Magnetic Properties of Materials - Zhejiang...

Magnetic Properties of Materials

Xin Wan (Zhejiang Univ.)

Lecture 10

Xin Wan (Zhejiang Univ.) Magnetic Properties of Materials Lecture 10 1 / 26

Bar Magnet vs Current-Carrying Coil

Two possible origins ofmagnetic dipole:

A magnet has two poles, andis known as a magnetic dipole.It produces a magnetic field.A current-carrying coil alsohas a magnetic dipolemoment. When placed in amagnetic field, the coil willtend to rotate, just like a barmagnet or a magnetic dipoleplaced in the field.

Magnetic Field of a CoilWe turn now to the other aspect of a current-carrying coil as a magnetic dipole.What magnetic field does it produce at a point in the surrounding space? Theproblem does not have enough symmetry to make Ampere’s law useful; so wemust turn to the law of Biot and Savart. For simplicity, we first consider only acoil with a single circular loop and only points on its perpendicular central axis,which we take to be a z axis. We shall show that the magnitude of the magneticfield at such points is

(29-26)

in which R is the radius of the circular loop and z is the distance of the point inquestion from the center of the loop. Furthermore, the direction of the mag-netic field is the same as the direction of the magnetic dipole moment ofthe loop.

Large z. For axial points far from the loop, we have z R in Eq. 29-26. Withthat approximation, the equation reduces to

Recalling that pR2 is the area A of the loop and extending our result to includea coil of N turns, we can write this equation as

Further, because and have the same direction, we can write the equation invector form, substituting from the identity NiA:

(current-carrying coil). (29-27)

Thus, we have two ways in which we can regard a current-carrying coil as amagnetic dipole: (1) it experiences a torque when we place it in an externalmagnetic field; (2) it generates its own intrinsic magnetic field, given, for dis-tant points along its axis, by Eq. 29-27. Figure 29-22 shows the magnetic field ofa current loop; one side of the loop acts as a north pole (in the direction of )m:

B:

(z) !m 0

2p

m:

z3

m !m:B

:

B(z) !m 0

2p

NiAz3 .

B(z) !m 0iR2

2z3 .

"

m:B:

B(z) !m 0 iR2

2(R2 # z2)3/2 ,

852 CHAPTER 29 MAGNETIC FIELDS DUE TO CURRENTS

N

S

i

i

B

µµ

Figure 29-22 A current loop produces a magnetic field like that of a bar magnet and thus hasassociated north and south poles.The magnetic dipole moment of the loop, its directiongiven by a curled–straight right-hand rule, points from the south pole to the north pole, inthe direction of the field within the loop.B

:

m:


Outline

The Magnetic Dipole MomentThe Field of a DipoleMagnetic MaterialsGauss’ Law for Magnetism


The Magnetic Dipole

A magnetic dipole can have different origins. It canbe

a current-carrying coil,a permanent magnet,a rotating sphere of charge, such as Earth, ora subatomic particles.

A complete description of magnetism needs quantummechanics. But for current purposes, we can modelmagnetic dipoles by current loops, and explain somemagnetic phenomena.


First, we can assign a magnetic dipole moment ~µfor a magnetic dipole. For example, the dipolemoment for a current-carrying coil is

~µ = Ni~A.An external magnetic field ~B will rotate the coil withtotal torque

~τ = ~µ× ~B.Assuming that the current is not changing, so themagnitude of the dipole moment is fixed. This,however, is tricky, because we are not discussing theenergy required to keep the current flowing.


Under this assumption, the dipole has an energy thatdepends on its orientation in the field:

UB = −~µ · ~B = −µB cos θ.

A magnetic dipole has itslowest energy when its dipolemomentum ~µ is lined up withthe magnetic field. It has itshighest energy when ~µ isdirected opposite the field.

