Bound to the Nucleus of an Atom No Longer Behave Like Point-Like Particles, but As Quantum-Mechanical Matter Waves

UIUC Physics 435 EM Fields & Sources I Fall Semester, 2007 Lecture Notes 19 Prof. Steven Errede

LECTURE NOTES 19

MAGNETIC FIELDS IN MATTER THE MACROSCOPIC MAGNETIZATION, Μ

There exist many types of materials which, when placed in an external magnetic field Bext ()r become magnetized — i.e. at the microscopic level ∃ internal atomic/molecular magnetic dipole moments matom or mmolecular , which, in the presence of the external aligning magnetic field Bext ()r produce magnetic torques τ (rmrBr) =×( ) ext ( ) which act on the individual atomic/molecular dipole moments, thereby causing a net alignment of the atomic/molecular magnetic dipole moments mmatom molecular which in turn results in a net, macroscopic magnetic polarization, also known as the magnetization, Μ (r ) . This is analogous to the situation associated with dielectric materials where electrostatic torques τ (rprEr) =×( )()ext act on individual atomic/molecular electric dipole moments ppatom molecular in an external electric field Erext () resulting in a net, macroscopic electric polarization, Ρ(r ) .

In the absence of an external applied magnetic field (i.e. Brext ( ) = 0) the macroscopic alignment of the atomic/molecular magnetic dipole moments mmatom molecular (in many, but not all magnetic materials) is random, due to fluctuations in the internal thermal energy of the material at finite temperature (e.g. room temperature). Thus, no net macroscopic magnetization M ()r exists in many such materials for Brext ( ) = 0 at finite (absolute) temperature, T.

We define the macroscopic magnetic polarization (a.k.a. magnetization) M ()r of a magnetic material in complete analogy to that associated with the macroscopic electric polarization P()r of a dielectric material: Macroscopic Electric Polarization P()r : ⎛⎞electric dipole moment 2 P()rr= ⎜⎟ at point SI Units of P : Coulombs/m ⎝⎠unit volume NN prmol( i ) Qd( r) P rnpr=≡i =ii i ()mol mol () ∑∑Volume, VVVolume, ii==11 # atoms/molecules n = mol unit volume

Macroscopic Magnetic Polarization/Magnetization M(r ) : ⎛⎞magnetic dipole moment M()rr= ⎜⎟ at point SI Units of M : Amperes/meter ⎝⎠unit volume NN mrmol( i ) Ia( r) M rnmr=≡i =ii i ()mol mol () ∑∑Volume, VVVolume, ii==11

© 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 19 Prof. Steven Errede Note that the magnetization Μ ()r has SI units the same as that for a surface current density, Kr() (Amperes/meter), whereas the electric polarization P(r ) has SI units the same as that for a surface charge density,σ ()r (Coulombs/m2).

There are (at least) four kinds of magnetism: Μdia ()r Bext (r ) 1.) DIAMAGNETISM: The induced macroscopic magnetization Μdia (r )is antiparallel to Bext (r ) . Due to the physics origin of diamagnetism at the microscopic scale – i.e. at the atomic/molecular scale, ALL substances are diamagnetic! However, diamagnetism is very a weak phenomenon – other kinds of magnetism (see below) can “over-ride”/mask out the diamagnetic behavior of a material.

Diamagnetism results from changes induced in the orbits of electrons in the atoms/molecules of a substance, due to the applied/external magnetic field. The direction of the change in orbital motion of the electrons is such that it to opposes the change in applied magnetic flux (this is nothing more than Lenz’s Law acting at the microscopic/atomic/molecular scale!).

Superconductors are examples of strong diamagnets – they are in fact perfect diamagnets, completely* screening out the applied external magnetic field Bext (r ) (* if no flux-pinning defects are present in the superconducting material). Note that Μdia (r )vanishes when Brext ()= 0.

Μ para ()r Bext (r ) 2.) PARAMAGNETISM: The induced macroscopic magnetization, Μ para (r )is parallel to Bext (r ) . Atoms or molecules that have a net orbital and/or intrinsic spin magnetic dipole moment m (e.g. atoms/molecules with unpaired electrons – such as A…, BaCaNaSrU, , , , , and also metals – due to the magnetic dipole moments m associated with intrinsic spins of the conduction electrons) are paramagnetic materials. The external applied magnetic field B (r ) exerts a torque on these ext atomic/molecular magnetic dipole moments m which tends to (partially) align them, giving rise to a net Μ para ()r which is parallel to Bext (r ) . The energy of alignment UrMext()=− mrBr ()i ( ) is a minimum when m is parallel to Bext (r ) . This is analogous to the net induced electric polarizationΡ()r which is parallel to Erext ( ) in dielectric materials, the energy of alignment UrEext()=− prEr ()i ()when p is parallel to Erext ( ) . Note that Μ para (r ) also vanishes when Brext ()= 0.

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 19 Prof. Steven Errede 3.) FERROMAGNETISM: The macroscopic magnetization,Μ ferro (r ) depends on the (entire) past history of exposure to Bext ()r !! There exists a non-linear hysteresis-type relation between Μ ferro ()r and Bext ()r . Iron and other ferromagnetic materials have a “macroscopic” crystalline domain structure (a typical scale length involves many thousands of atoms), within a domain (nearly) all of the atomic/molecular magnetic dipole moments m are aligned parallel to each other ⇒Μdomain ()r can be very large. However, the orientation of Μdomain over many domains is ≈ random, unless Brext ()≠ 0 . However, ferromagnetic materials have a critical temperature

(known as the Curie Temperature TC ) below which the domains can spontaneously align − a phase transition occurs in the material at this temperature! In the presence of an external applied magnetic field Bext the alignment of ferromagnetic domains tends to be parallel to Bext , but it is in fact (more) complicated than this, because it it history dependent!!! The alignment arises from quantum mechanics – intrinsic spin and the Pauli exclusion principle. Thus, Μ ferro ()r does not vanish when Brext ()= 0!!! History-Dependence / Hysteresis Relation Between Μ ferro and Bext for Ferromagnetic Materials

for TT< C (= Curie Temperature):

Ferromagnetic behavior vanishes for TT> The material then becomes paramagnetic. C The arrows indicate the path taken for Μ ferro : Bext starts at Bext = 0 , then goes to Bmax , then through 0, going to Bmin , then through 0 again and then going to Bmax , etc….

