UIUC Physics 435 EM Fields & Sources I Fall Semester, 2007 Lecture Notes 17 Prof. Steven Errede

LECTURE NOTES 17

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 ( 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 B()rBr= ∑ n () via n=0 B()rAr=∇× ().

Thus, we carry out a power / / 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 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- 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 term, the 2nd term (~ 1/r2) is the magnetic term, the 3rd term (~ 1/r3) is the magnetic 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 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 ) 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 . Note that mr× ˆ points in the +ϕˆ direction, because mr× ˆˆ=×=+ mzrˆ msinθϕˆ .

⎛⎞μ msinθ valid for r characteristic o ˆ ˆ ∴ Ardipole ()= ⎜⎟2 ϕ i.e. Afcnrdipole = ( ,θ )ϕ ⎝⎠4π r size of m , i.e. ra .

μ ⎛⎞m valid for r characteristic Br=∇× Ar =o 2cosθ rˆ + sinθθ Then: dipole() dipole () ⎜⎟3 ( ) 4π ⎝⎠r size of m , i.e. ra .

Compare this result to the of an electric dipole with electric dipole moment p = qd : 1 ⎛⎞p Er=+2cosθ rˆ sinθθ valid for r characteristic dipole () ⎜⎟3 () 4πε o ⎝⎠r size of p = qd , i.e. rd .

They have the same form!!

We can also write Bdipole ()r in coordinate-free form by using: mmzm==ˆ ⎡⎤cosθ rmˆˆˆ − sinθθ = mrrmii + θ θ { cosθ = zrˆiˆ and sinθ = zˆiθˆ } ⎣⎦()( ) Then: 3()mri ˆˆ r−= m 3 m cosθ r ˆ + m sinθθ − m cos θ r ˆ =+2cosmrmθ ˆ sinθθ

Then, in coordinate-free form the magnetic field associated with a physical magnetic dipole moment, mIa= is: μ ⎛⎞1 B rmrrmo 3 iˆˆ valid for r characteristic dipole ()=−⎜⎟3 ⎣⎡ () ⎦⎤ 4π ⎝⎠r size of m , i.e. ra .

Compare this result to the coordinate-free form of the electric field associated with a physical electric dipole with dipole moment, p :

11⎛⎞ valid for r characteristic Er=−3 prrpi ˆˆ dipole () ⎜⎟3 ⎣⎡ () ⎦⎤ 4πε o ⎝⎠r size of p = qd , i.e. rd .

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For completeness’ sake, we give the coordinate-free form of the magnetic field associated with a point magnetic dipole moment, m :

pointμμoo⎛⎞1 8 3 B rmrrmmr3iˆˆ n.b. valid for all r. dipole ()=−+⎜⎟3 ⎣⎦⎡⎤() δ () 43π ⎝⎠r

Note that the δ- term compensates for the singularity at r = 0 associated with the first term, and arises from calculating the average magnetic field over an infinitesimally small sphere of infinitesimal ε that entirely contains the current density associated with the magnetic dipole moment m (See Griffiths Problem 5.59, p. 254). In , this δ-function term is responsible for hyperfine splitting of bound energy levels in atoms!

Compare this result to the coordinate-free form of the electric field associated with a point electric dipole moment, p (See P435 Lect. Notes 8, p. 8, and/or Griffiths Problem 3.42, p.157):

11⎛⎞ 1 Erpoint=−−3 prrpi ˆˆ prδ 3 n.b. valid for all r. dipole () ⎜⎟3 ⎣⎦⎡⎤() () 43πεoo⎝⎠r ε where again the δ-function term compensates for the singularity at r = 0 associated with the first term, and arises from calculating the average electric field over an infinitesimally small sphere of infinitesimal radius ε that entirely contains the charge densities associated with the electric dipole moment p .

