Electrodynamics Lecture 8: Multipole Moment Expansion

phy 712 electrodynamics 9 9 50 am mwf olin 105 n.w
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Explore the multipole moment expansion of electrostatic potential in spherical and Cartesian coordinates. Dive into Poisson and Laplace equations in spherical polar coordinates, spherical harmonic functions, Legendre polynomials, and associated Legendre functions. Gain insight into useful identities and examples related to charge densities and electrostatic potential.

  • Electrodynamics
  • Multipole Moment
  • Spherical Coordinates
  • Cartesian Coordinates
  • Electrostatic Potential
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  1. PHY 712 Electrodynamics 9-9:50 AM MWF Olin 105 Plan for Lecture 8: Start reading Chapter 4 Multipole moment expansion of electrostatic potential A. Spherical coordinates B. Cartesian coordinates 02/02/2018 PHY 712 Spring 2018 -- Lecture 8 1

  2. 02/02/2018 PHY 712 Spring 2018 -- Lecture 8 2

  3. Poisson and Laplace equation in spherical polar coordinates = sin cos x r = sin sin y r = cos z r http://www.uic.edu/classes/eecs/eecs520/textbook/node32.html 02/02/2018 PHY 712 Spring 2018 -- Lecture 8 3

  4. Poisson and Laplace equation in spherical polar coordinates -- continued ( ) , , Laplace equation electrosta for potential tic : r 2 1 r 1 r 1 1 ( ) + + = sin 0 r 2 2 2 2 sin sin r ( ) ( ) , , = , ( ) r R r Y lm lm lm Spherical harmonic functions : 1 1 ( ) ( ) ( ) + , = l l + , sin 1 Y Y lm lm 2 2 sin sin 02/02/2018 PHY 712 Spring 2018 -- Lecture 8 4

  5. Properties of spherical harmonic functions ( ) ( ( , Y ) )( ) ) m = * , 1 , (standard Condon-Shortley convention) Y Y ( lm l m ) ( ( ) ( ) = * * sin d Y , d d Y , Y , ll' mm' lm l'm' lm l'm' Completeness: ( ) ( ) ( ) ( ) ( ) = * r r ' ' ' cos cos ' ' Y , Y , lm lm lm Relationship to Legendre po lynomials: + 2 1 l ( ) ( ) = cos Y , P 0 l l 4 Relationship to Associated Legendre polynomials: ( ( ) ) + + ! ! l l m m 2 1 l ( ) ( ) = m im cos Y , P e lm l 4 02/02/2018 PHY 712 Spring 2018 -- Lecture 8 5

  6. Legendre and Associated Legendre functions Legendre differenti equation al : ( 1 ) d d ( ) l l + + = 2 1 ( ) 0 x P x l dx dx Associated Legendre differenti equation al : ( 1 ) 2 d d m ( ) l l + + = 2 m 1 ( ) 0 x P x l 2 1 dx dx x For 0 m ) ( 1 ) m d ( / 2 m m = 2 m ( ) 1 ( ) P x x P x l l m dx )( 1 ) ) ! l m ( m = m m ( ) ( ) P x P x ( l l + ! l m 02/02/2018 PHY 712 Spring 2018 -- Lecture 8 6

  7. Useful identity: l 1 4 r ( ) ( ) lm * = l ' ' Y Y , , lm lm + + 1 r r ' 2 1 l r ( ) r Example isolated for charge density with electrosta potential tic vanishing r r for : r ( ) r 1 ' ( ) r = 3 ' d r 4 ' 0 l 1 4 r ( ) r ( ) ( ) lm * = 3 l ' ' ' ' d r Y Y , , lm lm + + 1 4 2 1 l r 0 02/02/2018 PHY 712 Spring 2018 -- Lecture 8 7

  8. Some spherical harmonic functions: ( ) = 4 1 r Y 00 3 )( ) r i = sin Y e ( 1 1 8 3 ( ) r = cos Y 10 4 15 )( ) r i = 2 2 sin Y e ( 2 2 32 15 )( ) r i = cos sin Y e ( 2 1 8 5 3 1 ( ) r = 2 cos Y 20 4 2 2 02/02/2018 PHY 712 Spring 2018 -- Lecture 8 8

  9. Example: General r form of electrosta potential tic boundary v with alue ( ) r isolated for , charge r r density : ( ) r 1 ' ( ) r = 3 ' d r 4 ' 0 l 1 4 l r ( ) r ( ) ( ) lm * = 3 l ' ' ' ' d r Y Y , , lm lm + + 1 4 2 1 r 0 , ( ) r ( ) r ( ) lm = Suppose that Y lm lm 1 1 1 l ( ) r ( ) ( ) r ( ) ' r r lm + = + ' 2 1 l l l ' ' ' ' Y r dr r r dr , lm lm lm + + 1 2 1 l r 0 r 0 02/02/2018 PHY 712 Spring 2018 -- Lecture 8 9

