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Sears and
Zemansky’s
University
Physics
28 Magnetic of Field and Magnetic Forces
28-1 INTRODUCTION
The most familiar aspects of magnetism are those associated
with permanent magnets, which attract unmagnetized iron
objects and can also attract or repel other magnets. A
compass needle aligning itself with the earth's magnetism is
an example of this interaction.
As we did for the electric force, we will describe magnetic
forces using the concept of a field. A magnetic field is
established by a permanent magnet, by an electric current in
a conductor, or by other moving charges. This magnetic field,
in turn, exerts forces on moving charges and current-carrying
conductors. In this chapter we study the magnetic forces and
torques exerted on moving charges and currents by magnetic
fields.
28-2 MAGNETISM
Permanent magnets were found to exert forces on each other
as well as on pieces of iron that were not magnetized. It was
discovered that when an iron rod is brought in contact with a
natural magnet, the rod also becomes magnetized. When
such a rod is floated on water or suspended by a string from
its center, it tends to line itself up in a north-south direction.
The needle of an ordinary compass is just such a piece of
magnetized iron.
Before the relation of magnetic interactions to moving
charges was understood, the interactions of permanent
magnets and compass needles were described in terms of
magnetic poles. If a bar-shaped permanent magnet, or bar
magnet, is free to rotate, one end points north. This end is
called a north pole or N-pole; the other end is a south pole or
S-pole. Opposite poles attract each other, and like poles repel
each other (Fig. 28-1).
The earth itself is a magnet. Its north geographical pole is
close to a magnetic south pole, which is why the north pole
of a compass needle points north.
The earth's magnetic axis is not quite parallel to its
geographic axis (the axis of rotation), so a compass reading
deviates somewhat from geographic north. This deviation,
which varies with location, is called magnetic declination or
magnetic variation. Also, the magnetic field is not horizontal
at most points on the earth's surface; its angle up or down is
called magnetic inclination. At the magnetic poles the
magnetic field is vertical.
The concept of magnetic poles may appear similar to that of
electric charge, and north and south poles may seem
analogous to positive and negative charge. But the analogy
can be misleading. While isolated positive and negative
charges exist, there is no experimental evidence that a single
isolated magnetic pole exists; poles always appear in pairs. If
a bar magnet is broken in two, each broken end becomes a
pole.
The existence of an isolated magnetic pole, or magnetic
monopole, would have sweeping implications for theoretical
physics. Extensive searches for magnetic monopoles have
been carried out, but so far without success.
The first evidence of the relationship of magnetism to moving
charges was discovered in 1819 by the Danish scientist Hans
Christian Oersted. He found that a compass needle was
deflected by a current-carrying wire, as shown in Fig. 28-4.
Similar investigations were carried out in France by Andr6
Ampere. A few years later, Michael Faraday in England and
Joseph Henry in the United States discovered that moving a
magnet near a conducting loop can cause a current in the
loop. We now know that the magnetic forces between two
bodies shown in Figs. 28-1 and 28-2 are fundamentally due to
interactions between moving electrons in the atoms of the
bodies (above and beyond the electric interactions between
these charges).
Inside a magnetized body such as a permanent magnet, there
is a coordinated motion of certain of the atomic electrons; in
an unmagnetized body these motions are not coordinated.
e
q

28-3 MAGNETIC FIELD
To introduce the concept of magnetic field properly, let's
review our formulation of electric interactions in Chapter
22, where we introduced the concept of electric field. We
represented electric interactions in two steps:
1. A distribution of electric charge at rest creates an electric
field E. in the surrounding space.
2. The electric field exerts a force F =Eq on any other charge
q that is present in the field.
We can describe magnetic interactions in a similar way:
1. A moving charge or a current creates a magnetic field in
the surrounding space (in addition to its electric field).
2. The magnetic field exerts a force F on any other moving
charge or current that is present in the field.
In this chapter we'll concentrate on the second aspect of the
interaction: Given the presence of a magnetic field, what
force does it exert on a moving charge or a current?
In this chapter we'll concentrate on the second aspect of the
interaction: Given the presence of a magnetic field, what
force does it exert on a moving charge or a current? In
Chapter 29 we will come back to the problem of how
magnetic fields are created by moving charges and currents.
Magnetic field is a vector field, that is, a vector quantity
associated with each point in space. We will use the symbol
B for magnetic field. At any position the direction of B is
defined as that in which the north pole of a compass needle
tends to point.
What are the characteristics of the magnetic force on a
moving charge? First, its magnitude is proportional to the
magnitude of the charge. If a 1- C charge and a 2- C charge
move through a given magnetic field with the same velocity,
the force on the 2- C charge is twice as great as that on the
1- C charge. The magnitude of the force is also proportional
to the magnitude, or "strength," of the field;
If we double the magnitude of the
field (for example, by using two
identical bar magnets instead of
one) without changing the charge
or its velocity, the force doubles.
The magnetic force also depends
on the particle's velocity. This is
quite different from the electricfield force, which is the same
whether the charge is moving or
not. A charged particle at rest
experiences no magnetic force.
Furthermore, the magnetic force
F does not have the same
direction as the magnetic field B,
but instead is always
perpendicular to both B and the
velocity V.
Figure 28-5 shows these relationships. The direction of F is
always perpendicular to the plane containing V and B. Its
magnitude is given by
F  q v B  q vB sin 
(28-1)
where qis the magnitude of the charge and Φ is the angle
measured from the direction of V to the direction of B, as
shown in the figure. This description does not specify the
direction of F completely; there are always two directions,
opposite to each other, that are both perpendicular to the
plane of V and B. To complete the description, we use the
same right-hand rule that we used to define the vector
product in Section 1 - 11. (It would be a good idea to review
that section before you go on.) Draw the vectors V and B with
their tails together, as in Fig. 28-5b. Imagine turning V until it
points in the direction of B. Wrap the fingers of your right
hand around the line perpendicular to the plane of V and B so
that they curl around with the sense of rotation from V and B.
Your thumb then points in the direction of the force F on a
positive charge. (Alternatively, the direction of the force F on
a positive charge is the direction in which a right-handthread screw would advance if turned the same way.) This
discussion shows that the force on a charge q moving with
velocity V in a magnetic field B is given, both in magnitude
and in direction, by



