15 November 2002 Physics 217, Fall 2002 1Today in Physics 217: magnetism in matter Magnetism Magnetization and bound currents Ampère’s Law and magnets Linear magnetic media Cautions about H Example: Hand Ampère’sLaweFpdfFFF15 November 2002 Physics 217, Fall 2002 2MagnetismThere are three forms of magnetism in matter, all having to do with spins of atomic electrons.  Electrons have spin, and spinning charges have magnetic dipole moments that can be aligned by external magnetic fields. However, the spin of electrons, and the pairings of electrons within atoms, are manifestations of quantum-mechanicaleffects, so magnetism is not really understandable simply from our present classicalviewpoint.Anyway, here are the three kinds of magnetism:15 November 2002 Physics 217, Fall 2002 3Magnetism (continued)FerromagnetismThe layman’s magnetism, this is the only strongform of magnetism. • It comes from aligned spins of unpaired delectrons in transition metals, especially nickel, iron and cobalt (hence the name), and rare earths like samarium.ParamagnetismAnalogous to electric polarization, and much weaker than ferromagnetism; can almost understand classically.• Produced by interaction of Bwith dipole moments of unpaired sor pelectrons. Aluminum, for example, has one pelectron in its valence shell, and is paramagnetic. So is molecular oxygen, which has two unpaired spins among its bonding electrons.15 November 2002 Physics 217, Fall 2002 4Magnetism (continued)DiamagnetismComes from the interaction of Bwith induced magnetic dipoles in atoms; also much weaker than ferromagnetism.• Contrary to the way electric polarization works, diamagnetism is characterized by magnetization in the direction oppositethat of the applied field.• Seen best in materials in which all the electrons are paired off (Ne, ).Whatever the type, we can define a magnetization, in analogy with the electric polarization P:2N ,...magnetic dipole momentunit volume=M15 November 2002 Physics 217, Fall 2002 5Magnetization and bound currentsLike P, which we interpret in terms of bound charge, Mcan be interpreted in terms of bound currents, which we can characterize by consideration of themagnetic potential for a dipole.r’rr()′Mrτ′dSV22ˆˆ11dddτττ××′==′′=×′′ ′=×+×∫∫∫mMAMMMVVVrrrrrrr———Use Product Rule #7:15 November 2002 Physics 217, Fall 2002 6Magnetization and bound currents (continued)Now, for any vector function v= v(r) and a constant vector C, we can write the divergence theorem as() ()()() ()().dddddddddττττ⋅× = × ⋅⋅ × +⋅ ×  = × ⋅⋅× = ×⋅=⋅ ××= ×∫∫∫∫∫∫∫∫∫vC vC aCvvC vCaCv vCaCavvavvvvvv—————Product Rule #6triple product rule #11Thus .ddτ′×′′=×+∫∫MaAMvVSrr—Cis constant015 November 2002 Physics 217, Fall 2002 7Magnetization and bound currents (continued)Thuswhere we have defined the bound current densities:[In MKS,111 1 1,bbddddccττ′×′′=×+′′=+∫∫∫∫MaAMJKavvVSVSrrrr—() ()()(cf. ,,ˆˆ).bbbbccρσ′′ ′=− ⋅=×=⋅′=×PJr MrPnKr MnSS——() () ()00ˆ,,11.]44bbbbddµµτππ′′ ′ ′=× =×′′=+∫∫Jr Mr Kr MnAJ KavSVSrr—15 November 2002 Physics 217, Fall 2002 8Ampère’s Law revisitedWith bound currents we can do to Ampère’s Law about what we did to Gauss’s law before:Or, in MKS,()()()free44 4444, 4 (cf. 4 )bfffccc cccππ ππππππ×= = + = + ××− =⇒×= ≡− =+BJ JJ J MBM JHJ HBMDEP————01,.fµ×= ≡ −HJ H BM—15 November 2002 Physics 217, Fall 2002 9Linear magnetic mediaAs before, when the definition of Dled to that of the electric susceptibility and the dielectric constant ε, we can defineIn these terms,or, in MKS,µis called the relative permeability.Contrary to ε, µis the same in cgs and MKS, though , equally dimensionless, is not.eχ(cf. )meχχ==MB PEmχ=MHThat would be too sensible.()414 (cf. ).mππχµ ε=+ = + ≡ =BH MH H D E()()001.mµµχµ=+=+ ≡BHM HHmχ15 November 2002 Physics 217, Fall 2002 10Linear magnetic media (continued)The run of for the three forms of magnetism:Ferromagnetism: and verylarge. For instance, pure, annealed iron hasParamagnetism: Typically,Diamagnetism:Typically somewhat smaller than is typical in paramagnetism.The consequence is that µis so close to unity for anything that isn’t ferromagnetic, that one could easily avoid noticing magnetism in everyday life. mχ0mχ>10000.µ≈0 but 1.mχ>()6cgs 10 .mχ−≈0, 1.mmχχ<1.µ≅15 November 2002 Physics 217, Fall 2002 11Linear magnetic media (continued)Practical realization of the three different kinds of linear magnetic media, compared to electric polarization:Capacitor anddielectricSolenoid andmagnetForce on dielectricForce:paramagneticdiamagneticferromagnetic15 November 2002 Physics 217, Fall 2002 12Cautions (?) about HOne must be careful about the use of H, for many of the same reasons that we are cautious about D. But Hdoes turn out to be somewhat more generally useful than D. Reason: measurability. We can easily measure potential differences and free currents (with voltmeters and ammeters), so these relate best to the fields Eand H, not Dand B.  Rule of thumb: His useful whenever the situation is symmetrical enough to use Ampère’s Law: infinite cylinders, planes and solenoids; toroids. Remember, though, Bis the fundamental field. There isn’t a Biot-Savart law or Lorentz force law for H.15 November 2002 Physics 217, Fall 2002 13Example: Hand Ampère’s Law Example 6.3: An infinite solenoid carries a current I, has nturns per unit length, and is filled with a magnetic material with relative permeability µ. What is the magnetic field Binside, and what are the bound currents associated with the material?Because there are bound currents we can’t compute Bdirectly, but we can compute H. ABCurrent: outin15 November 2002 Physics 217, Fall 2002 14Example: Hand Ampère’s Law (continued)Clearly the field should be zero outside, axial inside, and perpendicular to the two vertical sides of the Ampèrean loop.In general, the bound current will reside on the surface (unless free current flows in the medium), and will follow the free current., enclosed4 Integrate over area, use Stokes' theorem:444 4ˆˆ,.ˆˆˆ4,0.ffbmmbmcdIcHIn nI nIcc ccc nIccππππ