The Magnetic Dipole MomentAs we have just discussed, a torque acts to rotate a current-carrying coil placed ina magnetic field. In that sense, the coil behaves like a bar magnet placed in themagnetic field.Thus, like a bar magnet, a current-carrying coil is said to be a mag-netic dipole. Moreover, to account for the torque on the coil due to the magneticfield, we assign a magnetic dipole moment to the coil. The direction of is thatof the normal vector to the plane of the coil and thus is given by the same right-hand rule shown in Fig. 28-19. That is, grasp the coil with the fingers of your righthand in the direction of current i; the outstretched thumb of that hand gives thedirection of .The magnitude of is given by

m ! NiA (magnetic moment), (28-35)

in which N is the number of turns in the coil, i is the current through the coil, andA is the area enclosed by each turn of the coil. From this equation, with i inamperes and A in square meters, we see that the unit of is the ampere – squaremeter (A !m2).

Torque. Using , we can rewrite Eq. 28-33 for the torque on the coil due to amagnetic field as

t ! mB sin u, (28-36)

in which u is the angle between the vectors and .We can generalize this to the vector relation

(28-37)

which reminds us very much of the corresponding equation for the torqueexerted by an electric field on an electric dipole—namely, Eq. 22-34:

In each case the torque due to the field—either magnetic or electric—is equal tothe vector product of the corresponding dipole moment and the field vector.

Energy. A magnetic dipole in an external magnetic field has an energy thatdepends on the dipole’s orientation in the field. For electric dipoles we haveshown (Eq. 22-38) that

In strict analogy, we can write for the magnetic case

(28-38)

In each case the energy due to the field is equal to the negative of the scalar prod-uct of the corresponding dipole moment and the field vector.

A magnetic dipole has its lowest energy (! "mB cos 0 ! "mB) when its di-pole moment is lined up with the magnetic field (Fig. 28-20). It has its highestenergy ( mB cos 180 mB) when is directed opposite the field. From Eq. 28-38, with U in joules and in teslas, we see that the unit of can be thejoule per tesla (J/T) instead of the ampere – square meter as suggested by Eq. 28-35.

Work. If an applied torque (due to “an external agent”) rotates a magneticdipole from an initial orientation ui to another orientation uf, then work Wa isdone on the dipole by the applied torque. If the dipole is stationary before andafter the change in its orientation, then work Wa is

Wa ! Uf " Ui, (28-39)

where Uf and Ui are calculated with Eq. 28-38.

m:B:

m:! #$! "m:

U(u) ! "m: ! B:

.

U(u ) ! "p: ! E:

.

t: ! p: % E:

.

t: ! m: % B:

,

B:

m:

m:

m:

m:m:

n:m:m:

82528-8 THE MAGNETIC DIPOLE MOMENT

Figure 28-20 The orientations of highest andlowest energy of a magnetic dipole (herea coil carrying current) in an external mag-netic field .The direction of the current igives the direction of the magnetic dipole moment via the right-hand ruleshown for in Fig. 28-19b.n:

m:

B:

i iµ µ

µ µ

Highestenergy

Lowestenergy

B

The magnetic moment vectorattempts to align with themagnetic field.


The Field of a Dipole

Pay attention to the following similarities:

~µ = Ni~A, ~p = q~dτB = ~µ× ~B, τE = ~p × ~E

UB = −~µ · ~B, UE = −~p · ~E

A significant difference between the electric andmagnetic field lines is that electric field lines start onpositive charges and end on negative charges,whereas magnetic field lines always form close loops.


It is not difficult to guess that both electric andmagnetic dipole fields decay as r−3 when we are farfrom the dipole.We now demonstrate this by calculating the fieldalong the whole axis of the dipole.

We have already learned that at the center of asingle-loop coil with a magnetic dipole momentµ = iA = iπR2,

B = µ0

4πi(2πR)

R2 = µ0

2πµ

R3 .


Meanwhile, recall that the electric field at anarbitrary point along the axis of an electric dipolewith moment p is

~E = 12πε0

~pz3 ,

for large z .An intelligent guess for the magnetic field at axialpoints far from the loop (z � R) is, then,

~B = µ0

2π~µ

z3 .


Therefore, an intelligent guess for the magnetic fieldat an axial point is

B(z) = µ0

2πµ

(R2 + z2)3/2 ,

which we will prove in the following.Note that the problem does not have enoughsymmetry to make Ampere’s law useful; so we mustturn to the law of Biot and Savart.


