EMLecture8

# EMLecture8 - Lecture 8 Notes 95.657 Electromagnetic Theory...

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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: Φ( r , θ , ϕ)= l = 0 m =− l l ( A lm r l + B lm r l 1 ) Y lm , ϕ) 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: Φ( r , θ , ϕ)= l = 0 m =− l l B lm Y lm , ϕ) r l + 1 - For later convenience, we redefine the arbitrary constant, B lm = 1 4 πϵ 0 4 π 2 l + 1 q lm so that: Φ( r , θ , ϕ)= 1 4 πϵ 0 l = 0 m =− l l 4 π 2 l + 1 q l m Y lm , ϕ) 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  0  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.

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1 x x ' = 4 l = 0 m =− l l r ' l r l 1 1 2 l 1 Y l m *  ' , ' Y lm  ,  so that: = 1 4  0 l = 0 m =− l l 4 2 l 1 [ Y lm *  ' , ' r ' l  x ' d x ' ] Y lm  ,  r l 1 - Comparing this solution to the one above, it becomes apparent that: q lm = Y l m * ' , ϕ ' ) r ' l ρ( 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 π q - The l = 1 terms are proportional to the components of the electric dipole moment p .
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