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Unformatted text preview: Lecture 8 Notes, 95.657 Electromagnetic Theory I, Fall 2011 Dr. Christopher S. Baird, UMass Lowell 1. Multipole Expansion of the Potential- Consider a localized charge density completely contained within some region R.- Very far away from the region R , the charge density behaves more and more like a sphere or a point charge.- Far away from the region R then we can make an expansion of the potential in spherical harmonics and keep only the first few terms and it will still be a valid approximation to the solution. - This is useful when the charge density is localized but too complex to be approached in an exact way.- Because we want the potential far away from the charge density, where there is no charge, we can use the spherical coordinates solution to the Laplace equation when a valid solution is required on the full azimuthal range: ( r , , )= l = m = l l ( A lm r l + B lm r l 1 ) Y lm ( , ) where Y lm , = 2 l 1 4 l m ! l m ! e i m P l m cos are the spherical harmonics- The region we are interested in includes infinity, but not the origin. To ensure the solution approaches zero at infinity, we require A l = 0. The solution now becomes: ( r , , )= l = m = l l B lm Y lm ( , ) r l + 1- For later convenience, we redefine the arbitrary constant, B lm = 1 4 4 2 l + 1 q l m so that: ( r , , )= 1 4 l = m = l l 4 2 l + 1 q l m Y lm ( , ) r l + 1- This equation is called a multipole expansion. The l = 0 term is called the monopole term, l = 1 are the dipole terms, etc.- We must now determine the coefficients q lm to fully solve the problem.- The solution in integral form was already obtained as Coulomb's law for the potential: = 1 4 x ' x x ' d x '- We expand the 1 / x x ' factor into spherical harmonics, remembering that we are interested in the solution far away from the charge so that we want the x > x ' case. 1 x x ' = 4 l = m = l l r ' l r l 1 1 2 l 1 Y l m * ' , ' Y lm , so that: = 1 4 l = m = l l 4 2 l 1 [ Y l m * ' , ' r ' l x ' d x ' ] Y lm , r l 1- Comparing this solution to the one above, it becomes apparent that: q l m = Y l m * ( ' , ' ) r ' l ( x ' ) d x '- These coefficients are called the spherical multipole moments. Their physical significance can be seen by representing the first few terms explicitly in Cartesian coordinates.- The l = 0 term is just proportional to the total charge q, which is known as the monopole moment, and has no angular dependence: q 00 = 1 4 x ' d x ' q 00 = 1 4 q- The l = 1 terms are proportional to the components of the electric dipole moment p ....
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