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Lecture Notes 17: Multipole Expansion of the Magnetic Vector Potential, A UIUC Physics 435 EM Fields & Sources I Fall Semester, 2007 Lecture Notes 17 Prof. Steven Errede LECTURE NOTES 17 MULTIPOLE EXPANSION OF THE MAGNETIC VECTOR POTENTIAL Ar( ) As we saw in the case of electrostatics, we carried out a multipole expansion of the scalar ∞ electrostatic potential Vr()= ∑ Vn () r that was valid for distant observation points (field points) n=0 Pr() far from a localized electrostatic source charge density distribution ρTOT ()r′ , which in turn ∞ Er=−∇ Vr enabled us to a corresponding solution for Er()= ∑ En () r via ( )(). n=0 ∞∞ 11 n Vr()== V () r() r′ P (cos Θ′′′ )ρ () r dτ ∑∑nn()n+1 ∫v′ nn==004πε o r with: cos Θ=′′rrˆˆi and rr== r′′ r Likewise, we can similarly/analogously carry out the same kind of multipole expansion for ∞ the magnetic vector potential Ar()= ∑ An () r , obtaining an expression for the magnetic vector n=0 potential that is valid for distant observation / field points Pr( ) far from a localized magnetostatic source current density distribution – e.g. a filamenary/line current I ()r′ , a surface current density Kr()′ , or a volume current density Jr( ′) , which Obtaining a solution for Ar( ) ∞ then enables us to obtain a corresponding solution for the magnetic field B()rBr= ∑ n () via n=0 B()rAr=∇× (). Thus, we carry out a power series / Taylor series / binomial expansion in rr′ with rr′ for Ar() {as we did in the electrostatics case for Vr( ) } where rr ( ′) is the distance from the origin (located near to the charge / current source distribution). For rr ′ , the multipole moment expansion will be dominated by the lowest-order non-vanishing multipole; higher-order terms in the expansion can be neglected/ignored. © Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois 1 2005-2008. All Rights Reserved. UIUC Physics 435 EM Fields & Sources I Fall Semester, 2007 Lecture Notes 17 Prof. Steven Errede Suppose we have a filamentary/line current loop, as shown in the figure below: (Drawing not to scale, rr ′ ) Pr( ) Observation / Field Point r I xˆ r ≡ rr− ′ ϑ zˆ Θ′ I rr= =−rr′ (with rr ′ ) yˆ cos Θ=′′rrˆˆi r′ I ddr′(= ′) Contour of integrationC′ Sr( ′) Current Source Point As we found before for the case of electrostatics, we can write a power-series expansion of 1/r (for rr ′ ) as: n 111∞ ⎛⎞r′ ==ΘP ()cos ′ with cosΘ=′′rrˆˆi 22 ∑⎜⎟n r rr+−′′′2cos rr Θrrn=0 ⎝⎠ Ordinary Legendre′ polynomial of 1 st kind, of order n Then, for a filamentary/line current source distribution with steady current I: ⎛⎞μμI ()rd′′ ⎛⎞ d ′′( r) Ar()==oo I (for I rI′ = =∀constant r′) ⎜⎟′′ ⎜⎟ ( ) ⎝⎠44ππ∫∫CCr ⎝⎠ r ∞ ⎛⎞μ 1 n Ar()=Θo I() r′′′ P (cos ) d () r with cosΘ′ = rrˆˆi ′ ⎜⎟∑ n+1 ∫C′ n ⎝⎠4π n=0 r ⎛⎞μ ⎧⎫11 1312 ⎛⎞2 Ar()=+Θ+Θ−+o I⎨⎬ d′′() r r ′()()cos ′ d ′′ r() r ′ cos ′ d ′′() r ... ⎜⎟ ∫∫CC′′23 ∫ C ′⎜⎟ ⎝⎠422π ⎩⎭rr r⎝⎠ The first term (~ 1/r) in the expansion is the magnetic monopole term, the 2nd term (~ 