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chapter 3 electrody -...

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Boundary-Value Problems in Electrostatics II Reading:  Jackson 3.1 through 3.3,  3.5 through 3.10 Legendre Polynomials These functions appear in the solution of Laplace's eqn in cases with azimuthal symmetry. Consider a point charge  q  located at ( x y z ) = (0, 0,  a ). x y z a d r Potential   at point ( r,   ) 1 q
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where is the  “generating   function”  for Legendre polynomials. then  g ( t x ) can be expanded in a binomial series. To derive the binomial expansion,  start with Maclaurin's Thm: , etc. 2
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e.g.,  n  = 5: Binomial Thm: 3
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We would like to express this as a power series in  t , with coeffs that depend on  : To this end, the following theorem is useful: Proof:   Start with 4
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A few examples: 5
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=> Since this vanishes for all values of  t , each power of  t  must vanish separately    (recurrence relation) 6
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This can be used to find higher order Legendre polynomials.   For example, given  P 0 ( x ) = 1 and  P 1 x ) =  x , we find (with  n  = 1) , as we found with the explicit formula. With repeated  application, we find: Note that  P n ( =1) = 1 7
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8
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9
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Differentiating recurrence relation wrt  : 10
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= 11
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(“Legendre's differential eqn”) Legendre polynomials satisfy Legendre's eqn   (hence the name). Legendre's eqn may also be written as: (interchanging  m  and  n ) Subtract eqns and integrate from  x  = -1 to 1: 12 0
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=>  Legendre polynomials are orthogonal on interval [-1, 1]. What about  n  =  ? (cross terms vanish because of orthogonality) 13
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Legendre polynomials are also complete on [-1, 1]. Legendre series: with Next, we'll derive a useful alternative formula for computing Legendre polynomials, called  “Rodrigues' formula” : 14
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(as derived earlier) (1) (2) (3) (1)  Differentiating  x 2 n -2 r   n  times yields (2)  The sum is extended from [ n /2] to  n .   In each of these terms the        exponent of  x  is <  n   =>   n  differentiations yields zero.  So, we're just        adding zero. 15
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Laplace eqn in  spherical coords: Separation of variables: If the region under consideration includes the full range of azimuth (0     < 2 ), then  Q ( ) must be periodic (with period 2 ). =>  const must be negative: with  m  an         integer Thus,  16
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Eqn for  P ( ): (“generalized Legendre eqn”) 17
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When  m  = 0 (as for cases with azimuthal symmetry, i.e., quantities do  not depend on the azimuthal angle  ), this reduces to the Legendre eqn. We know that Legendre polynomials satisfy Legendre's eqn for non-negative integer values of  .   Other values of   are excluded on physical grounds, since in these cases the soln diverges at  x  = +1 or -1 (  = 0 or  ).  The same is  true of the 2 nd
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