F  q v  B (magnetic force on a moving charged particle). (28-2)
This is the first of several vector products we will encounter
in our study of magnetic field relationships. It's important to
note that Eq. (28-2) was not deduced theoretically; it is an
observation based on experiment. When a charged particle
moves through a region of space where both electric and
magnetic fields are present, both fields exert forces on the
particle. The total forceis the vector sum of the electric and
magnetic forces:




F  q( E  v  B).
(28-4)
28-4 MAGNETIC FIELD LINES AND MAGNETIC FLUX
We draw the lines so that the line through any point is
tangent to the magnetic field vector B at that point. Just as
with electric field lines, we draw only a few representative
lines; otherwise, the lines would fill up all of space. Where
adjacent field lines are close together, the field magnitude is
large; where these field lines are far apart, the field
magnitude is small. Also, because the direction of B at each
point is unique, field lines never intersect.
MAGNETIC FLUX AND GAUSS'S LAW FOR MAGNETISM
We define the magnetic flux B through a surface just as we
defined electric flux in connection with Gauss's law in
Section 23-3. We can divide any surface into elements of area
dA For each element we determine B , the component of B
normal to the surface at the position of that element, as
shown. From the figure,
B  B cos 
where  is the angle between the
direction of B and a line
perpendicular to the surface. We
define the magnetic flux dB
through this area as


d B  B dA  B cos dA  B d A
(28-5)
The total magnetic flux through the surface is the sum of the
contributions from the individual area elements:


 B   B dA  B cos dA  B d A
(magnetic flux through a surface).
(28-6)
Magnetic flux is a scalar quantity. In the special case in which
B is uniform over a plane surface with total area A, B and
Φare the same at all points on the surface, and
 B  B A  BA cos 
(28-7)
If B happens to be perpendicular to the surface, then cos
=1 and Eq. (28-7) reduces to B = BA. We conclude that the
total magnetic flux through a closed surface is always zero.
Symbolically,


 B d A  0
(magnetic flux through closed surface). (28-8)
This equation is sometimes called Gauss's law for
magnetism.
28-5 MOTION OF CHARGED PARTICLES IN A
MAGNETIC FIELD
When a charged particle moves in a magnetic field, it is
acted on by the magnetic force given by Eq. (28-2), and the
motion is determined by Newton's laws. Figure 28-13 shows
a simple example. A panicle with positive charge q is at point
O, moving with velocity V in a uniform magnetic field B
directed into the plane of the figure. The electors V and B are
perpendicular, so the magnetic force

 
F  q v  B has magnitude F  qvB and a direction as shown
in the figure. The force is always perpendicular to V, so it cannot change the magnitude of the velocity, only its direction.
To put it differently, the magnetic force never has a
component parallel to the particle's motion, so the magnetic
force can never do work on the particle.
This is true even if the magnetic field is not uniform. Motion of a
charged particle under the action of a magnetic field alone is
always motion with constant speed.
v2
F  q vB  m ,
R
(28-10)
where m is the mass of the particle. Solving Eq. (28-10) for
the radius R of the circular path, we find
mv
R
qB
(radius of a circular orbit in a magnetic field).
(28-11)
qB qB
v
 v

(28-12)
R
mv
m
f   / 2
The number of revolutions per unit time is
This frequency f is independent of the radius R of the path. It
is called the cyclotron frequency; Motion of a charged
particle in a non-uniform magnetic field is more complex.
Figure 28-15 shows a field produced by two circular coils
separated by some distance. Particles near either coil
experience a magnetic force toward the center of the region;
particles with appropriate speeds spiral repeatedly from one
end of the region to the other and back. Because charged
particles can be trapped in such a magnetic field, it is called a
magnetic bottle.