4.) ANTI-FERROMAGNETISM (a.k.a. FERRIMAGNETISM) In some magnetically-ordered materials ∃ an anti-parallel alignment of intrinsic spins, due to two (or more) inter-penetrating crystalline structures, such that no spontaneous magnetization in the bulk material occurs. Ferrimagnetism/antiferromagnetism occurs for temperatures TT< Ne' el . Materials exhibiting antiferromagnetic properties are relatively uncommon – e.g. URu2Si2.

© 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 19 Prof. Steven Errede

FORCES & TORQUES ON MAGNETIC DIPOLES When a magnetic dipole with magnetic dipole moment m is placed in an external magnetic field B a torque on the magnetic dipole τ =×mB will occur, just as we saw for the case of ext M ext an electric dipole with electric dipole moment p when it is placed in an external electric field Eext giving rise to a torque on the electric dipoleτ Eext= p× E .

As we also learned for the case of an electric dipole in a uniform external electric field, similarly, for a magnetic dipole placed in a uniform external magnetic field, there is no net force acting on the magnetic dipole.

For a magnetic dipole with magnetic dipole moment m (e.g. arising from a current loop) placed in an uniform external magnetic field Bext the net force on the is zero: net FIdrBrIdrBrmextext=×=×=′′() () ′ ′′ () () ′ 0 ∫∫CC′′( ) ≡ 0 cf w/ that for an electric dipole placed in a uniform external electric field Eext : net Fp=+= FrFrqErqEr++() −− () ext () + − ext () − = qErEr( ext () + − ext () −) =0

The nature of the magnetic (electric) torque τ M =×mBext (τ Eext=×p E ) is such that it tends to align mp() with (i.e. parallel to) the applied/external Bext (Eext ) respectively. ⇒ The effect(s) of magnetic torque explains paramagnetism, with Μ para B ext . One might be tempted to believe that paramagnetism should be a universal phenomenon, common to all materials. However, paramagnetism is connected to the intrinsic magnetic dipole moment of an unpaired electron and/or its orbital magnetic dipole moment. Because of the Pauli exclusion principle (identical fermions, here, electrons) cannot be in the exact same quantum state, hence pairs of electrons can only be in the same quantum state with one of them spin-up, and the other spin down. Thus, torques on paired magnetic dipole moments (or more correctly, the B -fields associated with the paired electron magnetic dipole moments me ) cancel.

⇒ Paramagnetism only arises in atoms/molecules with an odd number of electrons – the outermost electron is unpaired ⇒ hence it (alone) is subject to magnetic torque(s).

As we saw in the case for an electric dipole with electric dipole moment p in a non-uniform external electric field E , a non-zero force acts on the electric dipole. Similarly, for a magnetic ext dipole, with magnetic dipole moment m in a non-uniform external magnetic field Bext experiences a non-zero force:

Fm() r=∇() mr ()ii B ext () r =() mr() ∇ B ext () r {last step valid iff mr( )= constant vector} Fp() r=∇() pr ()ii E ext () r =() pr() ∇ E ext () r {last step valid iff p(r ) = constant vector}

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 19 Prof. Steven Errede

Similarly, work (= potential energy) of a magnetic (electric) dipole moment in an external magnetic (electric) field, Bext()E ext are (respectively) given by: WPEmBmm==−.. i ext vs. WPEpEpp==−.. i ext

The Physics of Diamagnetism

Atomic electrons orbit/revolve around the nucleus of the atom at some mean / average / characteristic radius, R. Atomic electrons bound to the nucleus of an atom no longer behave like point-like particles, but as quantum-mechanical matter waves. However, an orbiting atomic electron “wave” still constitutes a circulating current:

evee ev ev e IQM ~ == λe CR= 2π for ground state λπe CRGnd State2 Gnd State

Conventional zˆ Current, I Bext= Bz o ˆ

R e−

vv= ϕˆ ee mmzee=− ˆ

Classically, a circulating point electric charge has:

e ev I = with τ ==CR2π ⇒ I ==e I Class orbit vv Class QM τ orbit ee 2π R

⎛⎞eve 2 1 Then: mIa==−⎜⎟π R zevRzˆˆ=− ()e ⎝⎠2π R 2 due to e− charge With no external magnetic field applied Bext = 0, thus the forces acting on the atomic electron are: FFelectrostatic= centripetal 1 Ze2 v2 1 Ze2 v2 e ˆˆe −=−=−2 rmaˆˆe centipetal mr e ⇒ Equation A: 2 rm= e r 4πε o R R 4πε o R R

me = mass of electron Z = nuclear electric charge # {+Ze = nuclear charge}

© 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 19 Prof. Steven Errede With an external magnetic field present Bext ≠ 0, thus the forces acting on the atomic electron are: net FFEM=+= electrostatic FF B centripetal′ 1 Ze2 net ˆ FFEM=+=−−× electrostatic F B2 revB() e ext 4πε o R Suppose Bext = Bz0 ˆ and vvee= ϕˆ (as shown in above pix) Then: ϕˆˆ×=×zrˆ ϕθθθ()cosˆ − sin ˆ θ = 90 sinθ = sin 90 = 1 =−ϕθˆ ×ˆ =−() −rˆ =+rˆ cosθ = cos90 = 0

rˆ×=θˆ ϕˆ θˆ×=−rˆ ϕˆ

θϕˆ×=ˆ rˆ ϕθˆ ×=−ˆ rˆ

ϕˆ ×=rˆ θˆ rˆ×=−ϕˆ θˆ

1 Ze2 v′2 net ′ ′ e Then: FevBrEM=−2 − e o ˆ ==−Fmrcentripetal e ˆ 4πε o R R 1 Ze2 v′2 ′ e Then for Bext ≠ 0 we have Equation B: 2 +=eveo B m e 4πε o R R Note that since we have an additional term on LHS of Equation B, then we see that: vBeext′ ()()≠≠00 vB eext =.

