Physics For Scientists And Engineers 6E - part 238

Ferromagnetism

A  small  number  of  crystalline  substances  exhibit  strong  magnetic  effects  called
ferromagnetism. Some examples of ferromagnetic substances are iron, cobalt, nickel,
gadolinium,  and  dysprosium.  These  substances  contain  permanent  atomic  magnetic
moments  that  tend  to  align  parallel  to  each  other  even  in  a  weak  external  magnetic
field.  Once  the  moments  are  aligned,  the  substance  remains  magnetized  after  the
external  field  is  removed.  This  permanent  alignment  is  due  to  a  strong  coupling
between neighboring moments, a coupling that can be understood only in quantum-
mechanical terms.

All ferromagnetic materials are made up of microscopic regions called

domains,

regions within which all magnetic moments are aligned. These domains have volumes
of about 10

to 10

and contain 10

to 10

atoms. The boundaries between

the various domains having different orientations are called

domain walls. In an

unmagnetized sample, the magnetic moments in the domains are randomly oriented
so that the net magnetic moment is zero, as in Figure 30.29a. When the sample is
placed in an external magnetic field

, the size of those domains with magnetic

moments aligned with the field grows, which results in a magnetized sample, as in
Figure 30.29b. As the external field becomes very strong, as in Figure 30.29c, the
domains in which the magnetic moments are not aligned with the field become very
small. When the external field is removed, the sample may retain a net magnetization
in the direction of the original field. At ordinary temperatures, thermal agitation is not
sufficient to disrupt this preferred orientation of magnetic moments.

A typical experimental arrangement that is used to measure the magnetic proper-

ties of a ferromagnetic material consists of a torus made of the material wound with N
turns of wire, as shown in Figure 30.30, where the windings are represented in black
and are referred to as the primary coil. This apparatus is sometimes referred to as a
Rowland ring. A secondary coil (the red wires in Fig. 30.30) connected to a galvanome-
ter is used to measure the total magnetic flux through the torus. The magnetic field

in the torus is measured by increasing the current in the toroid from zero to I. As the
current changes, the magnetic flux through the secondary coil changes by an amount
BA, where A is the cross-sectional area of the toroid. As shown in Chapter 31, because
of this changing flux, an emf that is proportional to the rate of change in magnetic
flux is induced in the secondary coil. If the galvanometer is properly calibrated, a value
for

B corresponding to any value of the current in the primary coil can be obtained.

The magnetic field

B is measured first in the absence of the torus and then with the

torus in place. The magnetic properties of the torus material are then obtained from a
comparison of the two measurements.

Now consider a torus made of unmagnetized iron. If the current in the primary

coil is increased from zero to some value I, the magnitude of the magnetic field
strength H increases linearly with I according to the expression H " nI. Furthermore,

SECTION 3 0.8 • Magnetism in Matter

949

(c)

(b)

(a)

Figure 30.29 (a) Random orienta-

tion of atomic magnetic dipoles in

the domains of an unmagnetized

substance. (b) When an external

field

is applied, the domains

with components of magnetic

moment in the same direction as

grow larger, giving the sample a

net magnetization. (c) As the field

is made even stronger, the domains

with magnetic moment vectors not

aligned with the external field

become very small.

Figure 30.30 A toroidal winding arrangement used to measure the

magnetic properties of a material. The torus is made of the material

under study, and the circuit containing the galvanometer measures

the magnetic flux.

the magnitude of the total field B also increases with increasing current, as shown by
the curve from point O to point a in Figure 30.31. At point O, the domains in the iron
are randomly oriented, corresponding to B

0. As the increasing current in the

primary coil causes the external field

to increase, the aligned domains grow in size

until nearly all magnetic moments are aligned at point a. At this point the iron core is
approaching saturation, which is the condition in which all magnetic moments in the
iron are aligned.

Next, suppose that the current is reduced to zero, and the external field is

consequently eliminated. The B-versus-H curve, called a

magnetization curve, now

follows the path ab in Figure 30.31. Note that at point b,

B is not zero even though the

external field

is zero. The reason is that the iron is now magnetized due to the

alignment of a large number of its magnetic moments (that is,

B " B

). At this point,

the iron is said to have a remanent magnetization.

If the current in the primary coil is reversed so that the direction of the external

magnetic field is reversed, the magnetic moments reorient until the sample is again
unmagnetized at point c, where B " 0. An increase in the reverse current causes the
iron to be magnetized in the opposite direction, approaching saturation at point d
in  Figure  30.31.  A  similar  sequence  of  events  occurs  as  the  current  is  reduced
to zero  and  then  increased  in  the  original  (positive)  direction.  In  this  case  the
magnetization  curve  follows  the  path  def. If  the  current  is  increased  sufficiently,
the magnetization  curve  returns  to  point  a,  where  the  sample  again  has  its
maximum magnetization.