B
线圈
线圈
hot plasmas
28-6 APPLICATIONS OF MOTION OF CHARGED
PARTICLES
VELOCITY SELECTOR
E
v
B
(28-13)
THOMSON'S e/m EXPERIMENT
In one of the landmark experiments in physics at the end of
the nineteenth century, J. J. Thomson (1856-1940) used the
idea just described to measure the ratio of charge to mass
for the electron. For this experiment, carried out in 1897 at
the Cavendish Laboratory in Cambridge, England, Thomson
used the apparatus shown in Fig. 28-19. The speed v of the
electrons is determined by the accelerating potential V, just
as in the derivation of Eq.(24-24). The kinetic energy 1/2mv2
equals the loss of electric potential energy eV , where e is the
magnitude of the electron charge:
1 2
mv  eV
2
or
v
2eV
m
(28-14)
The electrons pass straight through when Eq.(28-13) is
satisfied; combining this with Eq.(28-14), we get
E
2eV

B
m
e
E2

m 2VB 2
(28-15)
All the quantities on the right side can be measured, so the
ratio e/m of charge to mass can be determined. It is not
possible to measure e or m separately by this method, only
their ratio.
MASS SPECTROMETERS
28-7 MAGNETIC FORCE ON A CURRENT-CARRYING
CONDUCTOR
We can compute the force on a current-carrying conductor
starting with the magnetic force



on a single moving charge.
F  q v B
B
Fm
q
j
Figure 28-21
Figure 28-21 shows a straight segment of a conducting wire,
with length l and cross-section area A; the current is from
bottom to top. The wire is in a uniform magnetic field B,
perpendicular to the plane of the diagram and directed into
the plane. Let's assume first that the moving charges are
positive. Later we'll see what happens when they are negative.
The drift velocity vd is upward, perpendicular to B. The
average force on each charge is



F  q vd  B
directed to the left as shown in the figure;
since vd and B are perpendicular, the magnitude of the
force is F  qvd B
We can derive an expression for the total force on all the
moving charges in a length l of conductor with cross-section
area A, using the same language we used in Eqs.(26-2) and
(26-3) of Section 26-2. The number of charges per unit volume
is n; a segment of conductor with length l has volume Al and
and contains a number of charges equal to nAl . The total
force F on all the moving charges in this segment has
magnitude
F  (nAl )(qvd B)  (nqvd A)(lB ).
(28-16)
From Eq. (26-3) the current density is J  nqvd The product JA
is the total current I, so we can rewrite Eq. (28-16) as
F  I lB
F  I lB  I lB sin 
(28-17)
(28-18)



F  I lB
(magnetic force on a
(magnetic force on a
straight wire segment).
(28-19)
If the conductor is not straight, we can divide it into
infinitesimal segments dl. The force dF on each segment is



d F  I d lB
(magnetic force on an infinitesimal wire segment).
(28-20)
Then we can integrate this along the wire to find the total
force on a conductor of any shape.
28-8 FORCE AND TORQUE ON A CURRENT LOOP
fab
b
B
fbc
a
n
Φ
fad
d
fcd
c
The total force on the loop is zero because the forces on opposite
sides cancel out in pairs. The net force on a current loop in a
uniform magnetic field is zero, however, the net torque is not
in general equal to zero.
F '  IbB sin( 90    )  IbB cos 
  2F (b / 2) sin   ( IBa )(b sin  )
(28-22)
  IBA sin 
(28-23)
The product IA is called the magnetic dipole moment or
magnetic moment of the loop, for which we use the symbol
=IA
(28-24)
It is analogous to the electric dipole moment introduced in
Section 22-9. In terms of  the magnitude of the torque on a
current loop is
  B sin 



   B
(28-25)
(vector torque on a current loop). (28-26)
An arrangement of particular interest is the solenoid, a helical
winding of wire, such as a coil wound on a circular cylinder
(Fig. 28-30). If the windings are closely spaced, the solenoid
can be approximated by a number of circular loops lying in
planes at right angles to its long axis. The total torque on a
solenoid in a magnetic field is simply the sum of the torques
on the individual turns. For a solenoid with N turns in a
uniform field B, the magnetic moment is  = NIA and
  NIAB sin 
(28-28)
Where  is the angle between the axis of the solenoid and the
direction of the field.
28-10 The Hall Effect
The reality of the forces acting on the moving charges in a
conductor in a magnitude field is striking demonstrated by
the Hall effect, an effect analogues to the transverse
deflection of an electron beam in a magnetic field in vacuum.
Z
+
Y