Consider a point P on thecentral axis of the loop, adistance z from its plane.From the symmetry, thevector sum of all theperpendicular componentsd~B due to all the loopelements d~s is zero. Thisleaves only the axial(parallel) components

d~B‖ = µ0

4πidsr 2 cosα.

and the other side as a south pole, as suggested by the lightly drawn magnet inthe figure. If we were to place a current-carrying coil in an external magneticfield, it would tend to rotate just like a bar magnet would.

85329-5 A CURRENT-CARRYING COIL AS A MAGNETIC DIPOLE

Checkpoint 3The figure here shows four arrangements of circular loops of radius r or 2r, centeredon vertical axes (perpendicular to the loops) and carrying identical currents in the di-rections indicated. Rank the arrangements according to the magnitude of the netmagnetic field at the dot, midway between the loops on the central axis, greatest first.

(a) (b) (c) (d)

Figure 29-23 Cross section through a currentloop of radius R.The plane of the loop isperpendicular to the page, and only theback half of the loop is shown.We use thelaw of Biot and Savart to find the magneticfield at point P on the central perpendicu-lar axis of the loop.

α

z

Pα

⊥ dB

dBdB<

R

ds

r

r

The perpendicular componentsjust cancel. We add only the parallel components.

Proof of Equation 29-26Figure 29-23 shows the back half of a circular loop of radius R carrying a currenti. Consider a point P on the central axis of the loop, a distance z from its plane.Let us apply the law of Biot and Savart to a differential element ds of the loop,located at the left side of the loop. The length vector for this element pointsperpendicularly out of the page.The angle u between and in Fig. 29-23 is 90°;the plane formed by these two vectors is perpendicular to the plane of the pageand contains both and From the law of Biot and Savart (and the right-handrule), the differential field produced at point P by the current in this elementis perpendicular to this plane and thus is directed in the plane of the figure,perpendicular to , as indicated in Fig. 29-23.

Let us resolve into two components: dB, along the axis of the loop andperpendicular to this axis. From the symmetry, the vector sum of all the per-

pendicular components due to all the loop elements ds is zero. This leavesonly the axial (parallel) components dB, and we have

For the element in Fig. 29-23, the law of Biot and Savart (Eq. 29-1) tells usthat the magnetic field at distance r is

We also havedB, ! dB cos a.

Combining these two relations, we obtain

(29-28)

Figure 29-23 shows that r and a are related to each other. Let us express each interms of the variable z, the distance between point P and the center of the loop.The relations are

(29-29)r ! 2R2 " z2

dB, !m 0 i cos a ds

4pr2 .

dB !m 0

4p

i ds sin 90#

r2 .

ds:

B ! ! dB, .

dB!

dB!

dB:

r

dB:

ds:.r

rds:ds:

Figure 1: The back half ofa circular loop of radius Rcarrying a current i .


From geometry, we have r 2 = R2 + z2 and

cosα = Rr = R√

R2 + z2 .

Therefore,

B =∫

B‖ = µ0iR4π(R2 + z2)3/2

∫ds.

Because ∫ ds is simply the circumference 2πR of theloop, we reach the desired relation, with µ = iπR2,

B(z) = µ0

2πµ

(R2 + z2)3/2 .


Magnetic Materials

A magnetic material can be regarded as a collectionof magnetic dipole moments (of atomic origin), eachwith a north and a south pole. They respond to anexternal magnetic field, they generate magnetic field,and thus they interact with each other.Depending on the magnetic dipole moments of theatoms and on the interactions among the atoms,magnetic properties of the materials can be classifiedinto paramagnetism, diamagnetism, ferromagnetism,among others.


Paramagnetism

Paramagnetism occurs in materials whose atomshave permanent magnetic dipole moments ~µ.In the absence of an external magnetic field, theseatomic dipole moments are randomly oriented, andthe net magnetic dipole moment of the material iszero.In an external magnetic field ~Bext, the magneticdipole moments tend to line up with the field, whichgives the sample a net magnetic dipole moment.


We can define a vector quantity magnetization ~Mas the net magnetic dipole moment per unit volume.In 1895 Pierre Curie discovered experimentally that

M = C Bext

T ,

where T is the temperature in kelvins. This is knownas Curie’s law, and C is called the Curie constant.Increasing Bext tends to align the atomic dipolemoments in a sample and thus to increase M.Increasing T tends to disrupt the alignment viathermal agitation and thus to decrease M.


