Physics For Scientists And Engineers 6E - part 239

SECTION 3 0.9 • The Magnetic Field of the Earth

953

Estimate the saturation magnetization in a long cylinder of
iron, assuming one unpaired electron spin per atom.

Solution The  saturation  magnetization  is  obtained  when
all  the  magnetic  moments  in  the  sample  are  aligned.  If
the sample  contains  n atoms  per  unit  volume,  then  the
saturation magnetization M

has the value

where # is the magnetic moment per atom. Because the
molar mass of iron is 55 g/mol and its density is 7.9 g/cm

Example 30.11 Saturation Magnetization

the value of n for iron is 8.6 & 10

atoms/m

. Assuming

that each atom contributes one Bohr magneton (due to one
unpaired spin) to the magnetic moment, we obtain

This is about half the experimentally determined saturation
magnetization for iron, which indicates that actually two
unpaired electron spins are present per atom.

8.0 & 10

A/m

(8.6 & 10

atoms/m

)(9.27 & 10

A(m

/atom)

30.9 The Magnetic Field of the Earth

When we speak of a compass magnet having a north pole and a south pole, we should
say more properly that it has a “north-seeking” pole and a “south-seeking” pole. By this
we mean that one pole of the magnet seeks, or points to, the north geographic pole of
the Earth. Because the north pole of a magnet is attracted toward the north
geographic pole of the Earth, we conclude that

the Earth’s south magnetic pole is

located near the north geographic pole, and the Earth’s north magnetic pole is
located near the south geographic pole. In fact, the configuration of the Earth’s
magnetic field, pictured in Figure 30.36, is very much like the one that would be
achieved by burying a gigantic bar magnet deep in the interior of the Earth.

If a compass needle is suspended in bearings that allow it to rotate in the vertical

plane as well as in the horizontal plane, the needle is horizontal with respect to the
Earth’s  surface  only  near  the  equator.  As  the  compass  is  moved  northward,  the
needle  rotates  so  that  it  points  more  and  more  toward  the  surface  of  the  Earth.
Finally, at a point near Hudson Bay in Canada, the north pole of the needle points
directly downward. This site, first found in 1832, is considered to be the location of
the south magnetic pole of the Earth. It is approximately 1 300 mi from the Earth’s
geographic  North  Pole,  and  its  exact  position  varies  slowly  with  time.  Similarly,  the
north magnetic pole of the Earth is about 1 200 mi away from the Earth’s geographic
South Pole.

Figure 30.36 The Earth’s

magnetic field lines. Note that a

south magnetic pole is near the

north geographic pole, and a

north magnetic pole is near the

south geographic pole.

North

geographic

pole

South

magnetic

pole

Geographic

equator

South

geographic

pole

North

magnetic

pole

Magnetic equator

954

CHAPTE R 3 0 • Sources of the Magnetic Field

Although the magnetic field pattern of the Earth is similar to the one that would be

set up by a bar magnet deep within the Earth, it is easy to understand why the source of
the Earth’s magnetic field cannot be large masses of permanently magnetized material.
The Earth does have large deposits of iron ore deep beneath its surface, but the high
temperatures  in  the  Earth’s  core  prevent  the  iron  from  retaining  any  permanent
magnetization.  Scientists  consider  it  more  likely  that  the  true  source  of  the  Earth’s
magnetic  field  is  convection  currents  in  the  Earth’s  core.  Charged  ions  or  electrons
circulating in the liquid interior could produce a magnetic field just as a current loop
does. There is also strong evidence that the magnitude of a planet’s magnetic field is
related  to  the  planet’s  rate  of  rotation.  For  example,  Jupiter  rotates  faster  than  the
Earth,  and  space  probes  indicate  that  Jupiter’s  magnetic  field  is  stronger  than  ours.
Venus, on the other hand, rotates more slowly than the Earth, and its magnetic field is
found to be weaker. Investigation into the cause of the Earth’s magnetism is ongoing.

There is an interesting sidelight concerning the Earth’s magnetic field. It has been

found that the direction of the field has been reversed several times during the last
million years. Evidence for this is provided by basalt, a type of rock that contains iron
and that forms from material spewed forth by volcanic activity on the ocean floor. As
the lava cools, it solidifies and retains a picture of the Earth’s magnetic field direction.
The rocks are dated by other means to provide a timeline for these periodic reversals
of the magnetic field.

Figure 30.37 A map of the continental United States showing several lines of constant

magnetic declination.

°W

°E

Quick Quiz 30.10

If we wanted to cancel the Earth’s magnetic field by

running an enormous current loop around the equator, which way would the current
have to to be directed: (a) east to west or (b) west to east?

Because of this distance between the north geographic and south magnetic poles, it

is only approximately correct to say that a compass needle points north. The difference
between true north, defined as the geographic North Pole, and north indicated by a
compass varies from point to point on the Earth, and the difference is referred to as
magnetic declination. For example, along a line through Florida and the Great Lakes, a
compass indicates true north, whereas in the state of Washington, it aligns 25° east of
true north. Figure 30.37 shows some representative values of the magnetic declination
for the continental United States.

Summary

955

The

Biot–Savart law says that the magnetic field dB at a point P due to a length

element d

s that carries a steady current I is

(30.1)

where #

is the

permeability of free space, r is the distance from the element to the

point P, and ˆ

r is a unit vector pointing from ds toward point P. We find the total field

at P by integrating this expression over the entire current distribution.