B
+
l

E

E
–
–
+
–
q

fm
b
–
I
+
d

v
I
a
X
To describe this effect, let’s consider a conductor in the form
of a flat strip, as shown in Fig. The current is in the direction
of the +x-axis, and there is a uniform magnetic field B
perpendicular to the plane of the strip, in the y-axis. The drift
velocity of the moving charges has magnitude vd. A moving
charge is driven toward the lower edge of the strip by the
magnetic force Fz  q vd B
If the charge carriers are positive charge, as in Fig, an excess
positive charge accumulates at the lower edge of the strip,
leaving an excess negative charge at its upper edge.
In terms of the coordinate axes in fig, the electrostatic field Ee
for the positive charge q case is in the +z-direction; its zcomponent Ez is positive. The magnetic field is in the +ydirection, and we write it as By. The magnetic force is qvdBy.
The current density Jx is in the +x-direction. In the steady
state, when the forces qEz and qvdBy are equal in magnitude
and opppsite in direction.
qE z  qvd B y  0
or
E z  vd B y
This confirms that when q is positive, Ez is positive. The
current density Jx is JX = nqvd. Eliminating vd between these
equations, we find
nq 
J x By
Ez
28-30
Note that this result is valid for both positive and negative q.
When q is negative, Ez is positive, and conversely.
29 Sources of Magnetic
29-1 INTRODUCTION
In a word, yes. Our analysis will begin with the magnetic field
created by a single moving point charge. We can use this analysis
to determine the field created by a small segment of a currentcarrying conductor. Once we can do that, we can in principle find
the magnetic field produced by any shape of conductor. Then we
will introduce Ampere's law, the magnetic analog of Gauss's law in
electrostatics. Ampere's law lets us exploit symmetry properties in
relating magnetic fields to their sources. Moving charged particles
within atoms respond to magnetic fields and can also act as
sources of magnetic field. We'll use these ideas to understand
how cer-tain magnetic materials can be used to intensify magnetic
fields as well as why some materials such as iron act as
permanent magnets. Finally, we will study how a time-varying
electric field, which we will describe in terms of a quantity called
displacement current, can act as a source of magnetic field.
29-2 MAGNETIC FIELD OF A MOVING CHARGE
Let's start with the basics, the magnetic field of a single point
charge q moving with a constant velocity v, As we did for
electric fields, we call the location of the charge the source
point and the point P where we want to find the field the field
point. In Section 22-6 we found that at a field point a distance
r from a point charge q, the magnitude of the electric field E
caused by the charge is proportional to the charge magnitude
q and 1/r2, and the direction of E (for positive q) is along the
line from source point to field point. The corresponding
relationship for the magnetic field B of a point charge q
moving with constant velocity has some similarities and
some interesting differences.
Experiments show that the magnitude of B is also
proportional to q and 1/r2. But the direction of B is not along
the line from source point to field point. Instead, B is
perpendicular to the plane containing this line and the
particle's velocity vector v, as shown in Fig. 29-1.
P
v
B

r
Φ
q
 0 q v sin 
B
2
4
r
Where 0/4 is a proportionality constant.
(29-1)


 0 q v r
B
4 r 2

(magnetic field of a point charge with constant velocity).
(29-2)
 0  4  10 7 N  s 2 / C 2  4  10 7 Wb / A  m  4  10 7 T  m / A
k
1
40
c2 
 (10 7 N  s 2 / C 2 )c 2
1
 0 0
(29-4)
29-3 MAGNETIC FIELD OF A CURRENT ELEMENT
Just as for the electric field, there is a principle of
superposition of magnetic fields: The total magnetic field
caused by several moving charges is the vector sum of the
fields caused by the individual charges.
dQ  nqAdl
 0 dQ v d sin   0 n q v d Adl sin 
dB 

2
2
4
4

r
r
 0 Idl sin 
dB 
4 r 2


0 I d l  r
dB
4 r 2

(magnetic field of a current element), (29-6)
Equations (29-5) and (29-6) are called the law of Biot and
Savart (pronounced "Bee-oh" and "Suh-var"). We can use
this law to find the total magnetic field B at any point in space
due to the current in a complete circuit. To do this, we
integrate Eq. (29-6) over all segments dl that carry current;
symbolically,


