Phys454 - SummaryFinal

2 2 1 em waves between parallel conducting planes te

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Unformatted text preview: , = and solve: = 2 TE Mode + Using , = 0 cos cos , = cos sin - sin cos 0 = - 2 - = , 2 , = - + - 2 0 , = sin cos + cos sin + cos cos Propagation characteristics: Dispersion relation: 2 2 2 = 2 2 + + phase = = > 2 2 2 - - Cutoff frequency: = For > , the lowest cutoff frequency is 2 - + 2 < In many applications the dimensions of the waveguide is chosen such that only TE 1,0 can propagate Group velocity: 10 = group = 2 2 = = 1 - 2 < Page 7 of 10 group phase = 2 Constructing a standing wave in the direction travelling in direction Fields 0 - cos + sin - 2 0 2 = - cos + sin - 2 1 = Cutoff wavelength: = cos = sin = Wavelength in direction = 2 1 2 = = = 2 2 0 0 1- = 1- 1- 2 2 2 = sin = 0 sin To get , use 2 sin - 1 1- 2 = - Guide impedance: = 0 0 At high frequencies: = If = , 0 0 = Electric dipole radiation: Retarded Potentials: 1 , = 0 , = 0 source point field point Asymptotic potential: , - - 4 - - , - 4 - 3 3 - || , - 3 , ~0 4 = - = , 3 Electric dipole moment: Potentials in terms of electric dipole: Asymptotic Radiation: 0 4 0 , ~ 4 0 2 2 , ~ 2 - 2 4 , ~ 0 2 , ~ - 2 4 Page 8 of 10 , , form an orthogonal triad = Power: 0 = 16 2 2 2 2 2 2 - 2 2 Hertzian dipole: 1 , = 2 = 60 3 = = 0 cos = , = 0 0 2 sin = - cos - 4 2 = 0 Total power through a sphere averaged over a period of oscillation: 2 2 0 4 2 avg = 120 3 2 Antenna resistance: = 2 = 1 0 6 0 2 Magnetic dipole radiation = 0 - 0 , ~ - ( ) 4 Using Lorentz Gauge conditions: 0 2 4 2 0 2 2 = - 4 2 2 2 = - If = 0 - , then rad 0 - - , ~ - sin 4 0 0 - - rad = 2 0 sin 4 0 0 - - , 0 = =- 2 sin 4 0 Half-Wave Linear Antenna Geometry Length of antenna: = Centre of antenna at = 0 Current in antenna: Potential and fields 2 Power: = 0 cos sin 0 0 cos 2 , = - sin - 2 sin2 0 0 cos 2 = cos - 2 sin = 2 2 0 0 cos 2 = 2 2 cos2 - 4 sin 2 0 0 = 4 cos 0 2 1.22 2 sin 73 = 2 Page 9 of 10 The Larmor Formula: Radiation from a Point Charge Charge density and currents: Radiation fields: , = 3 - , = 3 - , = - , = Power: 0 4 , = vector from position of at the retarded time to the field point - = - 2 2 2 = - 16 2 0 3 2 Larmor Formula: 1 22 2 = 2 = 40 3 3 Classical electron theory of scattering Point-charge electron bound to origin by Hooke's law force Equation of motion neglecting magnetic forces: 2 = - - - 0 - 2 2 0 = - 0 - , 0 = - 2 - = 2 = 1 cos + 2 sin 2 2 0 2 0 - 2 1 = 2 2 2 2 + 2 2 0 - 0 3 1 = 2 2 0 - 2 2 + 2 2 Power: By Larmour formula, avg = Scattering cross section 2 2 2 2 1 + 2 4 0 4 = 3 3 2 2 - 2 120 120 0 2 + 2 2 = inc average of Poynting vector of the incident wave For incident plane wave, = = At 0 , 82 2 4 2 , = 2 2 - 2 2 + 2 2 3 40 2 0 Rayleigh = 82 3 0 4 2 0 0 2 At 0 , - 82 = 6.6 10-29 m2 3 This is the cross section for scattering of light from a free electron; it is observed from plasmas Thompsons = For 2 , this classical theory breaks down. Scattering is explained by Compton scattering. Page 10 of 10...
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