Homework8 - Homework 8 Answers, 95.657 Fall 2011,...

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Homework 8 Answers, 95.657 Fall 2011, Electromagnetic Theory I Dr. Christopher S. Baird, UMass Lowell Problem 1 Jackson 4.1 parts (a) and (b) only Calculate the multipole moments q lm of the charge distributions shown as parts a) and b). Try to obtain results for the nonvanishing moments valid for all l , but in each case find the first two sets of nonvanishing moments at the very least. a) b) SOLUTION: a) The charge density is written down in spherical coordinates as: = q a 2  r a  cos  [ −− 3 / 2 −−−/ 2 ] Plug this into the multipole moments definition and evaluate: q lm = Y lm *  ' , ' r ' l  x ' d x ' q lm = q a 2 Y lm *  ' , ' r ' l  r ' a  cos ' [  ' − ' 3 / 2 − ' − ' −/ 2 ] d x ' q lm = q a l 0 2 0 Y lm *  ' , '  cos ' [  ' − ' 3 / 2 − ' − ' −/ 2 ] sin ' d ' d ' q lm = q a l 2 l 1 4 l m ! l m ! P l m 0 0 2 e im ' [  ' − ' 3 / 2 − ' − ' −/ 2 ] d ' q lm = q a l 2 l 1 4 l m ! l m ! P l m 0 [ 1 i m −− 1 m − 1 m i m ] q lm = q a l 2 l 1 4 l m ! l m ! P l m 0  1 −− 1 m  1 i m y z x q q -q -q y z x q q - 2 q a a a a a a
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lm = 2 q a l 2 l 1 4 l m ! l m ! P l m 0  1 i m if m is odd, q lm = 0 if m is even This is the solution valid for all l . Let us write our the first few multipoles explicitly to see what this means. The monopole moment is: q 00 = 0 because m is even This makes sense because the total charge is zero. The dipole moments are: q 1, 1 = qa 3 2 1 i q 1,0 = 0 q 11 = q a 3 2 − 1 i Put these together into the Cartesian dipole moment vector: p = p x i p y j p z k p = 2 3 q 1, 1 q 11 i i q 1, 1 q 11 j 2 q 10 k p = 2 q a i j The quadrupole moments are: q 2, 2 = 0 q 2, 1 = 0 q 2,0 = 0 q 21 = 0 q 2,2 = 0 The problem asks us for the first two sets of non-vanishing moments, so we have to keep going. The non-zero octapole moments are:
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Homework8 - Homework 8 Answers, 95.657 Fall 2011,...

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