The law is actually an approximation that is validonly when the ratio Bext/T is not too large.

derived from Eq. 28-38, is the difference in energy !UB (" 2mBext) between par-allel alignment and antiparallel alignment of the magnetic dipole moment of anatom and the external field. (The lower energy state is #mBext and the higher en-ergy state is +mBext.) As we shall show below, , even for ordinary tem-peratures and field magnitudes. Thus, energy transfers during collisions amongatoms can significantly disrupt the alignment of the atomic dipole moments,keeping the magnetic dipole moment of a sample much less than Nm .

Magnetization. We can express the extent to which a given paramagneticsample is magnetized by finding the ratio of its magnetic dipole moment to itsvolume V. This vector quantity, the magnetic dipole moment per unit volume, isthe magnetization of the sample, and its magnitude is

(32-38)

The unit of is the ampere–square meter per cubic meter, or ampere per meter(A/m). Complete alignment of the atomic dipole moments, called saturation ofthe sample, corresponds to the maximum value Mmax " Nm /V.

In 1895 Pierre Curie discovered experimentally that the magnetization of aparamagnetic sample is directly proportional to the magnitude of the externalmagnetic field and inversely proportional to the temperature T in kelvins:

(32-39)

Equation 32-39 is known as Curie’s law, and C is called the Curie constant. Curie’slaw is reasonable in that increasing Bext tends to align the atomic dipole momentsin a sample and thus to increase M, whereas increasing T tends to disrupt thealignment via thermal agitation and thus to decrease M. However, the law is actu-ally an approximation that is valid only when the ratio Bext/T is not too large.

Figure 32-14 shows the ratio M/Mmax as a function of Bext/T for a sample ofthe salt potassium chromium sulfate, in which chromium ions are the para-magnetic substance. The plot is called a magnetization curve. The straight linefor Curie’s law fits the experimental data at the left, for Bext/T below about0.5 T/K. The curve that fits all the data points is based on quantum physics. Thedata on the right side, near saturation, are very difficult to obtain because theyrequire very strong magnetic fields (about 100 000 times Earth’s field), even atvery low temperatures.

M " CBext

T.

B:

ext

M:

M "measured magnetic moment

V.

M:

K $ !UB

960 CHAPTER 32 MAXWELL’S EQUATIONS; MAGNETISM OF MATTER

M/M

max

1.0

04.03.02.01.0

0.25

0.50

0.75

Bext/T (T/K)

Curie’slaw

1.30 K2.00 K3.00 K4.21 K

Greater Bext at same T gives greater dipole alignment.

Approximately linear

Quantum theory

Figure 32-14 A magnetization curve for potas-sium chromium sulfate, a paramagnetic salt.The ratio of magnetization M of the salt tothe maximum possible magnetization Mmax isplotted versus the ratio of the applied mag-netic field magnitude Bext to the temperatureT. Curie’s law fits the data at the left; quan-tum theory fits all the data. Based on mea-surements by W. E. Henry.

Checkpoint 6The figure here shows two paramagnetic sphereslocated near the south pole of a bar magnet.Are(a) the magnetic forces on the spheres and (b) themagnetic dipole moments of the spheres directed toward or away from the bar magnet?(c) Is the magnetic force on sphere 1 greater than, less than, or equal to that on sphere 2?

S N 1 2

In a sufficiently strong ~Bext, all dipoles in a sample ofN atoms and a volume V line up with ~B, hence ~Msaturates at Mmax = Nµ/V .


Diamagnetism

Paramagnetic substances are always attracted by amagnet, while diamagnetic substances are repelled bya strong magnet.Diamagnetism occurs in all materials, but the weakeffect is only observable in materials having atomicdipole moments of zero.Such a material can be modeled by equal numbers ofelectrons orbiting counterclockwise or clockwise. Anexternal magnetic field will either accelerate ordecelerate these electrons, leading to a net magneticdipole moment.