Subtract Equation A from Equation B: m e 22 ′ eveee′′ B0 =−() v v ⇒>vvee for B ext =+ Bz0 ˆ vvee′ <=− for B ext Bz0 ˆ R ( ) >0 >0 ′22 If the change in ve , Δ≡vvveee()′ − is small, then: vvee−=−()()() vvvv eeee′′ +=Δ+ vvv eee ′

But: vvee′ =+Δ v e (since Δ≡vvveee()′ − )

22 ∴ vee′ −=Δ+Δ+=Δ+Δ+=Δ+Δ v vv eeee()() v v vv eeee( vv) v ee(2 v v e) =Δ+Δ22vv v2 vv Δ ee e ee neglect m ∴ ev′ B=+Δ e() v v B e ()2 v Δ v eee00R ee 2 v m ev B e e Δv e 0 R e

eBo R or: Δve 2me

6 © 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 19 Prof. Steven Errede

ev ev′ But if: I ==>ee and Ivv′′ and 22ππRRee ′ ev()ee− v evΔ e eBo R Then: Δ=III′ − = = but: Δve 22π R π R 2me eB2 R eB2 ∴ Δ=I o = o 4π mRe 4π me eB2 Δ=I 0 4π me

Then: mIaIR==π 2 and mIaIR′′== ′π 2 (aR= π 2 )

Thus: Δ=mmm′′ − =() I − Ia =Δ=Δ IaIRπ 2

2 22 2 ⎛⎞eB 2 eBR ∴ Δ=ΔmIRπ =⎜⎟o π R = o ⎝⎠4π me 4me But recall that mmz=− ˆ

i.e. m points down. 22 eBRo Therefore: Δ=−mzBBzˆˆ, ext = o 4me

⎛⎞eR22 Or: Δ=−mB⎜⎟ext ⎝⎠4me

The point is, that for diamagnetic materials, the change in the magnetic dipole moment m , Δm is opposite to the direction of B - i.e. if B = Bzˆ increases, then m also increases, but in the ext ext o opposite direction to try to cancel/buck the external/applied magnetic field, Bext . This is a simply a manifestation of Lenz’s Law at the atomic scale!!!

This is what phenomenon of diamagnetism is due to, at least from a ≈ semi-classical perspective.

The induced dipole moments in diamagnetic materials (essentially every material) point in the direction opposite to the applied magnetic field. The macroscopic magnetization Μ resulting from diamagnetism is relatively speaking very small. Diamagnetism (except in superconductors) is extremely weak.

© 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 19 Prof. Steven Errede

THE MAGNETIC VECTOR POTENTIAL Ar( ) , THE MAGNETIC FIELD B()rAr=∇× ( ) OF A MAGNETIZED OBJECT WITH MAGNETIZATION Μ ()r

Recall that the magnetic vector potential Ar( ) of a magnetic dipole with magnetic dipole moment m ()Amp-m2 is: ⎛⎞μo m× rˆ Ardipole ()= ⎜⎟2 {SI Units: Tesla-meters = Newtons/Ampere = F/I !!!} ⎝⎠4π r

Thus, in a magnetized object with macroscopic magnetization (magnetic dipole moment per unit volume) Μ ()r′ , each volume element dτ ′within the volume v′ has a magnetic dipole moment associated with it of: mr( ′) =Μ( r′′) dτ .

Thus, the infinitesimal contribution to the magnetic vector potential Ar( ) due to the magnetic dipole moment mr()′ associated with the macroscopic magnetizationΜ ()r′ in the infinitesimal volume element dτ ′is:

⎛⎞μμmr( ′)× ˆˆ ⎛⎞Μ×( r′′) dτ oor r ′ dA() r ==⎜⎟22 ⎜⎟ with r = rr− ⎝⎠44ππr ⎝⎠ r

Then the total magnetic vector potential Ar( ) is obtained by integrating this expression over the entire volume v′ of the magnetized material:

′ ′ ⎛⎞μo Μ (rd) τ × rˆ Ar()== dAr () ⎜⎟ 2 ∫∫vv′′⎝⎠4π r

11ˆ ′′⎛⎞ r Now again: ∇=∇=⎜⎟ 2 ⎝⎠r rr− ′ r

⎛⎞μ ⎡⎤ ⎛⎞1 Thus: Ar()=Μ×∇o () r′ ′′ dτ ⎜⎟∫v′ ⎢⎥⎜⎟ ⎝⎠4π ⎣⎦⎝⎠r Integrating by parts, and using ∇×=∇×−×∇( fAf) ( AAf) ( ) :

⎛⎞μ ⎪⎪⎧⎫1 ⎡⎤Μ ()r′ Then: Ar()=∇×Μ−∇×o ⎨⎬⎡⎤′ () r′′ dτ ′ dτ ′ ⎜⎟∫∫vv′′⎣⎦ ⎢⎥ ⎝⎠4π ⎩⎭⎪⎪rr⎣⎦

Then using: ∇× Vrd()τ =− Vr () × daVr () = Arbitrary Vector Point Function ∫∫vS ( ) (See Griffiths Problem 1.60 (b), page 56)

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 19 Prof. Steven Errede

⎛⎞μ ⎧⎫11 Thus: Ar=∇×Μ+Μ×o ⎡⎤′ r′′ dτ ⎡ r ′ da ′⎤ () ⎜⎟⎨⎬′′⎣⎦() ⎣ () ⎦ ⎝⎠4π ⎩⎭∫∫vsr r

But: da′′= ndaˆ

μoo11 μ Then: Ar()=∇×Μ+⎡⎤′′′() r dτ ⎡⎤ Μ×=() r ′ nˆ da ′ ABound () r 44ππ∫∫vS′′⎣⎦ ⎣⎦ rr ′ ′ ≡ JrBound () ≡ KrBound () μμJr( ′′) Kr( ) Or: Ar()=+ooBound dτ ′ Bound da′ with =−rr′ Bound ′′ r 44ππ∫∫vSr r

Compare this result to that which we obtained for the magnetic vector potential Ar()associated with a free volume current density Jrfree ( ′) and a free surface/sheet current density Krfree ( ′) (see P435 Lecture Notes 16, page 6):

μμJr( ′′) Kr( ) Ar=+oofree dτ ′ free da′ with =−rr′ free () ′′ r 44ππ∫∫vSr r

Thus for a magnetized material with macroscopic magnetization (magnetic dipole moment per unit volume) Μ ()r′ contained within in the enclosing source volume v′ bounded by the surface S′ , the magnetic vector potential at the field/observation point Ar( ) arising from the sum total of the macroscopic magnetizationΜ ()r′ present in the material can be equivalently represented by contributions from an equivalent bound volume current density JrBound ( ′)()≡∇′′ ×Μ r and an equivalent bound surface current density Kr( ′′) ≡Μ( rn) ×ˆ where nˆ = outward unit Bound surface normal at the surface of the magnetized material.