0 I d l  r
B
4  r 2

(29-7)
P
I
Φ
B

r
Problem-Solving Strategy
MAGNETIC FIELD CALCULATIONS
1. Be careful about the directions of vector quantities. The
current element dl always points in the direction of the
current. The unit vector r is always directed from the
current element (the source point) toward the point P at
which the field is to be determined (the field point).
2. In some situations the dB at point P have the same
direction for all the current elements; then the magnitude
of the total B field is the sum of the magnitudes of the dB.
But often the dB have different directions for different
current elements. Then you have to set up a coordinate
system and represent each dB in terms of its components.
The integral for the total B is then expressed in terms of an
integral for each component. Sometimes you can use the
symmetry of the situations to prove that one component
must vanish. Always be alert for ways to use symmetry to
simplify the problem.
3. Look for ways to use the principle of superposition of
magnetic fields. Later in this chapter we'll determine the
fields produced by certain simple conductor shapes. If you
encounter a conductor of a complex shape that can be
represented as a combination of these simple shapes, you
can use superposition to find the field of the complex shape.
Examples include a rectangular loop and a semicircle with
straight-line segments on both sides.
29-4 MAGNETIC FIELD OF A STRAIGHT CURRENTCARRYING CONDUCTOR
An important application of the law of Blot and Savart is
finding the magnetic field produced by a straight currentcarrying conductor. This result is useful because straight
conducting wires are found in essentially all electric and
electronic devices. Figure 29-5 shows such a conductor with
length 2a carrying a current I.
Y
a
Idl Φ
r
B
O
P
X
I
-a
0 I
B
4
xdy
a ( x 2  y 2 ) 3 / 2
a
0 I
2a
B
4 x x 2  a 2
When the length 2a of the conductor is very great in
comparison to its distance x from point P, we can consider it
to be infinitely long. When a is much larger than x, x 2  a 2
is approximately equal to a; hence in the limit a  
0 I
0 I
B
B
2 r
2x
(a long, straight, current-caning conductor). (29-9)
29-5 FORCE BETWEEN PARALLEL CONDUCTORS
In Example 29-4 (Section 29-4) we showed how to use the
principle of superposition of magnetic fields to find the total
field due to two long current-carrying conductors. Another
important aspect of this configuration is the interaction force
between the conductors. This force plays a role in many
practical situations in which current-carrying wires are close
to each other, and it also has fundamental significance in
connection with the definition of the ampere. Figure 29-8
shows segments of two long, straight parallel conductors
respectively, in the same direction. Each conductor lies in the
magnetic field set up by the other, so each experiences a
force. The diagram shows some of the field lines set up by
the current in the lower conductor.
From Eq. (29-9) the lower conductor produces a B field that, at
the position of the upper conductor, has magnitude
0 I
B
2 r
From Eq. (28-19) the force that this field exerts on a length L of

 
the upper conductor is
F  I ' L B
where the vector L is in the direction of the current I‘ and has
magnitude L. Since B is perpendicular to the length of the
conductor and hence to L the magnitude of this force is
,
 0 II ' L
F  I ' LB 
2 r
and the force per unit length F/L is
F  0 II '

L 2 r
(two long, parallel, current-carrying conductors). (29-11)



Applying the right-hand rule to F  I ' L B
shows that the force on the upper conductor is directed downward
The attraction or repulsion between two straight, parallel, currentcarrying conductors is the basis of the official SI definition of the
ampere: One ampere is that unvarying current that, if present
in each of two parallel conductors of infinite length and one
meter apart in empty space, causes each conductor to
experience a force of exactly 2  10-7 newtons per meter of
length.
29-6 MAGNETIC FIELD OF A CIRCULAR CURRENT LOOP
We can use the law of Blot and Savart, Eq. (29-5) or (29-6), to
find the magnetic field at a point P on the axis of the loop, at a
distance x from the center. As the figure shows, dl and r are
perpendicular, and the direction of the field B caused by this
particular element dl lies in the xy-plane. Since r 2  x 2  a 2
the magnitude dB of the field due to element dl is
0 I
dl
dB 
4 ( x 2  a 2 )
(29-12)
The components of the vector dB are
0 I
dl
a
dBx  dB cos  
4 ( x 2  a 2 ) ( x 2  a 2 )1 / 2
0 I
dl
x
dB y  dB sin  
4 ( x 2  a 2 ) ( x 2  a 2 )1 / 2
(29-13)
(29-14)
0 I
 0 Ia
adl
Bx  

dl
2
2 3/ 2
2
2 3/ 2 
4 ( x  a )
4 ( x  a )
Bx 
 0 Ia 2
2( x  a )
2
2 3/ 2
(on the axis of a circular loop).
(29-15)
Now suppose that instead of the single loop in Fig. 29-10 we
have a coil consisting of N loops, all with the same radius. The
loops are closely spaced so that the plane of each loop is
essentially the same distance x from the field point P. Each loop
contributes equally to the field, and the total field is N times the
field of a single loop:
Bx 
 0 NIa 2
2( x  a )
2
Bx 
Bx 
2 3/ 2
 0 NI
(on the axis of N circular loops).
(at the center of N circular loops).
(29-16)
(29-17)
2a
0 
2 ( x 2  a 2 ) 3 / 2
(on the axis of any number of circular loops). (29-18)
29-7 AMPERE'S LAW
So far our calculations of the magnetic field due to a current
have involved finding the infinitesimal field dB due to a
current element, then summing all the dB’s to find the total
field.
Ampere's law is formulated not in terms of magnetic flux, but
rather in terms of the line integral of B around a closed path,
denoted by
 
