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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 θ + r 2 = R q 1 2 r R cos θ + ( r R ) 2 • Use h = r R and x = cos θ , 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 θ variables, we have V = K ∞ X l =0 r l P l (cos θ ) 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 π ∞ X l =0 ∑ i q i r l i P l (cos θ i ) R l +1 • Or if we have a continuous distribution ρ ( ~ r ), V = 1 4 π ∞ X l =0 R R R r l P l (cos θ ) ρ d τ R l +1 • Lowest order term l = 0, is just the total charge, V ∝ 1 R Q = Z Z Z ρ d τ Multipole expansion, continued • Next order term l = 1 is the dipole moment, V ∝ 1 R 2 p = Z Z Z r cos θρ d τ • Writing both the l = 0 (monopole) and l = 1 (dipole) terms, we have V = 1 4 π 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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This note was uploaded on 07/30/2011 for the course PHZ 3113 taught by Professor Staff during the Spring '03 term at University of Central Florida.
 Spring '03
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