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3. Physical Interpretation of Generating Function

3. Physical Interpretation of Generating Function. Leading term : (point charge). for r > a. . . for r < a. Expansion of 1 / | r  r |. Let :. . either r or r  on z -axis. Electric Multipoles. Electric dipole :. point dipole. Leading term :.

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3. Physical Interpretation of Generating Function

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  1. 3. Physical Interpretation of Generating Function Leading term : (point charge) forr > a.   forr < a.

  2. Expansion of 1 / | r  r | Let :  either r or r on z-axis

  3. Electric Multipoles Electric dipole : point dipole Leading term :

  4. (Linear) Multipoles Let = 2l-pole potential with center of charge at z = r. Mono ( 20 ) -pole : Di ( 21 ) -pole : Quadru ( 22 ) –pole : ( 2l ) –pole : Quadrupole Mathematica

  5. Multipole Expansion If all charges are on the z-axis & within the interval [zm, zm ] : for r > zm where is the (linear) 2l–pole moment. For a discrete set of charges qiat z = ai. 

  6. If one shifts the coord origin to Z. • lis independent of coord, i.e.,  Z • iff Multipole expansion for a general (r) are done in terms of the spherical harmonics.

  7. 4. Associated Legendre Equation Associated Legendre Eq. Let   Set  Mathematica

  8. Frobenius Series  with indicial eqs. or By definition,  Mathematica

  9. Series diverges at x = 1 unless terminated.  For s = 0 & a1= 0 (even series) : ( l,mboth even or both odd )  Mathematica Fors = 1 & a1=0 (odd series) : (l,mone even & one odd )  Plm = Associated Legendre function

  10. Relation to the Legendre Functions Generalized Leibniz’s rule :   

  11. Set Associated Legendre function : ()mis called the Condon-Shortley phase. Including it in Plmmeans Ylmhas it too.  Rodrigues formula :    Mathematica

  12. Generating Function & Recurrence     ( Redundant since Plm is defined only forl  |m| 0. ) &

  13.  as before    

  14. ( Redundant since Plm is defined only forl  |m|0. ) & 

  15. Recurrence Relations for Plm (1) = (15.88) (2) (1) : (3) (3)  (2) : (15.89)

  16. Table 15.3 Associated Legendre Functions Using one can generate all Plm (x)s from the Pl (x)s. Mathematica

  17. Example 15.4.1. Recurrence Starting from Pmm (x) no negative powers of (x1)      

  18. l = m l = m+k1 E.g., m = 2 :

  19. Parity & Special Values Rodrigues formula :  Parity Special Values : Ex.15.4-5

  20. Orthogonality Plm is the eigenfunction for eigenvalue of the Sturm-Liouville problem where Lm is hermitian  ( w = 1 ) Alternatively : 

  21. No negative powers allowed For p q , let  & only j = q ( x = + 1) or j = kq ( x = 1 ) terms can survive

  22. pq: For j > m :  For j < m + 1 : 

  23. pq:   Only j = 2mterm survives  

  24. Ex.13.3.3 B(p,q)  

  25.  For fixed m, polynomials { Ppm (x) } are orthogonal with weight ( 1  x2 )m. Similarly

  26. Example 15.4.2. Current Loop – Magnetic Dipole Biot-Savart law(for A , SI units) : By symmetry : Outside loop :  E.g. 3.10.4 Mathematica  

  27. For r > a :  

  28. For r > a :

  29. on z-axis : or   (odd in z)

  30. Biot-Savart law(SI units) : Cartesian coord:  

  31. For r > a :

  32. Electric dipole :  

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