Chapter 4 - CHAPTER 4: Multipoles, Electrostatics of...

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3 0 1( ) ( ) (1.17) 4| | In Ch. 3, we developed various methods of expansion for the solution of the Poisson equation. In this chapter, we dx ρ φ πε = x x xx continue the subject of electrostatics by taking a closer look at the source ( ). By the method of expansion, we first decompose ( ) in (1.17) into multipole fields and thereby express the source in x x multipole moments, then show that the atomic/molecular dipole moments account for the macroscopic properties of a dielectric medium and allow a conscise characterization of the medium by a single numbe r called the dielectric constant. x x 0 4.1 Multipole Expansion CHAPTER 4: Multipoles, Electrostatics of Macroscopic Media, Dielectrics
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ρ x x * 1 0 0 11 4 ( , ) ( , ) (3.70) 21 For outside the sphere enclosing , , . 1( ) () 4| l l lm lm l lml r YY l r rr πθϕ θ ϕ φ πε < + == > <> ∑∑ ′′ = −+ ⇒= Multipole Expansion in Spherical Coordinates: xx x x x x [] 2 3 *3 1 0 1 1 | (,) ( , ) ( ) (4.2) monopole ( 0) dipole ( 1) quadrupole ( 2) l lm lm l lm r r v dx Y Yr d x l r l l l θϕ ε + = + =+ +− ⎡⎤ =⇒ ⎢⎥ x x ±²²²²³²²²²´ 3 1 r 4.1 Multipole Expansion ( continued ) partial cancellation of monopoles partial cancellation of dipoles ( multipole moments) lm q
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4.1 Multipole Expansion ( continued ) Expansion in Cartesian coordinates is more useful for our purposes. We first summarize the formulae needed for the expansion. Taylor ex Multipole Expansion in Cartesian Coordinates : () 123 2 1 2 3 12 3 1 : [see Appendix A] ( ) , ( 1 ) where (2) i j j ii i xxx x i ij i j i j xx x x pansion ff f f aa a a a a ∂∂∂ = ∂∂ ∑∑ += +⋅ + + ⋅∇ = + + = ⋅∇ = xa x a x a a x a " 2 2 : ( ) [derived in Sec. 1.5] (3) ( (4) i nn x Other useful relations n nx x ′′ ∇− = −=
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() P 2 2 3 2 35 5 Now apply (1)-(4) to expand 1/ . 11 1 1 1 2 1 2 1 3 22 1 ij ij ij i j ij r rr r xx rx x r r xx xx r r r r r δ ∑∑ ′′ =− + + ⋅∂ =+ + + ∂∂ + + x " " ±²³ ²´ µ²¶ ²· " 2 1 3 (5) 2 i j ij r r +− + " 4.1 Multipole Expansion ( continued ) N 33 3 5 3 3 jj i j i j ii i ij x x j x r r r x = Use (2)-(4) i i j j = Use (1); Write r = x
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3 33 3 2 5 0 0 1 4 1 4 0: () ( ) || 1 (3 2 [ ij q Multipole moments with respect to dx r r xx r r πε ρ φ ρρ = = ′′ =+ +− ∫∫ p x x x x x ±²³² ´± ² ³ ² ´ 3 35 0 0 1 4 1 4 )() 1 2 1 ( ) (4.10) 2 : Multipole moments ] [] ij ij ij ij i j ij Q q Qxx r rr q r Note δρ + + + ⎡⎤ ⇒= + + + ⎢⎥ ⎣⎦ xx Q x px xQ x x I ±²²²²³²²²²´ " " µ²¶ ²· I " are defined with repect to a point of reference. In (4.10), it is the origin of coordinates ( 0). = x 4.1 Multipole Expansion ( continued ) monopole moment quadruple moment dipole moment What is the advantage of expressing this : way?
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Chapter 4 - CHAPTER 4: Multipoles, Electrostatics of...

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