 B d
l
We used line integrals to define work in Chapter 6 and to
calculate electric potential in Chapter 24.To evaluate this
integral. we divide the path into infinitesimal segments dl,
calculate the scalar product of B·dl for each segment, and
sum these products. In general, B varies from point to point,
and we must use the value of B at the location of each dl. An
alternative notation is
 B dl
||
where BІІ s the component of B parallel to dl at each point.
The circle on the integral sign indicates that this integral is
always computed for a closed path, one whose beginning
and end points are the same.
To introduce the basic idea of Ampere's law, let's consider
again the magnetic field caused by a long, straight conductor
carrying a current I. We found in Section 29-4 that the field at
a distance r from the conductor has magnitude and that the
magnetic field lines are circles centered on the conductor.
Let's take the line integral of B around one such circle with
radius r, as in Fig. 29-13a. At every point on the circle, B and
dl are parallel, and so B·dl = Bdl, since r is constant around
the circle, B is constant as well. Alternatively, we can say that
BІІ is constant and equal to B at every point on the circle.
Hence we can take B outside of the integral. The remaining
integral
dl is just the circumference of the circle, so

0 I
 B d l   B|| dl  B dl  2 r (2 r )   0 I


The line integral is thus independent of the radius of the
circle and is equal to 0 multiplied by the current passing
through the area bounded by the circle. In Fig. 29-13b the
situation is the same, but the integration path now goes
around the circle
in the opposite direction. Now B and dl are

antiparallel, so B d l   Bdl , and the line integral equals   0 ,
We get the same result if the integration path is the same as
in Fig. 29-13a, but the direction
of the current is reversed.


Thus the line integral  B d l equals  0 multiplied by the
current passing through the area bounded by the integration
path, with a positive or negative sign depending on the
direction of the current relative to the direction of
integration.
An integration path that does not enclose the conductor is
used in Fig. 29-13c. Along the circular arc ab of radius r1,
B and dl are parallel, and
B||  B1   0 I / 2 r1
along the circular arc cd of radius r2, B and dl are antiparallel and
B||   B2    0 I / 2 r2
The B field is perpendicular to dl at each point on the
straight sections bc and da, so B||  0
and these sections contribute zero to the line integral. The
total line integral is then
0 I
0 I
 B d l   B||dl  B1 a dl  (0)b dl  (B2 )c dl  (0)d dl  2 r1 ( r1 )  0  2 r2 ( r2 )  0  0


b
c
d
a
We can also derive these results for more general integration
paths, At the position of the line element dl, the angle
between dl and B is Φ, and


B d l  Bdl cos 
I
L
L
r
p
I
L

r
d

 B
r


dl
From the figure, dl cos   rd where d is the angle
subtended by dl at the position of the conductor and r is the
distance of dl from the conductor. Thus
0 I
0 I
 B d l   2 r (rd )  2  d
But  d is just equal to 2 the total angle swept out by the


radial line from the conductor to dl during a complete trip
around the path. So we get


 B d l   I
0
(29-19)
This result doesn't depend on the shape of the path or on the
position of the wire inside it. If the current in the wire is
opposite to that shown, the integral has the opposite sign.
But if the path doesn't enclose the wire .
29-8 APPLICATIONS OF AMPERE'S LAW
Problem-Solving Strategy
AMPERE’S LAW
1. The first step is to select the integration path you will use
with Ampere's law. If you want to determine B at a certain
point, then the path must pass through that point.
2. The integration path doesn't have to be any actual physical
boundary. Usually, it is a purely geometric curve; it may
be in empty space, embedded in a solid body, or some of
each.
3. The integration path has to have enough symmetry to
make evaluation of the integral possible. If the problem
itself has cylindrical symmetry, the integration path will
usually be a circle coaxial with the cylinder axis.
4. If B is tangent to all or some portion of the integration path
and has the same magnitude B at every point, then its line
integral equals B multiplied by the length of that portion of
the path.
5. If B is perpendicular to all or some portion of the path, that
portion of the path makes no contribution to the integral.
6. The sign of a current enclosed by the integration path is
given by a right-hand rule. Curl the fingers of your right
hand so that they follow the integration path in the direction
that you carry out the integration. Your right thumb then
points in the direction of positive current. If B is as described
in Step 4 and Iencl is positive, then the direction of B is the
same as the direction of the integration path; if instead Iencl is
negative, B is in the direction opposite to that of the
integration.