11.5 Electric currents in atoms 543

Initial states

Final states

(a), (c)

(a)

(b)

(c)

(d)

(b), (d)

m0 =qr2

v0

m0

m0 + ∆m

m0

m0 + ∆m

∆m upward in both cases

v0v0

B1 B1 B = B1 (downward)

v0 + ∆v

q q

v0 – ∆v∆m = B1

q2 r2

4 M

Figure 11.13.The change in the magnetic moment vector isopposite to the direction of B, for both directionsof motion.

The radius r being fixed by the length of the cord, the factor qr/2M isa constant. Let !v denote the net change in v in the whole process ofbringing the field up to the final value B1. Then

!v =! v0+!v

v0

dv = qr2M

! B1

0dB = qrB1

2M. (11.37)

Note that the time has dropped out – the final velocity is the same whetherthe change is made slowly or quickly.

The increased speed of the charge in the final state means an increasein the upward-directed magnetic moment m. A negatively charged bodywould have been decelerated under similar circumstances, which wouldhave decreased its downward moment. In either case, then, the appli-cation of the field B1 has brought about a change in magnetic momentopposite to the field. From Eq. (11.28), the magnitude of the change inmagnetic moment !m is

!m = qr2

!v = q2r2

4MB1. (11.38)

Likewise for charges, either positive or negative, revolving in theother direction, the induced change in magnetic moment is opposite tothe change in applied magnetic field. Figure 11.13 shows this for a posi-tive charge. It appears that the following relation holds for either sign ofcharge and either direction of revolution:

!m = − q2r2

4MB1 (11.39)

In this example we forced r to be constant by using a cord of fixedlength. Let us see how the tension in the cord has changed. We shall

11.5 Electric currents in atoms 543

Initial states

Final states

(a), (c)

(a)

(b)

(c)

(d)

(b), (d)

m0 =qr2

v0

m0

m0 + ∆m

m0

m0 + ∆m

∆m upward in both cases

v0v0

B1 B1 B = B1 (downward)

v0 + ∆v

q q

v0 – ∆v∆m = B1

q2 r2

4 M

Figure 11.13.The change in the magnetic moment vector isopposite to the direction of B, for both directionsof motion.

The radius r being fixed by the length of the cord, the factor qr/2M isa constant. Let !v denote the net change in v in the whole process ofbringing the field up to the final value B1. Then

!v =! v0+!v

v0

dv = qr2M

! B1

0dB = qrB1

2M. (11.37)

Note that the time has dropped out – the final velocity is the same whetherthe change is made slowly or quickly.

The increased speed of the charge in the final state means an increasein the upward-directed magnetic moment m. A negatively charged bodywould have been decelerated under similar circumstances, which wouldhave decreased its downward moment. In either case, then, the appli-cation of the field B1 has brought about a change in magnetic momentopposite to the field. From Eq. (11.28), the magnitude of the change inmagnetic moment !m is

!m = qr2

!v = q2r2

4MB1. (11.38)

Likewise for charges, either positive or negative, revolving in theother direction, the induced change in magnetic moment is opposite tothe change in applied magnetic field. Figure 11.13 shows this for a posi-tive charge. It appears that the following relation holds for either sign ofcharge and either direction of revolution:

!m = − q2r2

4MB1 (11.39)

In this example we forced r to be constant by using a cord of fixedlength. Let us see how the tension in the cord has changed. We shall

The change in the magnetic moment vector isopposite to the direction of ~B1, for both directions ofmotion.


FerromagnetismA ferromagnet has strong, permanent magnetism.What distinguishes ferromagnets from paramagnets isthat there is a strong interaction betweenneighboring atoms.The interaction keeps the dipolemoments of atoms aligned evenwhen the magnetic field isremoved.Iron, cobalt, nickel, gadolinium,dysprosium, and alloyscontaining these elements exhibitferromagnetism.

566 Magnetic fields in matter

To get the moment per cubic meter we multiply m by the density of iron,7800 kg/m3. The magnetization M is thus

M = (235 joules/tesla-kg)(7800 kg/m3)

= 1.83 · 106 joules/(tesla-m3). (11.76)

It is µ0M, not M, that we should compare with field strengths in tesla. In thepresent case, µ0M has the value of 2.3 tesla.

