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 HalfWave 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
Pointcharge 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 1029 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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This document was uploaded on 03/14/2014 for the course PHYS 454 at UBC.
 Winter '09
 AXEN
 Magnetism, Energy

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