Homework10

# Homework10 - Homework 10 Answers, 95.657 Electromagnetic...

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Homework 10 Answers, 95.657 Electromagnetic Theory I Dr. Christopher S. Baird, UMass Lowell Jackson 5.13 A sphere of radius a carries a uniform surface-charge distribution σ . The sphere is rotated about a diameter with constant angular velocity ω . Find the vector potential and magnetic-flux density both inside and outside the sphere. SOLUTION: The current density in spherical coordinates is: J = a sin  r a A = 0 4 J x ' x x ' d x ' A = 0 4  a sin '  r ' a ' x x ' d x ' Expand the denominator in spherical harmonics. A = 0 4 0 2 0 0  r ' a '4 l = 0 m =− l l 1 2 l 1 r l r l 1 Y lm *  ' , ' Y lm  ,  r ' 2 sin ' dr ' d ' d ' We must be careful and realize that ' because the primed variables are being integrated over. The best way to handle this is to use the expansion: ' =− sin ' i cos ' j A = 0   a 0 2 0 0  r ' a − sin ' i cos ' j l = 0 m =− l l 1 2 l 1 r l r ' l 1 Y l m *  ' , ' Y lm  ,  sin 2 ' dr ' d ' d ' for r < r ' (inside the sphere). A = 0   a 0 2 0 0  r ' a − sin ' i cos ' j l = 0 m =− l l 1 2 l 1 r ' l 2 r l 1 Y l m *  ' , ' Y lm  ,  sin 2 ' dr ' d ' d ' for r > r ' (outside the sphere). Now evaluate the delta functions and rearrange:

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## This note was uploaded on 03/21/2012 for the course PHY 2030 taught by Professor Avery during the Spring '11 term at University of Florida.

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Homework10 - Homework 10 Answers, 95.657 Electromagnetic...

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