Creation of a Magnetic Dipole Moment from N & S Magnetic Charges: We can create a magnetic dipole moment m (at least conceptually) in a manner completely analogous to that associated with making an electric dipole moment p = qd from two opposite electric charges +q and −q, but instead using N and S magnetic charges ±g for mgd= :

n.b. SI units of: Ampere-m2 ⇒ SI units of magnetic charge, g: Ampere-meters

zˆ +g (N pole) mmzgd==ˆ d −g (S pole)

We summarize below the magnetic dipole moments associated with filamentary/line, surface/sheet and volume current densities:

11 mrIrdIrd=×′′′() = ′′ × If I == Ir constant ∀ ′ 22∫∫CC′′ 11 mrKrdaKrKda=×′′′() = ′′ ׈ If KK==constant ∀ r′ 22∫∫SS′′⊥ ⊥ 1 mrJrd=×′′′()τ 2 ∫v′

© Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois 7 2005-2008. All Rights Reserved. UIUC Physics 435 EM Fields & Sources I Fall Semester, 2007 Lecture Notes 17 Prof. Steven Errede

A comparison of the magnetic dipole fields associated with a pure / ideal / point magnetic dipole moment, mpoint versus a physical / finite spatial extent magnetic dipole moment 2 2 mIaIRzphys ==π ˆ (e.g. aa==π R for a magnetic dipole loop of radius R)

Pure vs. Physical Magnetic Dipole

A comparison of the electric dipole fields associated with a pure / ideal / point electric dipole moment, ppoint versus a physical / finite spatial extent electric dipole moment pphys = qd

Pure vs. Physical Electric Dipole

8 © 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 Magnetic Vector Potential Arquad ( ) and Magnetic Field Bquad ()r

Associated with a Magnetic Quadrupole Moment Qm

We can create / build a magnetic quadrupole moment using two back-to-back magnetic in analogy to how an electric quadrupole was generated from two back-to-back electric dipoles - i.e. use the principle of linear superposition, e.g. using magnetic charges +gm = N, −gm = S poles, or using two identical current loops back-to-back to make a linear magnetic quadrupole:

zˆ zˆ +g N m Magnetic m1 =+ gdzˆ d + gm N Dipole #1 −gm S d “up” (shrink) −2gm 2S −g S d m Magnetic m2 =− gdzˆ d + gm N Dipole #2 +gm N “down”

SI units of magnetic charge: gm = Ampere-meters

zˆ

2 mIaIRz11==π ˆ Two identical magnetic dipole loops I carrying opposing equal currents I, each 2 aRz1 = π ˆ R of radius R and separation distance d = R. I

I R = d 2 aRz2 = π ˆ R I

2 mIaIRz22==−π ˆ

⎛⎞μ I 2 ′ o ′ ′′′ Then (for rr ): Arquad ()=Θ⎜⎟3 () rP2 (cos ) dr () for line currents, ⎝⎠4π r ∫C′ ⎛⎞μ I 2 ⎛⎞31 o ′ 2 ′′′ ′′ˆˆ =Θ−⎜⎟3 ()rdr⎜⎟cos () with cosΘ=rri ⎝⎠422π r ∫C′ ⎝⎠

© Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois 9 2005-2008. All Rights Reserved. UIUC Physics 435 EM Fields & Sources I Fall Semester, 2007 Lecture Notes 17 Prof. Steven Errede

The Magnetic Quadrupole Moment (in terms of discrete magnetic charges):

# discrete magnetic charges 1 n=3 Qrrrg≡−312 m ∑()ii i mi 2 i=1 xˆˆx 0 0 in Unit Dyadic: 1 = 0 yyˆˆ 0 Cartesian 0 0 zzˆˆ coordinates 2 rdz1 =+ ˆ ggmm=+ rrriii= i (i = 1,2,3) 1 rz2 = 0ˆ ggmm=−2 2 rdz=− ˆ gg=+ 3 mm3

⎛⎞=0 =0 31zzˆˆ − 1222 1 22 2() Thus: Qgdzzdgm =−−mm()31ˆˆ ⎜⎟ 30i zzˆˆ − 01 +−=gdzzdmm()312ˆˆ gd 22 ⎜⎟22 ⎝⎠ charge #1 +gm charge #3 +gm charge #2 − 2g at rdz1 =+ ˆ m at rdz3 =− ˆ at rz2 = 0 ˆ

