Conditions 0 0 0 1 0 0 2 0 cos cos 1 2 reflectivity

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Unformatted text preview: + 1 2 cos 0 1 2 cos - 2 1 cos 0 = 2 1 cos + 1 2 cos 0 0 = 0 = Reflectivity and transmitivity: - In non-magnetic materials, tan - tan + 0 2 1 Brewster's angle No reflection, and tan = Conservation of energy: cos + = 1 cos EM Waves in Conductors Maxwell's equations in metal = 0 = 0 = - EM waves in conductors 2 2 = + 2 , = 0 - = + 2 = + 2 = 1 + 2 1 = 2 2 1+ 2 1 2 +1 1 2 2 = 1+ 2 -1 , = 0 - 2 Wavelength and phase velocity: = Skin Depth: 1 - 2 , phase = 1 1 1 2 = 1 = 2 = Good conductor = 1 + = 21 4 2 2 1 = Skin Depth: 0 2 = 2 1 20 Page 4 of 10 = Magnetic field: Lags electric field by /4 0 0 Energy densities: 1 = 2 2 = 0 - = = Reflectivity of a good conductor Assuming 1 = 1 2 0 2 2 = 1 0 2 2 = 0 1 - 2 - = = 0 1 + 2 + 2 - 1 2 + 2 2 = = 2 + 1 2 + 2 2 1 - Classical model of dispersion: Electron moving in an electric field: 41 80 2 2 =1- 2 1 + 2 Where is the position of the electron = 0 - 2 = - - - 0 - 2 = - 0 - - 2 - Dispersion in a dielectric Dipole moment: = - = 2 = - 2 - Index of refraction: = 1 + 2 2 20 0 2 2 0 0 - 2 2 0 - 2 2 + 2 Inverse absorption length: -1 = 22 = 2 0 2 2 0 - 2 2 + 2 0 = Transmission lines: Impedance Characteristic Impedance: 0 = If , , then 0 = Input impedance: in = At = 0, At = , + + + 0 tanh = 0 tanh + 0 0 in = 0 tanh Page 5 of 10 in = 0 tanh 0 in in 0 = Load impedance Current and voltage: = 1 + - - - , = + ( + ) = + 0 = + - + - 1 1 + - = - = - - , + 0 = + ( + ) = + + right-moving wave = + - - - left-moving wave Phase velocity: - = Voltage reflection coefficient of load: = Coaxial cable: By Gauss's law, = , = 20 charge per unit length = 1 = = - - 0 = + + 0 1 - = 2 2 ln 20 1 20 = ln 2 1 0 0 2 = = , = = ln 2 2 1 EM Waves between parallel conducting planes (TE Mode) Fields and boundary conditions Fields: , = - , = - = + By Ampere-Maxwell law, = 2 2 = - 2 - 2 2 - 2 2 - = 1 sin + 2 cos , 2 = Boundary conditions = 0 sin Dispersion relation: 0 = = 0 2 = 2 + 2 2 EM Field solutions: - 0 0 , = sin + cos , = 0 sin ph = = 2 - Propagation characteristics 2 - cutoff = Page 6 of 10 Surface currents: Energy transport , = 2 0 sin2 0 2 2 = = 1 - 0 0 = = - sin - at = 0, 0 0 0 = - = -1 sin - at = , 0 0 cos2 - + sin cos sin - cos - gr = = 2 2 (, ) = 0 2 0 sin2 20 Power per unit length in the direction: = avg = 0 2 0 1- 40 2 The Rectangular Waveguide Geometry Interior is vacuum, exterior is conductor Wave propagates in the z direction Boundary conditions: = = 0 at = 0, , 0 and { = 0, , 0 } Start with , = , + , - Let = - , = Then 2 2 = - 2 , 2 = 2 - 2 Letting...
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This document was uploaded on 03/14/2014 for the course PHYS 454 at The University of British Columbia.

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