The magnetic force per unit length between two parallel wires separated by a

distance a and carrying currents I

and I

has a magnitude

(30.12)

The force is attractive if the currents are in the same direction and repulsive if they

are in opposite directions.

Ampère’s law says that the line integral of B ( ds around any closed path equals #

where I is the total steady current through any surface bounded by the closed path:

(30.13)

Using Ampère’s law, one finds that the magnitude of the magnetic field at a

distance r from a long, straight wire carrying an electric current I is

(30.14)

The field lines are circles concentric with the wire.

The magnitudes of the fields inside a toroid and solenoid are

(30.16)

(30.17)

where N is the total number of turns.

The

magnetic flux 1

through a surface is defined by the surface integral

(30.18)

Gauss’s law of magnetism states that the net magnetic flux through any closed

surface is zero.

The general form of Ampère’s law, which is also called the

Ampère–Maxwell law, is

(30.22)

This law describes the fact that magnetic fields are produced both by conduction
currents and by changing electric fields.

When a substance is placed in an external magnetic field

the total magnetic

field

B is a combination of the external field and a magnetic field due to magnetic

moments of atoms and electrons within the substance:

(30.29)

B " B

B(ds " #

I * #

B(d

B " #

I " #

(solenoid

)

B "

2$r

(toroid)

B "

2$r

B(d

s " #

2$a

B "

I d

s ! ˆr

S U M M A R Y

Take a practice test for

this chapter by clicking on
the Practice Test link at
http://www.pse6.com.

956

CHAPTE R 3 0 • Sources of the Magnetic Field

where

M is the magnetization vector. The magnetization vector is the magnetic

moment per unit volume in the substance.

The effect of external currents on the magnetic field in a substance is described by

the

magnetic field strength H " B

The magnetization vector is related to the

magnetic field strength as follows:

(30.32)

where 8 is the

magnetic susceptibility.

Substances can be classified into one of three categories that describe their

magnetic behavior.

Diamagnetic substances are those in which the magnetization is

weak and opposite the field

, so that the susceptibility is negative.

Paramagnetic

substances are those in which the magnetization is weak and in the same direction
as the field

, so that the susceptibility is positive. In

ferromagnetic substances,

interactions between atoms cause magnetic moments to align and create a strong
magnetization that remains after the external field is removed.

M " 8H

1. Is the magnetic field created by a current loop uniform?

Explain.

2. A current in a conductor produces a magnetic field that

can be calculated using the Biot–Savart law. Because
current is defined as the rate of flow of charge, what can
you conclude about the magnetic field produced by
stationary charges? What about that produced by moving
charges?

Explain why two parallel wires carrying currents in
opposite directions repel each other.

4. Parallel current-carrying wires exert magnetic forces on each

other. What about perpendicular wires? Imagine two such
wires oriented perpendicular to each other, and almost
touching. Does a magnetic force exist between the wires?

5. Is Ampère’s law valid for all closed paths surrounding a

conductor? Why is it not useful for calculating

B for all

such paths?

6. Compare Ampère’s law with the Biot–Savart law. Which is

more generally useful for calculating

B for a current-

carrying conductor?

7. Is the magnetic field inside a toroid uniform? Explain.
8. Describe the similarities between Ampère’s law in magnet-

ism and Gauss’s law in electrostatics.

A hollow copper tube carries a current along its length. Why
is

B " 0 inside the tube? Is B nonzero outside the tube?

10. Describe the change in the magnetic field in the space

enclosed by a solenoid carrying a steady current I if (a) the
length of the solenoid is doubled but the number of turns
remains the same and (b) the number of turns is doubled
but the length remains the same.

11. A flat conducting loop is located in a uniform magnetic

field directed along the x axis. For what orientation of the
loop is the flux through it a maximum? A minimum?

12. What new concept did Maxwell’s generalized form of

Ampère’s law include?

13. Many loops of wire are wrapped around a nail and the

ends of the wire are connected to a battery. Identify the
source of

M, of H, and of B.

14. A magnet attracts a piece of iron. The iron can then attract

another piece of iron. On the basis of domain alignment,
explain what happens in each piece of iron.

15. Why does hitting a magnet with a hammer cause the

magnetism to be reduced?

16. A Hindu ruler once suggested that he be entombed in a

magnetic coffin with the polarity arranged so that he
would be forever suspended between heaven and Earth. Is
such magnetic levitation possible? Discuss.

17. Why is

M " 0 in a vacuum? What is the relationship

between

B and H in a vacuum?

18. Explain why some atoms have permanent magnetic dipole

moments and others do not.

19. What factors contribute to the total magnetic dipole

moment of an atom?

20. Why is the susceptibility of a diamagnetic substance

negative?

21. Why can the effect of diamagnetism be neglected in a

paramagnetic substance?

22. Explain the significance of the Curie temperature for a

ferromagnetic substance.

23. Discuss the difference among ferromagnetic, paramag-

netic, and diamagnetic substances.

24. A current in a solenoid having air in the interior creates

a magnetic field

B " #

H. Describe qualitatively what

happens to the magnitude of

B as (a) aluminum,

(b) copper, and (c) iron are placed in the interior.

25. What is the difference between hard and soft ferromag-

netic materials?

26. Should the surface of a computer disk be made from a

hard or a soft ferromagnetic substance?

Q U E S T I O N S

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