It is more interesting to see how many electron spin moments this magne-tization corresponds to. Dividing M by the electron moment given in Fig. 11.14,namely 9.3 · 10− 24 joule/tesla, we get about 2 · 1029 spin moments per cubicmeter. Now, 1 m3 of iron contains about 1029 atoms. The limiting magnetiza-tion seems to correspond to about two lined-up spins per atom. As most of theelectrons in the atom are paired off and have no magnetic effect at all, this indi-cates that we are dealing with substantially complete alignment of those fewelectron spins in the atom’s structure that are at liberty to point in the samedirection.

A very suggestive fact about ferromagnets is this: a given ferromag-netic substance, pure iron for example, loses its ferromagnetic proper-ties quite abruptly if heated to a certain temperature. Above 770 ◦C,pure iron acts like a paramagnetic substance. Cooled below 770 ◦C, itimmediately recovers its ferromagnetic properties. This transition tem-perature, called the Curie point after Pierre Curie who was one of itsearly investigators, is different for different substances. For pure nickel itis 358 ◦C.

What is this ferromagnetic behavior that so sharply distinguishesiron below 770 ◦C from iron above 770 ◦C, and from copper at any tem-perature? It is the spontaneous lining up in one direction of the atomicmagnetic moments, which implies alignment of the spin axes of certainelectrons in each iron atom. By spontaneous, we mean that no exter-nal magnetic field need be involved. Over a region in the iron largeenough to contain millions of atoms, the spins and magnetic momentsof nearly all the atoms are pointing in the same direction. Well below theCurie point – at room temperature, for instance, in the case of iron –the alignment is nearly perfect. If you could magically look into theinterior of a crystal of metallic iron and see the elementary magneticmoments as vectors with arrowheads on them, you might see somethinglike Fig. 11.27.

Figure 11.27.The orderliness of the spin directions in a smallregion in a crystal of iron. Each arrow representsthe magnetic moment of one iron atom.

It is hardly surprising that a high temperature should destroy thisneat arrangement. Thermal energy is the enemy of order, so to speak.A crystal, an orderly arrangement of atoms, changes to a liquid, a muchless orderly arrangement, at a sharply defined temperature, the meltingpoint. The melting point, like the Curie point, is different for differentsubstances. Let us concentrate here on the ordered state itself. Two orthree questions are obvious.


Ferromagnetic Domains

Why isn’t every piece of iron anaturally strong magnet?

A ferromagnetic specimen, in itsnormal state, is made up of manymagnetic domains.The domains are so oriented thattheir external magnetic effects arelargely cancelled.

however, are not all aligned. For the crystal as a whole, the domains are so ori-ented that they largely cancel with one another as far as their external magneticeffects are concerned.

Figure 32-17 is a magnified photograph of such an assembly of domains in asingle crystal of nickel. It was made by sprinkling a colloidal suspension of finelypowdered iron oxide on the surface of the crystal. The domain boundaries, whichare thin regions in which the alignment of the elementary dipoles changes from acertain orientation in one of the domains forming the boundary to a differentorientation in the other domain, are the sites of intense, but highly localized andnonuniform, magnetic fields. The suspended colloidal particles are attracted tothese boundaries and show up as the white lines (not all the domain boundariesare apparent in Fig. 32-17). Although the atomic dipoles in each domain arecompletely aligned as shown by the arrows, the crystal as a whole may have onlya very small resultant magnetic moment.

Actually, a piece of iron as we ordinarily find it is not a single crystal but anassembly of many tiny crystals, randomly arranged; we call it a polycrystallinesolid. Each tiny crystal, however, has its array of variously oriented domains, justas in Fig. 32-17. If we magnetize such a specimen by placing it in an externalmagnetic field of gradually increasing strength, we produce two effects; togetherthey produce a magnetization curve of the shape shown in Fig. 32-16. One effectis a growth in size of the domains that are oriented along the external field at theexpense of those that are not. The second effect is a shift of the orientation of thedipoles within a domain, as a unit, to become closer to the field direction.