1/r2) is the magnetic dipole term, the 3rd term (~ 1/r3) is the magnetic quadrupole term, etc. for the multipole expansion of the magnetic vector potential Ar( ) . ∞ Thus, we see that: Ar()= ∑ An () r where n = order of the magnetic multipole, and: n=0 ⎛⎞μ 1 n o ′′′′ Arnn()=Θ⎜⎟n+1 () r P (cos )() Ird for filamentary/line currents I (r′) ⎝⎠4π r ∫C′ ⎛⎞μ 1 n o ′′′′′ Arnn()=Θ⎜⎟n+1 () r P (cos ) Krda () ⊥ for surface/sheet current densities Kr( ′) ⎝⎠4π r ∫S′ ⎛⎞μ 1 n o ′′′′ Arnn()=Θ⎜⎟n+1 () r P (cos )() Jrdτ for volume current densities Jr()′ ⎝⎠4π r ∫v′ 2 © Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois 2005-2008. All Rights Reserved. UIUC Physics 435 EM Fields & Sources I Fall Semester, 2007 Lecture Notes 17 Prof. Steven Errede The reader of these lecture notes may have already realized that since (empirically) there are no (N/S) magnetic charges / no magnetic monopoles have been (conclusively / convincingly) ever observed in our universe, i.e. all magnetic field phenomena arises from (relative) motional effects of electric charges that the n = 0 term in the multipole expansion of the magnetic vector potential Ar() does not exist in nature. Mathematically we can also see this for the n = 0 term: Ar()≡ 0 because e.g. dr′′( ) ≡ 0 around a closed contour of integration 0 ∫C′ This is a consequence of Maxwell’s equation ∇iBr( ) = 0 ∞ ⇒ Ar()= ∑ An () r n=1 Thus, the dominant term for magnetostatics is the (n = 1) magnetic dipole term, e.g. for a filamentary/line current I ()r′ : ⎛⎞μ I o ′ ′′′ ′ˆˆ ′ ˆ ′ ′′ ˆ Ar1 ()== Adipole () r⎜⎟2 rcos Θ di() r with cos Θ=== rr and r rrr, rr ⎝⎠4π r ∫C′ ⎛⎞μ I o ˆ ′′′ = ⎜⎟2 ()()rri d r ⎝⎠4π r ∫C′ ⎛⎞μ I o ′′′ = ⎜⎟3 ()()rri d r ⎝⎠4π r ∫C′ Now if C = any constant vector, then (see Griffiths 1.106, 7 & 8 p. 57): Cri′′ d= a×=−× C C a ∫C′ () 1 Where: adardr≡=′′′ × () = vector area of the contour loop ∫∫SC′′2 And: aan= ˆ where the unit normal nˆ associated with the vector area enclosed by the contour loop is defined by the right hand rule. Thus (here): rC= because the observation / field-point Pr( ) (by definition) is a constant vector, pointing from the defined origin ϑ to the observation / field point Pr(). Then: ()rrˆˆˆi′′ d=×=−× da ′ r r da ′ ∫∫CS′′ ∫ S ′ ⎛⎞μ mr× ˆ o ′ and thus: Ardipole ()= ⎜⎟2 where: mIdaIa≡ = = magnetic dipole moment of loop. ⎝⎠4π r ∫S′ (SI units of m = Amp – m2) © Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois 3 2005-2008. All Rights Reserved. UIUC Physics 435 EM Fields & Sources I Fall Semester, 2007 Lecture Notes 17 Prof. Steven Errede Griffiths Example 5.13: Determine the magnetic dipole moment m associated with a “book-end” shaped loop carrying steady current I as shown in the figure below: zˆ w I I B I I I w I ϑ yˆ I I A w xˆ Use the principle of linear superposition: Superpose two square current loops – one in the x-y plane (of square side w) and another one in the x-z plane (also of square side w). The side in common (line segment AB ) to both square loops have currents I flowing in