The effect just described, called

magnetic hysteresis, shows that the magnetiza-

tion of a ferromagnetic substance depends on the history of the substance as well as
on the magnitude of the applied field. (The word hysteresis means “lagging behind.”)
It is  often  said  that  a  ferromagnetic  substance  has  a  “memory”  because  it  remains
magnetized  after  the  external  field  is  removed.  The  closed  loop  in  Figure  30.31  is
referred  to  as  a  hysteresis  loop.  Its  shape  and  size  depend  on  the  properties  of
the ferromagnetic substance and on the strength of the maximum applied field. The
hysteresis  loop  for  “hard”  ferromagnetic  materials  is  characteristically  wide  like  the
one shown in Figure 30.32a, corresponding to a large remanent magnetization. Such
materials  cannot  be  easily  demagnetized  by  an  external  field.  “Soft”  ferromagnetic
materials,  such  as  iron,  have  a  very  narrow  hysteresis  loop  and  a  small  remanent
magnetization  (Fig.  30.32b.)  Such  materials  are  easily  magnetized  and  demagne-
tized. An  ideal  soft  ferromagnet  would  exhibit  no  hysteresis  and  hence  would  have
no remanent  magnetization.  A  ferromagnetic  substance  can  be  demagnetized  by
carrying  it  through  successive  hysteresis  loops,  due  to  a  decreasing  applied  magnetic
field, as shown in Figure 30.33.

950

CHAPTE R 3 0 • Sources of the Magnetic Field

(a)

(b)

Figure 30.32 Hysteresis loops for (a) a hard ferromagnetic

material and (b) a soft ferromagnetic material.

Figure 30.31 Magnetization curve

for a ferromagnetic material.

Figure 30.33 Demagnetizing a

ferromagnetic material by carrying

it through successive hysteresis

loops.

The magnetization curve is useful for another reason:

the area enclosed by the

magnetization curve represents the energy input required to take the material
through  the  hysteresis  cycle. The  energy  acquired  by  the  material  in  the
magnetization process originates from the source of the external field—that is, the emf
in the circuit of the toroidal coil. When the magnetization cycle is repeated, dissipative
processes within the material due to realignment of the magnetic moments result in an
increase  in  internal  energy,  made  evident  by  an  increase  in  the  temperature  of  the
substance. For this reason, devices subjected to alternating fields (such as AC adapters
for  cell  phones,  power  tools,  and  so  on)  use  cores  made  of  soft  ferromagnetic
substances, which have narrow hysteresis loops and correspondingly little energy loss
per cycle.

Magnetic computer disks store information by alternating the direction of

for portions  of  a  thin  layer  of  ferromagnetic  material.  Floppy  disks  have  the  layer
on a circular  sheet  of  plastic.  Hard  disks  have  several  rigid  platters  with  magnetic
coatings on each side. Audio tapes and videotapes work the same way as floppy disks
except  that  the  ferromagnetic  material  is  on  a  very  long  strip  of  plastic.  Tiny  coils
of wire  in  a  recording  head  are  placed  close  to  the  magnetic  material  (which  is
moving rapidly  past  the  head).  Varying  the  current  in  the  coils  creates  a  magnetic
field that  magnetizes  the  recording  material.  To  retrieve  the  information,  the
magnetized  material  is  moved  past  a  playback  coil.  The  changing  magnetism  of
the material  induces  a  current  in  the  coil,  as  shown  in  Chapter  31.  This  current
is then amplified  by  audio  or  video  equipment,  or  it  is  processed  by  computer
circuitry.

When the temperature of a ferromagnetic substance reaches or exceeds a

critical temperature called the

Curie temperature, the substance loses its

residual magnetization  and  becomes  paramagnetic  (Fig.  30.34).  Below  the
Curie temperature,  the  magnetic  moments  are  aligned  and  the  substance  is
ferromagnetic. Above the Curie temperature, the thermal agitation is great enough
to  cause  a  random  orientation  of  the  moments,  and  the  substance  becomes
paramagnetic.  Curie  temperatures  for  several  ferromagnetic  substances  are  given
in Table 30.3.

Paramagnetism

Paramagnetic  substances  have  a  small  but  positive  magnetic  susceptibility
(0 . 8 .. 1) resulting from the presence of atoms (or ions) that have permanent
magnetic  moments.  These  moments  interact  only  weakly  with  each  other  and
are randomly  oriented  in  the  absence  of  an  external  magnetic  field.  When  a
paramagnetic  substance  is  placed  in  an  external  magnetic  field,  its  atomic
moments tend  to  line  up  with  the  field.  However,  this  alignment  process  must
compete  with  thermal  motion,  which  tends  to  randomize  the  magnetic  moment
orientations.