Exchange coupling and domain shifting give us the following result:

96332-8 FERROMAGNETISM

Figure 32-17 A photograph of domainpatterns within a single crystal of nickel;white lines reveal the boundaries of thedomains. The white arrows superimposedon the photograph show the orientationsof the magnetic dipoles within the domainsand thus the orientations of the net mag-netic dipoles of the domains. The crystalas a whole is unmagnetized if the net mag-netic field (the vector sum over all thedomains) is zero.

Courtesy Ralph W. DeBlois

Figure 32-18 A magnetization curve (ab) for a ferromagnetic specimen and an associatedhysteresis loop (bcdeb).

B0

BM

cb

a

ed

A ferromagnetic material placed in an external magnetic field develops astrong magnetic dipole moment in the direction of . If the field is nonuniform,the ferromagnetic material is attracted toward a region of greater magnetic fieldfrom a region of lesser field.

B:

ext

B:

ext

HysteresisMagnetization curves for ferromagnetic materials are not retraced as we increaseand then decrease the external magnetic field B0. Figure 32-18 is a plot of BM

versus B0 during the following operations with a Rowland ring: (1) Starting withthe iron unmagnetized (point a), increase the current in the toroid untilB0 (! m 0in) has the value corresponding to point b; (2) reduce the current in thetoroid winding (and thus B0) back to zero (point c); (3) reverse the toroid currentand increase it in magnitude until B0 has the value corresponding to point d;(4) reduce the current to zero again (point e); (5) reverse the current once moreuntil point b is reached again.

The lack of retraceability shown in Fig. 32-18 is called hysteresis, and thecurve bcdeb is called a hysteresis loop. Note that at points c and e the iron core ismagnetized, even though there is no current in the toroid windings; this is thefamiliar phenomenon of permanent magnetism.

Hysteresis can be understood through the concept of magnetic domains.Evidently the motions of the domain boundaries and the reorientations of thedomain directions are not totally reversible. When the applied magnetic field B0

is increased and then decreased back to its initial value, the domains do notreturn completely to their original configuration but retain some “memory” oftheir alignment after the initial increase. This memory of magnetic materials isessential for the magnetic storage of information.

This memory of the alignment of domains can also occur naturally. Whenlightning sends currents along multiple tortuous paths through the ground,the currents produce intense magnetic fields that can suddenly magnetize any

An external magnetic field leads to a growth in sizeof the domains, and a shift of the orientation of thedipoles within a domain.


Hysteresis CurveMagnetization curves forferromagnetic materials arenot retraced; the lack ofretraceability is calledhysteresis, and the curvebcdeb is called a hysteresisloop.

however, are not all aligned. For the crystal as a whole, the domains are so ori-ented that they largely cancel with one another as far as their external magneticeffects are concerned.

Figure 32-17 is a magnified photograph of such an assembly of domains in asingle crystal of nickel. It was made by sprinkling a colloidal suspension of finelypowdered iron oxide on the surface of the crystal. The domain boundaries, whichare thin regions in which the alignment of the elementary dipoles changes from acertain orientation in one of the domains forming the boundary to a differentorientation in the other domain, are the sites of intense, but highly localized andnonuniform, magnetic fields. The suspended colloidal particles are attracted tothese boundaries and show up as the white lines (not all the domain boundariesare apparent in Fig. 32-17). Although the atomic dipoles in each domain arecompletely aligned as shown by the arrows, the crystal as a whole may have onlya very small resultant magnetic moment.

Actually, a piece of iron as we ordinarily find it is not a single crystal but anassembly of many tiny crystals, randomly arranged; we call it a polycrystallinesolid. Each tiny crystal, however, has its array of variously oriented domains, justas in Fig. 32-17. If we magnetize such a specimen by placing it in an externalmagnetic field of gradually increasing strength, we produce two effects; togetherthey produce a magnetization curve of the shape shown in Fig. 32-16. One effectis a growth in size of the domains that are oriented along the external field at theexpense of those that are not. The second effect is a shift of the orientation of thedipoles within a domain, as a unit, to become closer to the field direction.

