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Today in Physics 217: in matter

‰ Magnetism ‰ Magnetization and bound currents

‰ Ampère’s Law and Fe ‰ Linear magnetic media ‰ Cautions about H ‰ Example: H and Ampère’s Fp Law Fd Ff

15 November 2002 Physics 217, Fall 2002 1 Magnetism

There are three forms of magnetism in matter, all having to do with spins of atomic . ‰ Electrons have , and spinning charges have moments that can be aligned by external magnetic fields. ‰ However, the spin of electrons, and the pairings of electrons within , are manifestations of quantum- mechanical effects, so magnetism is not really understandable simply from our present classical viewpoint. Anyway, here are the three kinds of magnetism:

15 November 2002 Physics 217, Fall 2002 2 Magnetism (continued)

‰ The layman’s magnetism, this is the only strong form of magnetism. • It comes from aligned spins of unpaired d electrons in transition metals, especially nickel, iron and cobalt (hence the name), and rare earths like samarium. ‰ Analogous to electric polarization, and much weaker than ferromagnetism; can almost understand classically. • Produced by interaction of B with dipole moments of unpaired s or p electrons. Aluminum, for example, has one p 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 3 Magnetism (continued)

‰ Comes from the interaction of B with 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 opposite that of the applied field. • Seen best in materials in which all the electrons are paired off (Ne, N 2 ,... ). Whatever the type, we can define a magnetization, in analogy with the electric polarization P: magnetic dipole moment M = unit volume

15 November 2002 Physics 217, Fall 2002 4 Magnetization and bound currents

Like P, which we interpret in terms of bound charge, M can be interpreted in terms of bound currents, which we can characterize by consideration of the magnetic potential for a dipole. S mM××rrˆˆ V A == dτ ′ 22∫ rrV Mr()′ r 1 Use Product =×M —′′dτ ∫ r Rule #7: r V 1 M r’ =×+×——′′M dτ ′ ∫ rr dτ ′ V 

15 November 2002 Physics 217, Fall 2002 5 Magnetization and bound currents (continued)

Now, for any vector function v = v(r) and a constant vector C, we can write the divergence theorem as ∫∫— ⋅×()vCddτ =v vC × () ⋅ a 0 ()() () ∫∫⋅CvvC—— × +⋅ × ddτ =v vCa × ⋅ Product Rule #6 () Cv⋅×∫∫— ddτ =v vCa ×⋅ C is constant =⋅Cavv∫ d × triple product rule #1 ∫∫—×=vavddτ v ×. 1 Ma× d ′ Thus AM=×+—′′dτ . ∫∫rrv VS 15 November 2002 Physics 217, Fall 2002 6 Magnetization and bound currents (continued) 1 Ma× d ′ Thus AM=×+—′′dτ ∫∫rrv VS 11 1 1 =+JKaddτ ′′, cc∫∫rrbbv VS where we have defined the bound current densities:

Jrb ()′′=×c— Mr ′ (), (cf. ρb =−— ⋅P , ()′ =׈ σ =⋅Pnˆ ) Krb c MnS . b S ′′=× ′ ′ =׈ [In MKS, Jrbb()— Mr (),, Kr Mn ()S µµ11 AJ=+00ddτ ′′ Ka.] 44ππ∫∫rrbbv VS 15 November 2002 Physics 217, Fall 2002 7 Ampère’s Law revisited

With bound currents we can do to Ampère’s Law about what we did to Gauss’s law before: 44ππ 4 π ——×=BJ =() JJ + =() J +c × M ccfree bf c 4π —×−()BM4π = J c f 4π ⇒×=— HJ, HBMDEP ≡− 4ππ (cf. =+4 ) c f 1 Or, in MKS, —×=HJf ,. H ≡ BM − µ0

15 November 2002 Physics 217, Fall 2002 8 Linear magnetic media

As before, when the definition of D led to that of the electric susceptibility χ e and the constant ε, we can define MB==χχme(cf. PE) That would be too sensible. MH= χm In these terms,

BH=+414(cf.ππχµε MH =() +m ≡ H D = E). or, in MKS,

BHM=+=+≡µµχµ00()()1.m HH µ is called the relative permeability. Contrary to ε, µ is the same in cgs and MKS, though χ m , equally dimensionless, is not.

15 November 2002 Physics 217, Fall 2002 9 Linear magnetic media (continued)

The run of χ m for the three forms of magnetism:

‰ Ferromagnetism: χ> m 0 and very large. For instance, pure, annealed iron has µ≈10000.  ‰ Paramagnetism: χ>m 0 but  1. −6  Typically, χm ()cgs≈ 10 .  µ ≅ 1.  ‰ Diamagnetism: χχmm< 0,  1.  Typically somewhat smaller than is typical in paramagnetism. The consequence is that µ is so close to unity for anything that isn’t ferromagnetic, that one could easily avoid noticing magnetism in everyday life.

15 November 2002 Physics 217, Fall 2002 10 Linear magnetic media (continued)

Practical realization of the three different kinds of linear magnetic media, compared to electric polarization:

Capacitor and dielectric

Force on dielectric

Solenoid and : paramagnetic diamagnetic ferromagnetic

15 November 2002 Physics 217, Fall 2002 11 Cautions (?) about H

One must be careful about the use of H, for many of the same reasons that we are cautious about D. But H does 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 E and H, not D and B. ‰ Rule of thumb: H is useful whenever the situation is symmetrical enough to use Ampère’s Law: infinite cylinders, planes and solenoids; toroids. Remember, though, B is the fundamental field. There isn’t a Biot-Savart law or law for H.

15 November 2002 Physics 217, Fall 2002 12 Example: H and Ampère’s Law

Example 6.3: An infinite solenoid carries a current I, has n turns per unit length, and is filled with a magnetic material with relative permeability µ. What is the B inside, and what are the bound currents associated with the material? A

Current: out Because there are bound currents we B can’t compute B directly, but we can compute H. in

15 November 2002 Physics 217, Fall 2002 13 Example: H and Ampère’s Law (continued)

Clearly the field should be zero outside, axial inside, and perpendicular to the two vertical sides of the Ampèrean loop. 4π —×=HJ Integrate over area, use Stokes' theorem: c f 4π H ⋅=dIA v∫ c f , enclosed 44ππ 4 π HInAA=⇒=HzBHz nIˆˆ,. ==µµ nI cc c KMnHsbmm=×=ccˆˆˆχπχ ×=4, nIφ JMbm=×=×cc——χ H =0. In general, the bound current will reside on the surface (unless free current flows in the medium), and will follow the free current. 15 November 2002 Physics 217, Fall 2002 14 Example: H and Ampère’s Law (continued)

The same thing, in MKS units:

—×=HJf Integrate over area, use Stokes' theorem:

v∫ H ⋅=dIA f , enclosed HInAA=⇒=Hz nIˆˆ,. BH ==µµ nI z Bound charges:

KMnHsbmm=×=ˆˆˆχχ ×= nIφ , JMbm=×—— =×χ H =0.

15 November 2002 Physics 217, Fall 2002 15