Pierre Curie (1859–1906) and others since him have found experimentally that,

under a wide range of conditions, the magnetization of a paramagnetic substance is
proportional to the applied magnetic field and inversely proportional to the absolute
temperature:

(30.35)

M " C

SECTION 3 0.8 • Magnetism in Matter

951

Substance

Curie

(K)

Iron

1 043

Cobalt

1 394

Nickel

631

Gadolinium

317

893

Curie Temperatures for
Several Ferromagnetic
Substance

Table 30.3

Quick Quiz 30.9

Which material would make a better permanent magnet,

(a) one whose hysteresis loop looks like Figure 30.32a or (b) one whose hysteresis loop
looks like Figure 30.32b?

Paramagnetic

Ferromagnetic

Curie

Figure 30.34 Magnetization versus

absolute temperature for a ferro-

magnetic substance. The magnetic

moments are aligned below the

Curie temperature T

Curie

, where

the substance is ferromagnetic.

The substance becomes paramag-

netic (magnetic moments

unaligned) above T

Curie

This relationship is known as

Curie’s law after its discoverer, and the constant C is

called

Curie’s constant. The law shows that when B

0, the magnetization is zero,

corresponding to a random orientation of magnetic moments. As the ratio of magnetic
field to temperature becomes great, the magnetization approaches its saturation value,
corresponding to a complete alignment of its moments, and Equation 30.35 is no
longer valid.

Diamagnetism

When  an  external  magnetic  field  is  applied  to  a  diamagnetic  substance,  a  weak
magnetic  moment  is  induced  in  the  direction  opposite  the  applied  field.  This
causes diamagnetic  substances  to  be  weakly  repelled  by  a  magnet.  Although
diamagnetism  is  present  in  all  matter,  its  effects  are  much  smaller  than  those  of
paramagnetism  or  ferromagnetism,  and  are  evident  only  when  those  other  effects
do not exist.

We can attain some understanding of diamagnetism by considering a classical

model of two atomic electrons orbiting the nucleus in opposite directions but with
the  same  speed.  The  electrons  remain  in  their  circular  orbits  because  of  the
attractive  electrostatic  force  exerted  by  the  positively  charged  nucleus.  Because
the magnetic  moments  of  the  two  electrons  are  equal  in  magnitude  and  opposite
in direction,  they  cancel  each  other,  and  the  magnetic  moment  of  the  atom  is
zero. When  an  external  magnetic  field  is  applied,  the  electrons  experience  an
additional  magnetic  force  q

v ! B. This added magnetic force combines with the

electrostatic force to increase the orbital speed of the electron whose magnetic
moment is antiparallel to the field and to decrease the speed of the electron whose
magnetic moment is parallel to the field. As a result, the two magnetic moments of
the electrons no longer cancel, and the substance acquires a net magnetic moment
that is opposite the applied field.

As you recall from Chapter 27, a superconductor is a substance in which the

electrical resistance is zero below some critical temperature. Certain types of supercon-
ductors also exhibit perfect diamagnetism in the superconducting state. As a result, an
applied magnetic field is expelled by the superconductor so that the field is zero in its
interior. This phenomenon is known as the

Meissner effect. If a permanent magnet is

brought near a superconductor, the two objects repel each other. This is illustrated in
Figure 30.35, which shows a small permanent magnet levitated above a superconductor
maintained at 77 K.

952

CHAPTE R 3 0 • Sources of the Magnetic Field

(Left) Paramagnetism: liquid oxygen, a paramagnetic material, is attracted to the poles

of a magnet. (Right) Diamagnetism: a frog is levitated in a 16-T magnetic field at the

Nijmegen High Field Magnet Laboratory, Netherlands. The levitation force is exerted

on the diamagnetic water molecules in the frog’s body. The frog suffered no ill effects

from the levitation experience.

Leon Lewandowski

High Field Magnet Laboratory

, University of Nijmegen, The Netherlands.

Figure 30.35 An illustration of the

Meissner effect, shown by this mag-

net suspended above a cooled ce-

ramic superconductor disk, has

become our most visual image of

high-temperature superconductivity.

Superconductivity is the loss of all

resistance to electrical current, and

is a key to more efficient energy use.

In the Meissner effect, the magnet

induces superconducting current in

the disk, which is cooled to ' 321+F

(77 K). The currents create a

magnetic force that repels and

levitates the disk.

Photo courtesy Argonne National Laboratory

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