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Lecture 22

PHYS 242 Lecture 22: PHYS242_Lecture_22-24

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Queen's University
PHYS 242
Wolfgang Rau

69 ENPHPHYS 242 Fall 2014 L 22 2.2.2 Classical Theory of Cavity Radiation Since the black body radiation is universal, as discussed above, it is not surprising that it is possible to derive a spectral distribution for this radiation based purely on some basic knowledge about electromagnetic radiation and statistical physics without input from experiments. The universality is a very convenient feature since it allows us to use a very specific model for a cavity without having to worry about generalization: any system which we describe will produce the same spectrum. As our model system we choose a metallic cubic box with side length . Let us start with the discussion of a onedimensional box. Since it is a metallic box the electric field at the wall has to be 0 (if it is not 0 a current will flow along the wall until the electric field disappears). If we have an electromagnetic wave inside the box it has to comply with this condition. The electric field of the wave can be described by a sinfunction: sin sin 2 (72) with wavelength . This function is 0 at 2 for 0,1,2,. Therefore a wave can only exist inside the box if 2 with 1,2,3, (73) The corresponding condition for the frequencies ( ) is: 2 with 1,2,3, (74) We can solve this equation for , the number of halfwaves in the box: 2 (75) is at the same time the number of allowed frequencies inside the box with . The orientation of the electric field (polarization) is always perpendicular to the direction of motion of the wave, which leaves two independent options for a wave with a given frequency (if the wave travels in direction we may have the electric field in or in direction). So the number of allowed waves is twice the number of allowed frequencies. With this knowledge we can determine the number of allowed waves in the frequency range [, ]: (76) Now we can go from a onedimensional box to a threedimensional box and consider waves in , , and direction. For the number of allowed frequencies with we can still use equation (75) but now is given by (77) W. Rau
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