarized
97 Radiação quadripolar elétrica multipolar
98 Expansão multipolar dos campos radiativos em ondas esféricas com E e H transversais
99 Modos de propagação
100 Campo multipolar elétrico (E) ou campo transversal magnético (TM) =
101 Campo multipolar magnético (M) ou campo transversal elétrico (TE)
102 Solução Geral Xom = 0
103 Exercício 12 Mostre que: (a) sendo (b) (c) e (d) Mostre que de acordo com o item (c): se r << 1
104 Campos de radiação multipolares Densidade de energia Densidade de momento angular
105 Distribuição angular de potência Multipolo de ordem (l,m)
106
107 Fontes multipolares e radiação
108 Equações de onda de Helmholtz
109 Solução geral
110 Coefcientes multipolares Obs.:
111 Complementando... Obs.:
112 Exercício 13 Mostre que no limite de longos comprimentos de onda k r <<1, temos: (a) (b)
113 Exercício 14 Considere um dipolo elétrico oscilante onde (b) Mostre que a potência total irradiada pode ser escrita como:
114 Tipos e usos de antenas Habilidade de transferir energia eletromagnética codificada entre um transmissor e um receptor
115 Atmosphere
116
117
118 Não funciona, pois a condutividade do solo, a constante dielétrica, as propriedades da ionosfera, as características do terreno são predominantes no cálculo da intensidade do campo. Usa-se tabelas gráficos de manuais de engenharia de rádio.
119 Marconi
120 Onidirectional Antenna
121 Hertz Folded Dipole Yagi-Uda
122 Half-wave dipole
123 Antena Direcional
124
125 FIM
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