HW#1_solutions

HW#1_solutions - ECE594I Prof Brown Fall Quarter 2009 Solutions#1 1.0E-02 Brightness[W/m^2/Sr/GHz Rayleigh-Jeans Limit 1.0E-03 1.0E-04 1.0E-05

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ECE594I, Prof. Brown, Fall Quarter 2009 1 Solutions #1 1. (a) From the course lecture notes, the brightness function is given by 1 2 / 2 3 T k h B e d c h ν [W/m 2 /Sr]. A plot of this brightness function vs frequency is shown below for T = 300 K. (a) by inspection and iteration, it is easy to find that the maximum value of B occurs at 17.6 THz ( λ = 17.0 µ m). (b) The maximum brightness function is 5.105x10 -12 W/(m 2 -Hz). (c) The brightness function at ν = 1.0 THz is 8.48x10 -14 W/(m 2 -Hz), 60.2 times lower than the maximum ! 2. The integrated brightness function is defined and evaluated as follows: π σ / 15 2 1 2 1 2 4 4 4 2 0 3 4 2 / 0 2 3 T h T k c h e dx x h T k c h e d c h IB B x B T k h B = = = where σ = 5.67x10 -8 W/(m 2 -K 4 ) [note the good mnemonic ordering of numbers] (b) At T = 300 K, IB = 146.2 W/(m 2 -Sr). (c) In the (long-wavelength) Rayleigh-Jeans limit, B T = 2k B T( ν /c) 2 , so that 2 3 0 2 2 3 / ) ( 2 2 ) ( c T k d c T k IB C B B C C = = < . And using elementary calculus, we find 2 3 4 0 0 3 / ) ( 2 / ) ( ) ( ) ( ) ( c T k T d B d B d B IB C B C C
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This note was uploaded on 12/02/2009 for the course ECE 000 taught by Professor O during the Spring '09 term at UCSB.

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HW#1_solutions - ECE594I Prof Brown Fall Quarter 2009 Solutions#1 1.0E-02 Brightness[W/m^2/Sr/GHz Rayleigh-Jeans Limit 1.0E-03 1.0E-04 1.0E-05

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