On the interior of the magnetized material: 2 JrBound ()′′≡∇ ×Μ() r ′ = equivalent bound volume current density, SI units = Amps/m 1/m Amps/ m2 Amps/ m

On the surface(s) of the magnetized material:

Kr′′≡Μ rn ×ˆ = equivalent bound surface current density, SI units = Amps/m Bound () () Amps// m Amps m

μμJr()′′ Kr( ) Then: Ar=+ooBound dτ ′ Bound da′ with = rr− ′ Bound () ′′ r 44ππ∫∫vSr r

© 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 19 Prof. Steven Errede

MAGNETIC MATERIALS DIELECTRIC MATERIALS JrBound ()′′≡∇×Μ () r ⇔ ρBound (rr′) =−∇Ρi ( ′) Kr()′′≡Μ () rn ×ˆ ⇔ σ (rrn′′) =Ρ( )i ˆ Bound surface Bound surface

So again, instead of integrating over the macroscopic magnetization Μ ()r′ (or polarization Ρ()r′ ) arising from the direct contributions from the infinitesimal magnetic (and/or electric) dipoles mr()′ (and/or p()r′ ), we replace these by macroscopic bound volume and surface current distributions JrBound()′′ and Kr Bound ( ) (and/or ρBound (r′) and σ Bound ()r′ ); we can then obtain Ar() (and/or Vr()). Once Ar( ) (and/or Vr( ) ) is known, we can then obtain B(r ) from B()rAr=∇× () (and/or Er()from Er( ) =−∇ Vr( ) ) !

Note that for a magnetized material with macroscopic magnetization Μ ()r′ we can also obtain the equivalent bound current, IBound from:

IBound=+ J Bound() rda′′ K Bound( rd ′′) surface ∫∫S′′⊥⊥ C surface ⊥⊥ Consider the equivalent bound surface current K associated with a thin slab of Bound magnetized material that has been placed in uniform magnetic field B = Bzˆ , in turn producing ext o a uniform macroscopic magnetization (magnetic dipole per unit volume) Μ =Μo zˆ . At the microscopic level, atoms and/or molecules will tend to have their induced and/or permanent magnetic dipole moments lined up parallel/anti-parallel to Bext for paramagnetic / diamagnetic materials, respectively. Suppose that the material is paramagnetic, as shown in the figure below:

B = Bzˆ produces mIaIaz==ˆ ext o uniform magnetization Μ =Μo zˆ

It can be seen from the above figure that on the interior of the uniformly magnetized material the atomic/molecular microscopic currents will cancel each other (for uniform magnetization, Μ=Μo zˆ ) except on the periphery (i.e. the surface) of the magnetic material. For uniformly magnetized material(s), e.g.Μ =Μo zˆ : JrBound( ) ≡ ∇×Μ( r)() =∇× Μ o zˆ =0 .

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 19 Prof. Steven Errede

Then: IKrdBound= Bound()′′ surface for uniform magnetization, e.g. Μ=Μ zˆ . ∫Csurface′ ⊥ o ⊥

Example: Consider a cylindrical rod of radius a and length of magnetized material immersed in a uniform Bext= Bz o ˆ as shown in the figure below. Then the magnetization is uniform, e.g. Μ=Μo zˆ . Thus, no equivalent bound volume current density JrBound ( ) exists, because JrBound()=∇×Μ () r =∇× ( Μ o zˆ ) =0 for uniform magnetization, Μ =Μo zˆ . Uniform Μ=Μo zˆ zˆ Μ=nmmol mol Bext= Bz o ˆ produces uniformΜ=Μo zˆ

=Μo zˆ a

Μ= m Volume K IKrd= ()′′ Tot bound Bound∫Csurface′ Bound⊥ surface ⊥ 2 Μ= maTot π yˆ = Kbound ϕˆ ϕˆ Knn= Μ×ˆˆwith =ρˆ Boundsurface surface mTot = total dipole moment xˆ ρˆ =Μooo( zKˆ ×ρˆˆˆ) =Μϕϕ = of magnetized ∴ IKBound==Μ boundϕˆˆ o ϕ, cylinder i.e. KKBound=Μ oϕˆˆ = oϕ

If B ≠ uniform magnetic field, will result in a non-uniform magnetization, i.e. Μ≠uniform, ext which in turn also implies that the equivalent bound volume current density J Bound =∇×Μ≠0 . This means that at microscopic level the atomic/molecular current loops no longer cancel each other (completely) in the interior region of the magnetized material. Hence for Μ≠uniform: Volume JrBound()′′=∇×Μ () r ≠0 ⇒ I Bound = Jrda Bound ( ′) ∫S′ ⊥ ⊥

Similarly, we also expect for non-uniformΜ that Kr( ′′) = Μ×( rn) ˆ ≠0 and thus we Bound surface will also have an equivalent bound surface current: Surface then IKrdBound= Bound()′′ surface (for magnetized cylinder in above figure: ddz′ = ) ∫Csurface′ ⊥ ⊥ ⊥

Then using the principle of linear superposition: Tot Volume Surface IBound=+= I Bound I Bound J Bound( rda′′′) + K Bound( rd) surface ∫∫S′′⊥⊥ C surface ⊥⊥

Note that these equivalent bound currents are flowing in different places in/on the magnetized material – one is flowing inside the material, the other is flowing on the surface of the material.

© 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 19 Prof. Steven Errede

Note also that: ∇=∇×∇×Μ=iJrBound ()() () r0 always in magnetostatics, because ∇iJrBound ( ) is the LHS of the Continuity Equation for equivalent bound currents (i.e. conservation of bound charge): ∂ρBound (rt, ) ∇=−=iJrtBound (),0 if ρ (rt, ) ≠ fcn(t). ∂t Bound

Note also that (here): ∇×() ∇×Μ()rrr =∇( ∇i Μ()) −∇2 Μ() =0 i.e.: ∇∇Μ()i ()rr =∇Μ2 () {We will come back to this relation in the near future…}