opposite directions, hence the total current along line segment AB vanishes! zˆ I I Square Loop 2 w B Square Loop 1 (side w, area A = w × w = w2) I I I I (side w, area A = w × w = w2) I w I I ϑ yˆ I I A w xˆ 2 2 mIaIanIwy2222==ˆˆ = mIaIanIwz1111==ˆ =ˆ I I Loop #2 Loop #1 By the principle of linear superposition, the total magnetic dipole moment is: 2 mmmIaIatot =+=12 1 + 2 aIanwz111==ˆ ˆ 22 2 2 by the right-hand rule mIwzIwyIwyztot =+=ˆˆˆˆ() + aIanwy222==ˆˆ 22 2 mmtot==+= tot mm12 2 Iw 4 © Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois 2005-2008. All Rights Reserved. UIUC Physics 435 EM Fields & Sources I Fall Semester, 2007 Lecture Notes 17 Prof. Steven Errede 2 zˆ mIwyztot = ( ˆ + ˆ) 45o yˆ xˆ out of page Note that (here) the magnetic dipole moments m1, m2 and mtot are independent of the choice of origin because the magnetic monopole moment of this magnetic charge distribution is zero. Recall that the electric dipole moment p associated with an electric charge distribution is also independent of the choice of origin, but ONLY when the electric monopole moment (i.e. the net electric charge) associated with that electric charge distribution is zero. The magnetic dipole moments discussed thus far are obviously for a physical magnetic dipole – i.e. one with finite spatial extent. A pure / ideal magnetic dipole moment has NO spatial extent – its area a → 0 while its current I → ∞, keeping the product mIa= = constant. For rr ′ , we asymptotically realize the case for an ideal / pure / point magnetic dipole, e.g. magnetic moments of atoms, molecules, etc. have r′ few Ǻngstroms (~ few x 10−10 m) whereas r ~ 1 – few cm typically. The Magnetic Field Associated with a Magnetic Dipole Moment It is easiest to first calculate the magnetic vector potential Ar( ) and then calculate the corresponding magnetic field B()rAr=∇× ( ) associated with a magnetic dipole moment m by choosing (without any loss of generality) to have the origin ϑ at the location of the magnetic dipole, i.e. place m at r′ = 0 and also orient the magnetic dipole moment such that mmz= ˆ (i.e. align m ║ to the zˆ -axis). Then: cosΘ=′′rrˆˆi = cosθ (i.e. θ = the usual polar angle) and: r ≡ rr−=−=′ r0 r, rr ′ zˆ ϕˆ Pr( ) Observation / Field Point r = r Θ′ = θ θ m Sr()′ Source Point ϑ yˆ = Local Origin ϕ ϕˆ xˆ SI units of m = Amp-m2 Note: zrˆ =−cosθ ˆ sinθθ mIa= From the multipole moment expansion of the magnetic vector potential Ar( ) we have: © Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois 5 2005-2008. All Rights Reserved. UIUC Physics 435 EM Fields & Sources I Fall Semester, 2007 Lecture Notes 17 Prof. Steven Errede ⎛⎞μ mr× ˆ Ar= o mmzm==ˆ cosθ rˆ − sinθθ dipole () ⎜⎟2 ( ) ⎝⎠4π r zrˆ× ˆˆ=−×(cosθθθ r sin ) r ˆ And: mr×=ˆ msin Θ where Θ = opening angle between zrˆ and ˆ rˆ×=θ ϕˆ θ × rˆ =−ϕˆ rrˆˆ× = 0 Very Useful θϕ ×=ˆ rˆ ϕθˆ × =−rˆ θθ × = 0 ⇐ Table # 2 ϕˆ ×=rˆ θ rˆ×ϕˆ =−θ ϕϕˆˆ× = 0 But Θ=θ here, and thus mr×=ˆ msinθ .
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