# lec12-1 - ~ R V = 1 4 X l =0 i q i r l i P l (cos i ) R l...

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Legendre series The orthogonality over the interval - 1 < x < 1 can be used to make a series expansion of a function f ( x ) over the same interval f ( x ) = X l =0 c l P l ( x ) We use the orthogonality of the Legendre functions to ﬁnd integrals that determine the c l c l = 2 l + 1 2 Z 1 - 1 f ( x ) P l ( x ) dx

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Recurrence relations The recurrence relations below sometimes come in handy lP l ( x ) = (2 l - 1) xP l - 1 ( x ) - ( l - 1) P l - 2 ( x ) xP 0 l ( x ) - P 0 l - 1 ( x ) = lP l ( x ) P 0 l ( x ) - xP 0 l - 1 ( x ) = lP l - 1 ( x ) (1 - x 2 ) P 0 l ( x ) = lP l - 1 ( x ) - lxP l ( x ) (2 l + 1) P l ( x ) = P 0 l +1 ( x ) - P 0 l - 1 ( x ) Notice ﬁrst recursion relation implies, with P 0 ( x ) = 1 and P 1 ( x ) = x , that highest power of P l ( x ) is x l
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

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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
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Unformatted text preview: ~ 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 o faster with increasing R For example, the quadrupole moments contribute a potential 1 R 3...
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## lec12-1 - ~ R V = 1 4 X l =0 i q i r l i P l (cos i ) R l...

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