Griffiths Example 6.1: Determine the magnetic field B()r associated with a uniformly magnetized sphere of radius R with uniform magnetization Μ=Μo zˆ as show in the figure below. Choose the local originϑ to be at the center of the magnetized sphere:

zˆ ϕˆ Pr( ) Field/Observation Point Μ=Μo zˆ rˆ θ θˆ ϑ yˆ ϕ ϕˆ xˆ

Since the magnetization of the sphere is uniform, then: JrBound( ) = ∇×Μ( r)() =∇× Μ o zˆ =0 . However: Kr()=Μ () rn ×ˆˆ =Μ( zrˆ ×) =Μ sinθ ϕˆ where: nrˆˆ= Boundsurface o o surface Note: zrˆ×=ˆˆ()cosθ r − sinθθˆˆ ×=− r ˆ 0 sin θ( θ × r ˆ) =+ sin θ ϕˆ since: rrˆˆ× =×=−0, θˆ r ˆ ϕˆ

Now recall that we learned in Griffiths Example 5.11 (p. 236-7)/P435 Lecture Note 16 p. 18-19 (the charged spinning hollow sphere) that: Kvfree ==×=σ σω rR′ σωsin ϕ ϕˆ

Uniformly Magnetized Sphere: Charged Spinning Hollow Sphere: KBound=Μ o sinθ ϕˆ vs. KRfree = σωθϕsin ˆ ⇒ Μ=o σωR 22 2 BrR()<=μμ Μ=Μ zˆ ⇐ B ()rR<=μσω() Rzˆ inside33 o o o inside3 o ⎛⎞μ m ⎛⎞μ m BrR>=o 2cosθ rˆ + sinθθˆ BrR>=o 2cosθ rˆ + sinθθˆ outside ()⎜⎟3 () ⇐ outside ()⎜⎟3 () ⎝⎠4π r ⎝⎠4π r 44 44 mR=Μ=Μππ33 Rzˆ vs. mRRzRz==π 34()σωˆˆ π σω 33o 33

12 © 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 19 Prof. Steven Errede

For magnetic media, we have obtained the following relations: ∂ρ (rt, ) Bound current continuity equation: ∇=−iJrt(), Bound Bound ∂t

Equivalent bound volume current density JrBound ( ) =∇×Μ( r) , where Μ ()r = magnetization (a.k.a. magnetic dipole moment per unit volume) Μ=(rnmr) mo mol( ) = mrvolume Tot ( ) and the equivalent bound surface current density Kr( ) = Μ×( rn) ˆ ≠0 with corresponding relations Bound surface Volume Surface I Bound= Jrda Bound () and IKrdBound= Bound( ) surface ∫S ⊥ ∫Csurface ⊥ ⊥ ⊥

Using the principle of linear superposition: total current density = free current + bound current density: Jrtot() =+ J free( r) J Bound ( r) Krtot()=+ K free( r) K Bound ( r) Ampere’s Circuital Law becomes (in differential form) for the magnetic field B()r : ∇×B()rJrJrJr =μμμo Tot () = o free( ) + o Bound ( ) Note that this is the analog of Gauss’ Law (in differential form) for the electric field Er( ) : 11 ∇=iEr() ρρρTot() r =() free() r + Bound () r εεoo

Now: JrBound ()≡∇×Μ () r ∴ ∇×B()rJr =μμofree () + o( ∇×Μ () r)

or: ∇×B()rrJr −μμoofree() ∇×Μ () = () 1 or: ∇×B()rrJr −∇×Μ () = free () μo ⎧⎫1 or: ∇×⎨⎬B()rrJr −Μ () = free () ⎩⎭μo 1 SI Units of H = Amperes/meter We now define the auxiliary field: Hr()≡−Μ Br() () r – the same as that for Μ !!! μo

We could call Hr()the magnetic displacement, in analogy to the electric displacement:

Dr( ) =+Ρε o Er( ) ( r) But usually we just call H “the H -field”.

© Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois 13 2005-2008. All Rights Reserved. UIUC Physics 435 EM Fields & Sources I Fall Semester, 2007 Lecture Notes 19 Prof. Steven Errede Ampere’s Law for the H -field (in differential form) then becomes: 1 ∇×Hr() = J () r where: Hr()≡−Μ Br() () r free μ o Ampere’s Law for the H -field is the analog of Gauss’ Law for the D -field, in differential form:

∇=iDr()ρ free () r where: Dr( ) ≡+Ρε o Er( ) ( r)

n.b. both Hr() and Dr () are auxiliary fields, B(rEr) and ( ) are fundamental fields.

In integral form, these relations become:

enclosed 1 enclosed enclosed enclosed Hr()i d= I Br()i d==+ ITot I free I Bound ∫C free ∫C μ0

Dr()i daQ= enclosed ε Er( )i daQ==+enclosed Q enclosed Q enclosed ∫S free o∫S ToT free Bound

We also have the relations:

JrBound ()′′=∇×Μ () r ρBound (rr′) =−∇Ρi ( ′) Kr()′′=Μ () rn ×ˆ σ ()rrn′′=Ρ ()i ˆ Bound surface Bound surface Volume I = Jrda′′ Volume Bound Bound () ⊥ QrdBound= ρ Bound ( ′) τ ′ ∫S⊥ ∫v′ I surface = Krd()′′ QrdaSurface = σ ()′ ′ Bound∫C Bound ⊥ Bound∫S′ Bound ⊥ Volume I = Jrda′′ Volume free free () ⊥ Qrdfree= ρ free ( ′) τ ′ ∫S⊥ ∫v′ Surface Surface I free= Krd free ()′′ Qrda= σ ()′ ′ ∫C ⊥ free∫S′ free ⊥

And the-time dependent Continuity Equations – separate conservation of bound and free charge:

∂ρ ()rt, ∇=−iJrt(), free ⇐ Free charge is conserved. free ∂t n.b. There are actually two ∂ρ ()rt, separate bound charge continuity ∇=−iJrt(), Bound ⇐ Bound charge is conserved. equations here, because we have Bound ∂t bound charges in dielectric media and effective bound currents in magnetic media! Then using the principle of linear superposition:

JrtJTot(),,=+ free () rtJ Bound () rt , ∂ρ free (rt, ) ∂∂ρρ(rt,,) () rt ⇒ ∇=∇+∇=−−iiiJrtJ(),, () rtJ () rt , Bound =− Tot Tot free Bound ∂∂tt ∂ t ∂ρ ()rt, ⇒ ∇=−iJrt(), Tot ⇐ Total charge is conserved. Tot ∂t

14 © 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 19 Prof. Steven Errede

