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Unformatted text preview: 35 Cavity radiation and thermal noise Cavity radiance: Energy den sity E ( f ) = N ( f ) V W ( f ) = 8 f 2 c 3 hf e hf/KT 1 J/m 3 Hz . in a 3D cavity in thermal equi librium resides by equal amounts in the traveling wave components of the cavity modes arriving with speed c from the boundaries of the cavity subtending 4 sterads. Multiplying E ( f ) by c/ 4 we ob tain L ( f ) = 2 f 2 c 2 hf e hf/KT 1 W/m 2 /ster Hz , which is called radiance and rep resents the power density per unit solid angle of the waves traveling within the cavity. Radiance L ( f ) also represents the spectrum of power radiated per unit solid angle by a unit area of a blackbody surface at temper ature T (since nonreflective walls of a cavity will produce the same E ( f ) as partialreflecting walls as mentioned earlier). In a 1D cavity of some length L e.g. a TL with shorts at both ends as discussed in ECE 329 notes the resonant frequencies are f m = c m = c 2 L/m = c 2 L m, where m = 1 , 2 , 3 , which indicates that the mode density in f is N ( f ) = 2 L c modes Hz . Therefore the energy density in a 1D cavity in thermal equilibrium will be E ( f ) = N ( f ) L W ( f ) = 2 c hf e hf/KT 1 J/m Hz in analogy with the energy density of 3D cavities. This energy density will reside by equal amounts in the traveling wave components of the 1D resonant modes arriving with speed c from the opposite ends of the 1D resonator. Power spectral content P ( f ) of each of these traveling wave components can thus be calculated as c/ 2 times 1 E ( f ) , i.e., P ( f ) = hf e hf/KT 1 W Hz . 1 Note that per TEM plane wave, c ( 1 4 o  E  2 + 1 4 o  H  2 ) =  E  2 4 o + o  H  2 4 =  E  2 2 o , which confirms that the timeaveraged stored energy density times...
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This note was uploaded on 09/27/2011 for the course ECE 450 taught by Professor Staff during the Fall '08 term at University of Illinois, Urbana Champaign.
 Fall '08
 Staff
 Electromagnet

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