Chapter 30
Sources of the Magnetic Field
CHAPTER OUTLINE
30.1
The Biot–Savart Law
30.2
The Magnetic Force Between
Two Parallel Conductors
30.3
Ampère’s Law
30.4
The Magnetic Field of a
Solenoid
30.5
Magnetic Flux
30.6
Gauss’s Law in Magnetism
30.7
Displacement Current and the
General Form of Ampère’s
Law
30.8
Magnetism in Matter
30.9
The Magnetic Field of the
Earth
926
±
A proposed method for launching future payloads into space is the use of
rail guns,
in
which projectiles are accelerated by means of magnetic forces. This photo shows the ﬁring
of a projectile at a speed of over 3 km/s from an experimental rail gun at Sandia National
Research Laboratories, Albuquerque, New Mexico. (Defense Threat Reduction Agency
[DTRA])
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I
n the preceding chapter, we discussed the magnetic force exerted on a charged
particle moving in a magnetic ﬁeld. To complete the description of the magnetic
interaction, this chapter explores the origin of the magnetic ﬁeld—moving charges.
We begin by showing how to use the law of Biot and Savart to calculate the magnetic
ﬁeld produced at some point in space by a small current element. Using this formalism
and the principle of superposition, we then calculate the total magnetic ﬁeld due to
various current distributions. Next, we show how to determine the force between two
currentcarrying conductors, which leads to the deﬁnition of the ampere. We also
introduce Ampère’s law, which is useful in calculating the magnetic ﬁeld of a highly
symmetric conﬁguration carrying a steady current.
This chapter is also concerned with the complex processes that occur in magnetic
materials. All magnetic effects in matter can be explained on the basis of atomic
magnetic moments, which arise both from the orbital motion of electrons and from an
intrinsic property of electrons known as spin.
30.1
The Biot–Savart Law
Shortly after Oersted’s discovery in 1819 that a compass needle is deflected by a
currentcarrying
conductor,
JeanBaptiste
Biot
(1774–1862)
and
Félix
Savart
(1791–1841) performed quantitative experiments on the force exerted by an
electric current on a nearby magnet. From their experimental results, Biot and
Savart arrived at a mathematical expression that gives the magnetic ﬁeld at some
point in space in terms of the current that produces the ﬁeld. That expression is
based on the following experimental observations for the magnetic ﬁeld
d
B
at a
point
P
associated with a length element
d
s
of a wire carrying a steady current
I
(Fig. 30.1):
•
The vector
d
B
is perpendicular both to
d
s
(which points in the direction of the
current) and to the unit vector
r
ˆ
directed from
d
s
toward
P
.
•
The magnitude of
d
B
is inversely proportional to
r
2
, where
r
is the distance from
d
s
to
P
.
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 Spring '11
 WANG
 Magnetism, Force, Magnetic Field, Electric charge, Wire

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