Griffiths Example 6.2: A long copper rod of radius R carries a steady, uniformly-distributed free current I = Izˆ with JJz= ˆ as shown in the figure below. Determine H ρ free free free o free () inside and outside the copper rod. Note that copper is weakly diamagnetic, so at the microscopic level the magnetic dipoles of the copper atoms will align opposite/antiparallel to the magnetic field B ~ ϕˆ , resulting in a bound volume current I Volume running antiparallel to the free current Bound Volume I free . All currents are longitudinal (ie. . in the ± zˆ direction) . 2 2 enclosed ⎛⎞ρ 22 JIRofree= π with IRIfree()ρ ≤= free ⎜⎟ where ρ = x + y (in cylindrical coords.) free ⎝⎠R

Use Ampere’s Circuital Law for the H -field: zIˆ,,free J free Hr()i d= Ienclosed R ∫C free inside 1 ρ HR()ρ ≤= Iϕˆ H ()ρ = R 2π R2 free Hr( ) max = IR 2π 1 ~ ρ free HRIoutside ()ρ ≥= ϕˆ ~1 ρ 2πρ free ρ Note that: ρ = R HRHRinside()ρρ== outside () = ϑ yˆ ϕ ϕˆ

xˆ ρˆ

1 Now: Hr()≡−Μ Br() () r thus: B(rHrr) =+Μμo ( ) ( ) μo outside outside μo outside Then: BRHRI()ρ >=μρofree () >= ϕˆ Because Μ (ρ >≡R) 0 2πρ ( ) = same as Boutside for non-magnetized wire! What is Binside ()ρ ≤ R ? inside inside inside B ()ρμ≤=RH000 () ρμρμρ ≤+Μ≤= R () R( H ()() ≤+Μ≤ R ρ R) We don’t (yet) have the “tools” in hand to know/determine Μ (ρ ≤ R) - but we will, shortly…. when we have these, we can then determine Binside (ρ ≤ R).

© Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois 15 2005-2008. All Rights Reserved. UIUC Physics 435 EM Fields & Sources I Fall Semester, 2007 Lecture Notes 19 Prof. Steven Errede

Note that from Ampere’s Circuital Law for the H -field: Hr( )i d= Ienclosed ∫C free This relation says that if we measure I enclosed then we can compute H . This makes H more free useful e.g. than D (the electric displacement).

In the “old” days (e.g. 1800’s), it was easier to reliably measure a free current I ( in Amps) than voltage V ( in Volts). Reliably measuring a current required the use of galvanometer (an early type of ammeter – which is a very low input impedance device – ideally zero Ohms), whereas reliably measuring the voltage V (with respect to a local ground) required the use of a voltmeter with a very high input impedance (ideally infinite Ohms), which was very difficult to achieve back then! In the “old” days, a good galvanometer was easy to make a good ammeter, but a good voltmeter was very difficult to make. These days, garden-variety/“vanilla” digital voltmeters typically have input impedances of ~ 10 Meg-Ohms.

Measuring a current thus enabled the monitoring of the H -field, e.g. for an electro-magnet (big fields ⇒ big magnet coils ⇒ lots of current!)

The Magnetic Permeability μ and Magnetic Susceptibility χm of Linear Magnetic Materials

Recall that for linear dielectric materials in electrostatics, that: DE=+Ρ=ε o ε E where ε is the electric permittivity of the material and ε =+εχoe(1 ) . =+εχoe()1 E where χe is the electric susceptibility of the dielectric material. =+ε ooeE ε χ E ⇒ Ρ=ε oeχ E .

It would seem reasonable/logical/rational for linear magnetic materials in magnetostatics, that we could define a magnetic permeability μ and related magnetic susceptibility χm in a manner similar to that for howε and χe were defined for linear dielectric materials in electrostatics, i.e.: 11 HB≡−Μ= B with μμ=1( + χ) . μ μ om o However, the 1 μ factor really messes things up!!! For if HB= μ and we want to have 1 μ=1μχom()+ then HB= and mathematically there is no rigorous way to separate μχom()1+ the RHS of this relation into two separate pieces that would enable us to relate the magnetization Μ directly to the magnetic field B , analogous to obtaining the relation Ρ=εχoeE for linear dielectric media.

1 1111 If χm 1 then: ≈−1 χm and then: HBBBB≈ ()1−=−χχmm =−Μ 1+ χm μμμμoooo 1 Thus, we see that for χm 1, that Μ χm B in analogy to Ρ=ε oeχ E . μo

16 © 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 19 Prof. Steven Errede

However, there are many linear magnetic materials where μμom=+(11 χ) i.e. χm 1 so therefore we cannot use this approximation!!! We must do “something” else!!!

−7 We could e.g. re-define the magnetic permeability of free space, μπo =×410Henrys/meter

2 11 7 2 (=N/A ) in terms of its inverse, e.g. define: ξo ≡=×10 meters/Henry (= A /N), μπo 4 1 Then HB≡−Μ ⇒ HB≡−Μ=ξξo B, whereξ is a “new” inverse magnetic permeability μo * * defined such that ξ =1ξχom()− with “new” magnetic susceptibility χm such that ** * HB≡−Μ==−ξoomoomξξ B()1 χ BB =− ξ ξχ B and thus Μ=ξomχ B in analogy to Ρ=ε oeχ E for linear dielectrics. But note that since ξ ~ 1/ μ , then as (old) μ increases, the (new)ξ decreases (and vice-versa)! So this approach has some troubles also… see Appendix at end of this lecture note for a bit more info on this….

However, we shouldn’t get too hung-up on this, because e.g. we have already seen that there is a vast difference between the nature of the electric field E vs. the nature of the magnetic field B in terms of how they are specified by their respective divergences and curls, and thus we have absolutely every reason to believe that there is also a vast difference between the nature of the two auxiliary fields D and H in terms of how they are specified by their respective divergences and curls. Hence insisting on (or wanting) “symmetry” between relations associated with E vs. those for B is illusory. In fact, only the macroscopic matter fields Ρ (the electric polarization / electric dipole moment per unit volume) and Μ ( the magnetic polarization/magnetic dipole moment per unit volume) are analogous/similar fields (by deliberate construction on our part)!

1 What people (Maxwell, et al.) actually did was to start with the auxiliary relation: HB≡−Μ μo Multiply both sides by μo : μμooHB=−Μ, then rearrange: BH= μo ( +Μ) n.b. This latter relation erroneously causes people to (wrongly) think that the H -field is the fundamental field and therefore that the B -field is the auxiliary field. ⇒ WRONG !!! ⇐

For linear magnetic materials, the magnetic permeability μ can then be defined such that μ connects H to B via the relation HB= μ (the magnetic analog of DE= ε ).

