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Unformatted text preview: Expansion of 1 / r potential in Legendre polynomials In electrostatics and gravitation, we see scalar potentials of the form V = K d Take d =  ~ R ~ r  = R 2 2 Rr cos + r 2 = R q 1 2 r R cos + ( r R ) 2 Use h = r R and x = cos , and then we see we have the generating function! V = K R ( 1 2 hx + h 2 ) 1 / 2 = K R X l =1 h l P l ( x ) Then in terms of the r and variables, we have V = K X l =0 r l P l (cos ) R l +1 Multipole expansion If we have make charges q i at different coordinates ~ r i , then we can use this to find the electrostatic potential at ~ R V = 1 4 X l =0 i q i r l i P l (cos i ) R l +1 Or if we have a continuous distribution ( ~ r ), V = 1 4 X l =0 R R R r l P l (cos ) d R l +1 Lowest order term l = 0, is just the total charge, V 1 R Q = Z Z Z d Multipole expansion, continued Next order term l = 1 is the dipole moment, V 1 R 2 p = Z Z Z r cos d Writing both the l = 0 (monopole) and l = 1 (dipole) terms, we have V = 1 4 Q R + p R 2 + ... Higher order terms take into account more details of the distribution with contributions that fall off faster with increasing R For example, the quadrupole moments contribute a potential 1 R 3 Associated Legendre equation (1 x 2 ) y 00 2 xy + l ( l + 1) m 2 1 x 2 y = 0 The Legendre equation corresponds to m = 0 We again have l and m integer, and write the solutions...
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 Spring '03
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