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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: Φ( r , θ , ϕ)= ∑ l = ∞ ∑ m =− l l ( A lm r l + B lm r − l − 1 ) Y lm (θ , 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 = ∞ ∑ m =− l l B lm Y lm (θ , ϕ) r l + 1 For later convenience, we redefine the arbitrary constant, B lm = 1 4 πϵ 4 π 2 l + 1 q l m so that: Φ( r , θ , ϕ)= 1 4 πϵ ∑ l = ∞ ∑ m =− l l 4 π 2 l + 1 q l m Y lm (θ , 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 ∫ 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 * (θ ' , 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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This note was uploaded on 02/13/2012 for the course PHYSICS 95.657 taught by Professor Staff during the Fall '11 term at UMass Lowell.
 Fall '11
 Staff
 Charge, Mass

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