The magnetic susceptibility can then be defined as: μμ≡+om(1 χ) paralleling that done for linear dielectrics: ε ≡+ε oe()1 χ

Then we see that: B ==μμHHHHom()1 + χ = μ oom + μχ but: BH= μμμooo( +Μ) = H + Μ Then “viola”: Μ=χ H , which is not analogous to: Ρ=ε χ E because we don’t have a direct m oe relationship between Μ and (the fundamental field) B .

© Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois 17 2005-2008. All Rights Reserved. UIUC Physics 435 EM Fields & Sources I Fall Semester, 2007 Lecture Notes 19 Prof. Steven Errede However, since HB= μ then Μ=χmmHB =χμχ =⎣⎦⎡⎤ m(1 + χ m) B μ o.

Now if χm 1, then the factor ⎣⎦⎡⎤χmmm(1+≈χχ) , and μμ= omo(1+≈ χ) μ. Thus: Μ=χmmHB ≈χμo for χm 1.

The following table lists the magnetic susceptibilities for a few typical types of diamagnetic and paramagnetic materials. Note that systematically, χm 1for both types of magnetic materials (except for gadolinium).

In this course, we want to “hang on to” the following: Er() and Br () are fundamental fields. Dr() and Hr ()are auxiliary fields associated with the E & M properties of matter. ⎧Dr( ) = ε Er( ) ⎫ ⎪ ⎪ For linear dielectrics and linear magnetic materials: ⎨ 1 ⎬ ⎪Hr()= Br()⎪ ⎩⎭μ ε ε = electric permittivity of matter = Keoε Kerel=≡=+ε ()1 χ e ε o dielectric “constant” electric susceptibility (a.k.a. relative electric permittivity)

μ μ = magnetic permeability of matter = Kmoμ Kmrel=≡=+μ ()1 χ m μo relative magnetic permeability magnetic susceptibility

18 © 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 19 Prof. Steven Errede

For Diamagnetic Materials: dia dia dia dia diaμ dia χm < 0 ⇒ μμ= om() 1+ χ < μ om , K ==() 1+ χ m < 1 μo For Paramagnetic Materials: para para para para paraμ para χμ > 0 ⇒ μμχ=>==>om()1+ μ om , K ()1+ χ m 1 μo

For Ferromagnetic Materials: χ ferro 0 but is in fact dependent on past magnetic history of material !!! m A non-linear hystesis-type relation exists between Μ vs. H (and/or Μ vs. B ) for ferromagnetic materials. Magnetic materials that obey the relation BH= μ =+μχom(1 ) H =+Μ μμ oo H ⇒ Μ=χm H are known as linear magnetic materials, i.e. μ = the magnetic permeability of the magnetic material and μ = constant of proportionality betweenB and H , whereas χ = magnetic m susceptibility of the magnetic material and χm = constant of proportionality between Μ and H. If B becomes extremely large, then the relation between B and H , and Μ and H can/does ext 2 2 become non-linear, e.g. B =++μμμ()1 cc23 +… Hand Μ=χχχmmm(1 +aa23 + +…) H

Note that various crystalline magnetic materials are anisotropic, hence: B = μH and Μ=χm H

⎛⎞μ μμ ⎛⎞χχχmmm magnetic magnetic xxxyxz ⎜⎟xxxyxxz permeability susceptibility ⎜⎟ mmm μ = ⎜⎟μμμyx yy yz χχχχmyxyyyz= ⎜⎟ tensor tensor ⎜⎟ ⎜⎟ μμμ ⎜⎟mmm ⎝⎠zx zy zz ⎝⎠χχχzx zy zz

mm mmm Note also that: μij= μ ji and μxx++=μμ yy zz 0 and likewise: χij= χ ji and χχχxx++= yy zz 0 .

© Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois 19 2005-2008. All Rights Reserved. UIUC Physics 435 EM Fields & Sources I Fall Semester, 2007 Lecture Notes 19 Prof. Steven Errede

We have the Maxwell relation: ∇=iBr( ) 0 (no magnetic charges/no magnetic monopoles) 1 1 and the constitutive relation: Hr=−Μ Br r. Then: ∇=∇−∇ΜiiiHr Br r () () () () () () μo μo = 0 or: ∇=−∇ΜiiHr() () r ⇐ These divergences do not necessarily vanish!!! Often they don’t! (especially on the surfaces of magnetized materials)

Only when ∇Μi ()r =0 , does ∇=iHr( ) 0 (and not vice-versa!!!)

Consider a bar magnet (permanent magnet) with uniformΜ (r ) ≠ 0 inside, thus Br()≠ 0 inside or outside: Μ S N

Consider Ampere’s Circuital Law for H : Hr( )i d= Ienclosed ∫C free

But: ∃ no free current(s) in a bar magnet – does this mean that HrHrinside( )()= outside = 0!!!??? !!! NONSENSE !!!

Μ=Μ()rzo ˆ inside the bar magnet. B()r for a cylindrical bar magnet = B(r ) for a short solenoid (w/ no pitch angle).

inside Lines of B : B is in the same direction as Μ=Μo zˆ

1 Outside: HBout= out μ0

in Lines of H : Inside: H is in the opposite direction to Μ !!!

Compare these pix to that for ED, and Ρ for the bar electret – see P435 Lecture Notes 10, p. 33.

20 © 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 19 Prof. Steven Errede

Note that since JrBound ()=∇×Μ () r and Μ=(rHr) χm ( ) then: JrBound()=∇×χ m Hr( ) ,

However: ∇×Hr() = Jfree () r ⇒ JrBound( ) = χ m Jr free ( ) .

This relation says that unless a free current actually flows through a linear magnetic material of susceptibility χm , with free volume current density Jrfree ( ) , then (and only then) will there be / arise a corresponding non-zero equivalent bound volume current density JrBound ()which is related to Jrfree ()via JrBound()= χ m Jr free ( ) . This is analogous to the relationship that we found ⎛⎞1 between ρBound ()r and ρ free ()r for linear dielectric materials: ρρBound()rr=−⎜⎟1 − free () ⎝⎠Ke {See P435 Lecture Notes 10, page 21}. If Jrfree ( ) = 0 inside a magnetic material, then Jrbound ()= 0 inside the magnetic material, also. In this situation, any/all non-zero effective bound currents can only exist on the surfaces of the magnetic material!!!

