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Multipole Moments

Multipole Moments. Section 41. Expand potential f in powers of 1/R 0. Quadrupole potential. Can set r = 0 before taking derivative. The quadrupole potential has two factors. The factor S a x a x b is a symmetric tensor. It is given by the coordinates of all of the charges.

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Multipole Moments

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  1. Multipole Moments Section 41

  2. Expand potential f in powers of 1/R0.

  3. Quadrupole potential Can set r = 0 before taking derivative

  4. The quadrupole potential has two factors • The factor Saxaxb is a symmetric tensor. • It is given by the coordinates of all of the charges. • A symmetric tensor has 6 independent quantities. ARE THERE REALLY 6? • The rest of f(2) depends only on the field point R0.

  5. Laplace’s equation is satisfied by the potential of a point charge

  6. Every 3-D symmetric tensor can be diagonalized by finding principal aces. • Since Daa = 0, only two principle axes are independent.

  7. Suppose charges are symmetric about z • Then z is a principle axis. • Other two axes lie in the x-y plane. • Their location is arbitrary. • Dxx = Dyy = -Dzz/2 • Let Dzz be “D”. • Now we can express f(2) as product of D and P2(cosq), a Legendre polynomial.

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