7. In the integral  B d l , B is always the total magnetic field
at each point on the path. This field can be caused partly by
currents enclosed by the path and partly by currents outside.
If no net current is enclosed by the path. The field
at
points


on the path need not be zero, but the integral  B d l is
always zero.
29-9 Magnetic Materials
In discussing how currents cause magnetic fields, we have
assumed that the conductors are surrounded by vacuum.
Permanent magnets, magnetic recording tapes, and
computer disks depend directly on the magnetic properties of
materials. We will discuss three broad classes of magnetic
behavior that occur in materials; these are called
paramagnetism, diamagnetism, and ferromagnetism.
The Bohr Magneton
L
I
v
r
+q

-e
The equivalent current is the total charge passing any point
on the orbit per unit time, which is just the magnitude e of the
electron charge divided by the orbital period T:
e
ev
I 
T 2r
The magnetic moment  = IA is then
 
ev
evr
2

r 
29-25
2r
2
It is useful to express  in terms of the angular momentum L
of the electron.
e

L
2m
h  6.626 10
29-26
34
J s
Equation (29-26) shows that associated with the fundamental
unit of angular momentum is a corresponding fundamental
unit of magnetic moment. If L = h/2, then
e  h  eh

 
2m  2  4m
29-27
This quantity is called the Bohr magneton, denoted by B. Its
numerical value is
 B  9.274 1034 A  m2  9.274 1034 J / T
Paramagnetism
 total
M
V
 

B  B0  0 M
magnetization
29-28
29-29
A material showing the behavior just described is said to be
paramagnetic. The result is that the magnetic field at any
point in such a material is greater by a dimensionless factor
Km, called the relative permeability.
  K m 0
29-30
Diamagnetism
In some materials the total magnetic moment of all the atomic
current loops is zero when no magnetic field is present. But
even these materials have magnetic effects because an
external field alters electron motions within the atoms,
causing additional current loops and induced magnetic
dipoles comparable to the induced electric dipoles we
studied in section 25-6. In this case the additional field
caused by these current loops is always opposite in direction
to that of the external field. Such material are said to be
diamagnetic.
  0
B0
B
Paramagnetism


B0

i

pm

Diamagnetism


L

M
Ferromagnetism
There is a third class of materials, called ferromagnetic
materials, which includes iron, nickel, cobalt, and many
alloys containing these elements. In these materials, strong
interactions between atomic magnetic moments cause them
to line up parallel to each other in regions called magnetic
domains, even when no external field is present. Saturation
B
B
B~H
b
a
f
r ~ H
o
Magnetization
c
H
d
o
e
Hysteresis Loops
H
29-10 DISPLACEMENT CURRENT
Ampere's law, in the form stated in Section 29-7, is
incomplete. As we learn how it has to be modified for the
most general situations, we will uncover some profound and
fundamental aspects of the behavior of electric and magnetic
fields.
To introduce the problem, let's consider the process of
charging a capacitor (Fig.29-26). Conducting wires lead
current ic into one plate and out of the other; the charge Q
increases, and the electric field E between the plates
increases. The notation ic indicates conduction current to
distinguish it from another kind of current we are about to
encounter, called displacement current iD. We use lowercase
i’s and v’s to denote instantaneous values of currents and
potential differences, respectively, that may vary with time.
A
E
iD
I
R

q  Cv
where
A
d
( Ed )  EA   E
(29-33)
 E  EA is the electric flux through the surface.
As the capacitor charges, the rate of change of q is the
conduction current, i  dq / dt
Taking the derivative of
C
Eq. (29-33) with respect to time, we get
iC 
d E
dq

dt
dt
d E
iD  
dt
(29-34)
(displacement current)
(29-35)
dΦE
iD   0
dt

B
 
 B  dl  0 ic  iD 
encl
dE
jD  
dt
Generalized Ampere’s law
Displacement current density
P1
IC

P2
ID
 
r2
 B  dl  2rbB  0 2 ic
R
0 r
B
i
c
2
2 R

r
R
30 Electromagnetic Induction
30-1 INTRODUCTION
The answer is a
phenomenon known as
electromagnetic induction:
If the magnetic flux through
a circuit changes, an emf
and a current are induced in
the circuit. In a powergenerating station, magnets
move relative to coils of
wire to produce a changing
magnetic flux in the coils
and hence an emf.
Michael Faraday
Other key components of electric power systems, such as
transformers, also depend on magnetically induced emfs.
When electromagnetic induction was discovered in the 1830s,
it was a mere laboratory curiosity; today, thanks to its central
role in the generation of electric power, it is fundamentally
responsible for the nature of our technological society. The
central principle of electromagnetic induction, and the
keystone of this chapter, is Faraday's law. This law relates
induced emf to changing magnetic flux in any loop. including
a closed circuit. We also discuss Lenz's law, which helps us
to predict the directions of induced emfs and currents. This
chapter provides the principles we need to understand
electrical energy-conversion devices such as motors,
generators, and transformers. We will also present a neat
package of formulas, called Maxwell's equations, that
describe the behavior of electric and magnetic fields in any
situation. These equations pave the way for the analysis of
electromagnetic waves in Chapter 33.
26-5 Electromotive Force
In an electric circuit there must be a
device somewhere in the loop that acts
like the water pump in a water fountain.
In this device a charge travels “uphill”,
from lower to higher potential energy,
even though the electrostatic force is
trying to push it from higher to lower
potential energy. The direction of
current in such a device is from lower
to higher potential , just the opposite
of what happens in an ordinary
conductor. The influence that make
currents flow from lower to higher
potential is called electromotive force.
This is a poor term because emf is not
a force but an energy-per-unit-charge
quantity.


