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HW05'sol - Classical Electrodynamics II PHY 506 Spring 2011...

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1 C:\User\Teaching\505-506\S11\HW05'sol.doc PHY 506 Classical Electrodynamics II Spring 2011 Homework 5 with solutions Problem 5.1 (25 points). Use the Born approximation to calculate the differential cross-section of plane wave scattering by a dielectric sphere with 0 , of an arbitrary radius R . In the limits kR << 1 and kR >> 1 (where k is the wave vector), analyze the angular dependence of the differential cross-section, and calculate the full cross-section. Solution : According to Eqs. (8.62)-(8.63) of the lecture notes, the differential cross-section may be calculated as V r r d i I I k d d 3 2 2 2 2 4 exp ) ( , sin ) ( ) 1 ( ) 4 ( r q q q , where the phase integral I ( q ) is taken over the scatterer’s volume, and vector q = k k 0 is the wave vector change due to scattering. According to Fig. 8.5 (reproduced on the right), magnitude of q is related to the scattering angle (i.e. the angle between vectors k and k 0 ) as q = 2 k sin( /2). As this figure shows, angle (i.e. the angle between vectors
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