Exchange coupling and domain shifting give us the following result:

96332-8 FERROMAGNETISM

Figure 32-17 A photograph of domainpatterns within a single crystal of nickel;white lines reveal the boundaries of thedomains. The white arrows superimposedon the photograph show the orientationsof the magnetic dipoles within the domainsand thus the orientations of the net mag-netic dipoles of the domains. The crystalas a whole is unmagnetized if the net mag-netic field (the vector sum over all thedomains) is zero.

Courtesy Ralph W. DeBlois

Figure 32-18 A magnetization curve (ab) for a ferromagnetic specimen and an associatedhysteresis loop (bcdeb).

B0

BM

cb

a

ed

A ferromagnetic material placed in an external magnetic field develops astrong magnetic dipole moment in the direction of . If the field is nonuniform,the ferromagnetic material is attracted toward a region of greater magnetic fieldfrom a region of lesser field.

B:

ext

B:

ext

HysteresisMagnetization curves for ferromagnetic materials are not retraced as we increaseand then decrease the external magnetic field B0. Figure 32-18 is a plot of BM

versus B0 during the following operations with a Rowland ring: (1) Starting withthe iron unmagnetized (point a), increase the current in the toroid untilB0 (! m 0in) has the value corresponding to point b; (2) reduce the current in thetoroid winding (and thus B0) back to zero (point c); (3) reverse the toroid currentand increase it in magnitude until B0 has the value corresponding to point d;(4) reduce the current to zero again (point e); (5) reverse the current once moreuntil point b is reached again.

The lack of retraceability shown in Fig. 32-18 is called hysteresis, and thecurve bcdeb is called a hysteresis loop. Note that at points c and e the iron core ismagnetized, even though there is no current in the toroid windings; this is thefamiliar phenomenon of permanent magnetism.

Hysteresis can be understood through the concept of magnetic domains.Evidently the motions of the domain boundaries and the reorientations of thedomain directions are not totally reversible. When the applied magnetic field B0

is increased and then decreased back to its initial value, the domains do notreturn completely to their original configuration but retain some “memory” oftheir alignment after the initial increase. This memory of magnetic materials isessential for the magnetic storage of information.

This memory of the alignment of domains can also occur naturally. Whenlightning sends currents along multiple tortuous paths through the ground,the currents produce intense magnetic fields that can suddenly magnetize any

Note that at points c and e the iron core ismagnetized, even though there is no externalmagnetic field; this is the familiar phenomenon ofpermanent magnetism.This memory of magnetic materials is essential forthe magnetic storage of information.


The Divergence of ~BFor volume currents, the Biot-Savart law becomes

~B(x , y , z) = µ0

4π∫ ~J(x ′, y ′, z ′)×~r

r 3 dx ′dy ′dz ′,

where the length element id~s is replace by thevolume element JdV ′ ≡ ~J(x ′, y ′, z ′)dx ′dy ′dz ′ and

~r = (x − x ′)x + (y − y ′)y + (z − z ′)z .

Applying the divergence, we obtain

∇ · ~B = µ0

4π∫∇ ·

~J ×~rr 3

dV ′.


Since the divergence does not apply to J , which doesnot depend on (x , y , z), we can rewrite

∇ · ~B = −µ0

4π∫

J ·(∇×

~rr 3

)dV ′.

The curl of ~r/r 3 turns out to be zero, as it does nottwists around; it only spreads out.Therefore, we conclude

∇ · ~B = 0,

which contrasts with

∇ · ~E = ρ

ε0.


Gauss’ Law for MagnetismThe integral form of ∇ · ~B = 0 can be obtained byconstructing a closed Gaussian surface:∮

~B · d~A =∫

(∇ · ~B)dV = 0.

In the first equality we have used the fundamentaltheorem of divergences.The law asserts that the net magnetic flux ΦBthrough any closed Gaussian surface is zero. This is aformal way of saying that magnetic monopoles donot exist. The simplest magnetic structure that canexist is a magnetic dipole.


Summary

Classification of magnetism: paramagnetism,diamagnetism, ferromagnetism, . . . .Field of a magnetic dipole at axial points (z � R):

~B = µ0

2π~µ

z3

Gauss’ law for magnetism

∇ · ~B = 0∮~B · d~A = 0


Reading

Halliday, Resnick & Krane:

Chapter 35: Magnetic Properties of Materials


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