Then for rr ′ : 2 ⎛⎞μμ⎛⎞11()3cosΘ−′ 1 ⎛⎞ ⎛⎞ oo22′ ′ ˆˆ′ Aquad() r==Θ⎜⎟22cos gd m⎜⎟33 ⎜⎟ gd m ⎜⎟ P2 () cosΘ=rri ⎝⎠424ππ⎝⎠rr ⎝⎠ ⎝⎠ 2 2 3 QQmm==m 2 gd Amp-meters*meters = Amp-meters

23 3 In terms of current loops: QmdIadRdIRIm ==22 = 2ππ = 2 (R = d ) Amp-meters

2 23 Qgdmm= 2 or: Qm ==22 mdIadIRdIR = 2(ππ) = 2( ) { dR= here}. Magnetic dipole current loop separation distance, d

⎛⎞μ Q om ′ ′ ˆˆ′ Thus for rr ′ : Arquad ()=Θ⎜⎟3 P2 ()cos where cosΘ = rri ⎝⎠4π r

We can also write this in coordinate-free form {valid for rr ′ , d (R = d here)} as:

⎛⎞μμμQQQ⎪⎪⎧⎫⎡⎤31zzˆˆ−Θ− ⎛⎞⎡⎤ 3cos12 ′ ⎛⎞ omˆˆ om om ′ Arquad ()=××=⎜⎟333⎨⎬ r⎢⎥ r ⎜⎟⎢⎥ = ⎜⎟ P2 ()cos Θ ⎝⎠42424πππrrr⎩⎭⎪⎪⎣⎦ ⎝⎠⎣⎦ ⎝⎠

We can obtain Bquad ()r from: Bquad()rAr=∇× quad ( ) (…an exercise for the energetic student...)

10 © 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

Thus, we can write the multipole expansion of the magnetic vector potential Ar() as: ∞ ⎛⎞μμmrˆˆ× ⎛⎞rQˆˆ×× r Ar A r oom.... ()==∑ n () ⎜⎟23 + ⎜⎟ + n=1 ⎝⎠44ππrr ⎝⎠ magnetic dipole magnetic quadrupole term nn== 1 term 2

Once Ar()is determined, we can obtain B(r ) from B(rAr) =∇× ( ). We can write the magnetic quadrupole tensor Qm as:

⎛⎞QQQ ⎜⎟mmmxx yx zx QQQQm = ⎜⎟mmm ⎜⎟xy yy zy ⎜⎟QQQ ⎝⎠mmmxz yz zz Note that (as for the electric quadrupole moment tensor Q ) the magnetic quadrupole moment e tensor Qm has only six independent components because QQmm= and also note that ij ji QQQ++=0 i.e. Q (like Q ) is traceless. mmmxx yy zz m e

For a linear magnetic quadrupole (oriented along the zˆ -axis:

QQ= and thus: QQQgd=−222 =− = 2 mmxx yy mmmmzz xx yy

Thus, the magnetic quadrupole tensor for a linear magnetic quadrupole is of the form:

⎛⎞−100 For a linear magnetic quadrupole linear 2 ⎜⎟ Qgd=−2010 oriented along the zˆ -axis m m ⎜⎟ ⇐ ⎜⎟consisting of magnetic charges g . ⎝⎠002 m ⎛⎞−100 For two back-to-back magnetic linear 3 ⎜⎟ QIR=−2010π dipole loops carrying steady current or: m ()⎜⎟ ⇐ ⎜⎟I separated by a vertical distance d. ⎝⎠002

Compare these results for the magnetic quadrupole to that for the electric quadrupole (P435 Lecture Notes 8, p. 13-15)

© Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois 11 2005-2008. All Rights Reserved. UIUC Physics 435 EM Fields & Sources I Fall Semester, 2007 Lecture Notes 17 Prof. Steven Errede

Another Kind of Magnetic Quadrupole Using Four Bar :

12 © Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois 2005-2008. All Rights Reserved.