  10. Example: sin co s 1 2 8 q qr Va ( ) ( ) ( ) = Y , Y , r a ( ) r 1 1 11 = Suppose 3 Va 0 r a 1 1 + 1 l r ( ) r ( ) ( ) r ( ) r + = + ' 2 1 ' l l l ' ' ' ' Y , r dr r r dr lm lm lm + 1 2 1 l r 0 r lm a 0 For r 1 6 8 1 r q ( ) r a ( ) r ( ) ( ) = + 4 ' ' ' ' Y , Y , r dr r r dr 1 1 11 2 3 Va 0 r 0 For r a 1 6 8 1 r q ( ) a ( ) r ( ) ( ) = 4 ' ' Y , Y , r dr 1 1 11 2 3 Va 0 0 02/02/2018 PHY 712 Spring 2018 -- Lecture 8 10

  11. Example -- continued: 1 2 8 qx Va qr Va ( ) ( ) ( ) = 1 1 Y , Y , r a ( ) r 11 = Suppose 3 0 r a For r a 1 6 8 1 r q ( ) r a ( ) r ( ) ( ) = + 4 ' ' ' ' 1 1 Y , Y , r dr r r dr 11 2 3 Va 0 r 0 ( ) q ( ) = sin cos 2 2 3 5 r a r 6 Va a 0 For r 1 6 8 1 r q ( ) a ( ) r ( ) ( ) = 4 ' ' Y , Y , r dr 1 1 11 2 3 Va 0 0 5 a 2 5 q = sin cos 2 6 Va r 0 02/02/2018 PHY 712 Spring 2018 -- Lecture 8 11

  12. Example -- continued: For q Va a r a ( ) ( ) ( ) r = sin cos 2 2 r a r 3 5 6 0 For r 5 5 a q qa V x r 2 5 ( ) r = sin cos = 2 3 6 15 V r 0 0 02/02/2018 PHY 712 Spring 2018 -- Lecture 8 12

  13. Notion of multipole moment: In the spherical harmonic representa tion - - r moment the define of (confined) the charge distributi on : q ) ( lm ' ( ) 3 * l r ' ' , ' ' q d r r Y ) ( lm lm In the Cartesian representa tion - - monopole the define moment : q 3 r ' ' q d r ) ( p define the dipole moment : 3 p ' ' r r ' d r ) ( quadrupole the define moment components Q (i,j : x,y,z) ij ( ) 3 2 r ' 3 ' ' ' ' Q d r r r r ) ( ij i j ij 02/02/2018 PHY 712 Spring 2018 -- Lecture 8 13

  14. Significance of multipole moments form general Recall of electrosta potential tic boundary v with alue ( ) r isolated for , charge r r density : r ( ) r 1 ' ( ) r = 3 ' d r 4 ' 0 l 1 4 r ( ) r ( ) ( ) lm of * = 3 l ' ' ' ' d r Y Y , , lm lm + + 1 4 2 1 l r 0 ( ) r For outside extent the : r ( l ) 1 4 Y , 1 ( ) r ( ) ( ) lm 0 * = 3 l r ' ' ' ' ' lm d r r Y , lm + + 4 2 1 l r 0 ( l ) 4 q 1 Y , 1 lm qlm = lm 1 lm + + 4 2 l r 0 02/02/2018 PHY 712 Spring 2018 -- Lecture 8 14

  15. ( ) ( ) Multipole moments continued: = , 3 l * r ' ' ' ' ' q d r r Y ( ) , + lm lm r For outside the extent of r : 0 ( ) Y 4 2 1 q + ( ) r lm =4 lm 1 l 1 l r lm 0 Relationship between spherical harmonic and Cartesian forms of multipole moments: 1 ( ) 15 = q q = iQ 2 q Q Q 00 4 2 2 xx xy yy 288 ( ) 3 ( ) 15 = q p ip = q Q iQ 1 1 x y 8 2 1 xz yz 72 3 5 = q p = q Q 10 z 4 20 zz 16 02/02/2018 PHY 712 Spring 2018 -- Lecture 8 15

  16. Consider previous example: 1 2 8 qx Va qr Va ( ) ( ) ( ) = 1 1 Y , Y , r a 11 3 ( ) r = 0 r a We previously showed that for = = r a 1 6 8 1 r q ( ) a ( ) r ( ) ( ) 4 ' ' 1 1 Y , Y , r dr 11 2 3 Va 0 0 5 5 1 6 8 2 5 q a q a r ( ) ( ) ( ) = n cos si 1 1 Y , Y , 11 2 2 3 5 6 Va r V 0 0 5 1 2 8 q a = Note that: 1 1 q 3 5 Va 5 1 2 8 4 q a Va ( ) = + = p q 1 1 q 11 x 3 3 5 02/02/2018 PHY 712 Spring 2018 -- Lecture 8 16