MAGNETOSTATIC BOUNDARY CONDITIONS FOR MAGNETIC MEDIA

⊥ = normal (i.e. perpendicular) component relative to plane of interface

= parallel component relative to plane of interface (tangential component)

From ∇=iBr() 0 (no magnetic charges/no monopoles) in integral form: Br()i ndaˆ = 0 . ∫S Use a Gaussian pillbox for the enclosing surface S, vertically centered on the interface between the two magnetic media. We then shrink the height of pillbox to infinitesimally above/below the interface – then only the top/bottom portions of the surface integral will contribute anything. We thus obtain a condition on the perpendicular components of B(r ) above/below interface: Br⊥⊥above= Br above 21( ) surface( ) surface

We also have the constitutive relation: B(rHrr) μo =+Μ( ) ( ) and thus ∇=iBr() 0 ⇒ ∇=−∇ΜiiHr() () r. In integral form this relation becomes: H( r)()ii ndaˆˆ=− Μ r nda . ∫∫SS Using the same Gaussian pillbox, we obtain the following condition on the perpendicular components of Hr() and Μ ()r above/below interface:

© Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois 21 2005-2008. All Rights Reserved. UIUC Physics 435 EM Fields & Sources I Fall Semester, 2007 Lecture Notes 19 Prof. Steven Errede

⎡⎤⎡⎤HrHr⊥⊥above−=−Μ−Μ above ⊥⊥ above r below r ⎣⎦⎣⎦21() ( ) surface 21( ) ( ) surface

Ampere’s Law for B()r is: ∇×B(rJr) = μoTot( ) , with JrJTot( ) =+ free( rJ)() Bound r which in integral form becomes: Br()i d= μ Ienclosed , with IIIenclosed=+ enclosed enclosed . We can take ∫C oTot Tot free Bound a (rectangular) contour vertically centered above/below the interface between the two magnetic media; we then shrink the height of this contour to be infinitesimally above/below the interface, thus only the tangential portions of the line integral above/below the interface will contribute. We obtain the following condition on the tangential components of B(r ) above / below the interface, where KrKTot()=+ free () rK Bound ( r) and nˆ is shown in the figure above:

1 ⎡⎤above below BrBr21()−=() KrTot () ⎣⎦surface surface μo

We can write this more compactly/succinctly in vector form as:

1 above below ⎡⎤BrBr21()−=× () KrnTot () ˆ ⎣⎦surface surface μo

Since B()rAr=∇× (), this relation can also be equivalently written as:

1 ⎡⎤∂∂Arabove() Ar below ( ) ⎢⎥21 −=− Kr() μ ∂∂nn Tot surface o ⎣⎦⎢⎥surface

Similarly, Ampere’s Law for Hr()is: ∇×Hr( ) = Jfree ( r) which in integral form becomes: Hr()i d= Ienclosed . Again, we can take a (rectangular) contour vertically centered above / ∫C free below the interface between the two magnetic media; we then shrink the height of this contour to be infinitesimally above/below the interface, thus only the tangential portions of the line integral above/below the interface will contribute. We obtain the following condition on the tangential ⎡⎤above below components of Hr() above/below the interface: HrHr21()−=() Krfree () ⎣⎦surface surface which can also be written compactly/succinctly in vector form as: above below ⎡⎤HrHr21()−=×( ) Krnfree ( ) ˆ ⎣⎦surface surface

Since B()rHr= μ () or: Hr()= Br( ) μ and B(rAr) =∇× ( ) , this relation can also be equivalently written as:

above below ⎡⎤⎛⎞11∂∂Ar21() ⎛⎞ Ar( ) ⎢⎥⎜⎟ −=− ⎜⎟ Krfree () μμ∂∂nn surface ⎣⎦⎢⎥⎝⎠21 ⎝⎠ surface

22 © 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 19 Prof. Steven Errede

Appendix: If we wanted to/needed to define the macroscopic magnetization Μ (auxiliary macroscopic matter field) in terms of the fundamental field B , in analogy to Ρ=(rEr)()εχoe . We have seen (above) that given the constitutive relation Hr( ) =−Μ Br( ) μo ( r) we are unable to do so.

The problem here actually focuses squarely on μo , the magnetic permeability of free space:

2 1 1 Note that μo is actually derived/defined from: c = ⇒ μo ≡ 2 ε ooμ ε oc

−12 The electric permittivity of free space is ε o =×8.85 10 Farads/meter. The Farad is the SI unit of capacitance, C (in electrostatics) – the ability of something (in this case, the vacuum) to store energy in the electric field of that something. Note thatε o has the dimensions of capacitance/unit length – Farads/meter.

The numerical value of the magnetic permittivity of free space μo is defined from the experimental measurement of cms=×3108 (speed of light in free space) and the electric −12 permittivity of free spaceε o =×8.85 10 Farads/meter, thus: −7 2 2 2 μπo ≡×410Newtons/Ampere = (kg-meter/sec )/Ampere = Henrys/meter {1 Newton/ Ampere2 = 1 Henry = 1 Tesla-m2/Ampere = 1 Weber/Ampere}

The Henry is the SI unit of inductance, L (in magnetostatics) – the ability of something (in this case, the vacuum) to store energy in the magnetic field of that something. Note that μo has the dimensions of inductance/unit length – Henrys/meter. 7 1 10 2 However, if we alternatively define ξo ≡ = Amperes /Newton = meters/Henry, μo 4π

Thenξo = inverse magnetic permeability (magnetic “reluctance”??) of free space.

2 ξo 2 Then: c = or: ξoo= c ε ε o 1 Then the magnetic constitutive relation becomes: HBM= − ⇒ HBM=−ξ in analogy μ o o to DE=+Ρε o and we also have (for linear materials) HB= ξ in analogy to DE= ε . * However, here we will define ξ ≡−ξχom(1 ) in contrast to μ ≡+μχom(1 ) . ** * Then: HB≡−Μ==−ξoomoomξξ B()1 χ BB =− ξ ξχ B and thus Μ=ξomχ B in analogy to Ρ=εχoeE for linear dielectrics. * Since: Μ=χmmHB =χμχ =⎣⎦⎡⎤ m()1 + χ m B μ o we see that ξχom=+ χ m()1 χ m μ o * or: χmm=+χχ()1 m.