  17. General form of electrostatic potential in terms of multipole moments: ( ) r For outside extent the of : r ( l ) ( ) ) 1 4 Y , 1 ( ) r ( ) ( ) lm * = 3 l r ' ' ' ' ' lm d r r Y , lm + + q 4 2 1 l r 0 ( l 4 1 Y , 1 lm = lm 1 lm + + 4 2 l r 0 In terms of Cartesian expansion : r r p r 1 1 q ( ) r i , i j = + + Q ij 3 5 4 2 r r r j 0 02/02/2018 PHY 712 Spring 2018 -- Lecture 8 17

  18. Example of multipole expansion in evaluating energy of a very localized charge density (r) in a electrostatic field (r) (such as an nucleus in the field due to the electrons in an atom). ( ) ( ) r = 3 r W d r 1 ( ) r ( ) 0 ( ) r ( ) ( ) r 2 + + + 3 r r d r = = 0 0 2 r r ( ) r r 2 1 0 ( ) 0 ( ) 0 i , = + + p E q Q ij 6 j i j 02/02/2018 PHY 712 Spring 2018 -- Lecture 8 18

  19. Simple examples of multipole distributions z y z x ( ) ( ) = = ( ) ( ) = + 3 3 r r z r z q d d 2 p p p q d y z = 0 x x y ( ) ( ) r ( ) ( ) ( ) r = + + 3 3 3 r z r z 2 q d d = = = 2 4 2 2 Q qd Q Q zz xx yy 02/02/2018 PHY 712 Spring 2018 -- Lecture 8 19

  20. Another example of multipole distribution 2 q r ( ) r = / 2 r a sin e 3 64 a a 5 4 3 1 3 ( ) , = = 2 2 Note that : cos 1 sin Y 20 2 2 2 5 r 1 5 2 2 4 2 4 2 4 ( ( ) ) ( ) ( ) = , = , , 2 sin Y Y Y 20 00 20 3 3 3 3 ( ) ( ) r ( ) ( ) 00 ( ( ) ) ( ) ( ) 20 = , + , r r Y Y 00 00 20 20 Y ( ) = , + , r Y r 00 20 1 4 l 1 l ( ) r ( ) ' r r + = + ' 2 1 l l l r ' ' ' ' r dr r r dr lm lm lm + + 1 4 2 1 r 0 r 0 2 2 2 2 4 q q r ( ) r ( ) r = = / / r a r a 4 e e 00 20 3 3 3 64 3 5 64 a a a a 02/02/2018 PHY 712 Spring 2018 -- Lecture 8 20

  21. Another example of multipole distribution -- continued 2 3 1 3 q r r r ( ) r = + + + / r a 4 1 1 e 00 2 3 4 4 4 24 r a a a 0 2 2 3 4 5 6 4 qa r r r r r ( ) r = + + + + + / r a 1 1 e 20 3 2 3 3 5 4 5 2 6 24 144 r a a a a a 0 For in terms ; Legendre for polynomial : s r + 2 1 l 2 1 r 6 q a ( ) ( ) ( ) r ( ) = , c os Y P cos P 0 l l 4 2 4 3 r 0 For in terms ; 0 Legendre for polynomial : s r 2 1 q r ( ) r ( ) cos P 2 4 3 4 120 a a 0 02/02/2018 PHY 712 Spring 2018 -- Lecture 8 21

  22. Another example of multipole distribution -- continued Legendre for in terms ; 0 For r polynomial : s 2 1 q r ( ) r ( ) cos P 2 4 3 4 120 a a 0 Implicatio ns electric for quadrupole interactio n : ( ) r r 2 ( ) 1 0 ( ) i , = = 2 2 2 cos cos 3 3 2 1 2 1 r P z r = + W Q 2 2 2 ij 6 ( ) j i j = 2 2 2 2 1 r z x y 2 2 For in terms ; 0 of Cartesian coordinate s r 2 2 2 1 2 q z x y ( ) r 3 4 4 240 a a 0 ( ) 2 ( ) 2 ( ) 2 2 2 2 0 0 1 0 1 q = = = 3 2 4 120 x y z a 0 02/02/2018 PHY 712 Spring 2018 -- Lecture 8 22

  23. Another example of multipole distribution -- continued Electric quadrupole interactio n : ( ) r ( ) 2 ( ) 2 ( ) 2 r i x y z 2 2 2 2 1 0 1 0 0 0 = 6 = + + W Q Q Q Q ij xx yy zz 6 , i j j 1 2 1 2 = = For symmetric nuclei, Q Qq Q Q zz xx yy 2 q Q W 3 4 240 a 0 02/02/2018 PHY 712 Spring 2018 -- Lecture 8 23

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