 Fc
Fe 






I
R
Fc

q
30-2 INDUCTION EXPERIMENTS
To explore further the common
elements in these observations,
let's consider a more detailed
series of experiments with the
situation shown in Fig. 30-2.
We connect a coil of wire to a
galvanometer, then place the
coil between the poles of an
electromagnet whose magnetic
field we can vary. Here's what
we observe:
1. When there is no current in
the electromagnet, so that B=0,
the galvanometer shows no
current.
2. When the electromagnet is turned on, there is a momentary
current through the meter as B increases.
3. When B levels off at a steady value, the current drops to
zero, no matter how large B is.
4. With the coil in a horizontal plane, we squeeze it so as to
decrease the cross-section area of the coil. The meter detects
current only during the deformation, not before or after. When
we increase the area to return the coil to its original shape,
there is current in the opposite direction, but only while the
area of the coil is changing.
5. If we rotate the coil a few degrees about a horizontal axis,
the meter detects current during the rotation, in the same
direction as when we decreased the area. When we rotate the
coil back, there is a current in the opposite direction during
this rotation.
6. If we jerk the coil out of the magnetic field, there is a
current during the motion, in the same direction as when we
decreased the area.
7. If we decrease the number of turns in the coil by unwinding
one or more turns, there is a current during the unwinding, in
the same direction as when we decreased the area. If we wind
more turns onto the coil, there is a current in the opposite
direction during the winding.
8. When the magnet is turned off, there is a momentary
current in the direction opposite to the current when it was
turned on.
9. The faster we carry out any of these changes, the greater
the current.
10. If all these experiments are repeated with a coil that has
the same shape but different material and different resistance,
the current in each case is inversely proportional to the total
circuit resistance. This shows that the induced emfs that are
causing the current do not depend on the material of the coil
but only on its shape and the magnetic field.
30-3 FARADAY'S LAW
The common element in all
induction effects is changing
magnetic flux through a
circuit. Before stating the
simple physical law that
summarizes all of the kinds
of experiments described in
Section 30-2, let's first
review the concept of
magnetic flux B (which we
introduced in Section 28-4).
For an infinitesimal area
element dA in a magnetic
field B (Fig. 30-3), the
magnetic flux dB through
the area is




d B  B d A  B dA cos 
 B   B d A   B dA cos 
 
 B  B A  BA cos 
(30-1)
(30-2)
Faraday's law of induction states:
The induced emf in a closed loop equals the negative of
the time rate of change of magnetic flux through the loop.
In symbols, Faraday's law is
d B
 
dt
(Faraday's law of induction).
(30-3)

n Φ 0

n Φ 0

n Φ 0

n Φ 0
N
S
N
S
S
N
S
N
dΦ
0
dt
dΦ
0
dt
dΦ
0
dt
dΦ
0
dt
DIRECTION OF INDUCED EMF
We can find the direction of an induced emf or current by
using Eq.(30-3) together with some simple sign rules.
Here's the procedure:
1. Define a positive direction for the vector area S.
2. From the directions of S and the magnetic field B,
determine the sign of the magnetic flux B and its rate of
change dB / dt.
3. Determine the sign of the induced emf or current. If the
flux is increasing, so dB / dt is positive, then the induced
emf or current is negative; if the flux is decreasing, dB / dt
is negative and the induced emf or current is positive.
4. Finally, determine the direction of the induced emf or
current using your right hand. Curl the fingers of your right
hand around the S, vector, with your right thumb in the
direction of S. If the induced emf or current in the circuit is
positive, it is in the same direction as your curled fingers; if
the induced emf or current is negative, it is in the opposite
direction.
If we have a coil with N identical turns, and if the flux varies
at the same rate through each turn, the total rate of change
through all the turns is N times as large as for a single turn.
If B is the flux through each turn, the total emf in a coil with
N turns is
d B
(30 -4)
  N
dt
Problem-Solving Strategy
1. To calculate the rate of change of magnetic flux, you first
have to understand what is making the flux change. Is the
loop or coil moving? Is it changing orientation? Is the
magnetic field changing? Remember that it's not the flux
itself that counts, but its rate of change.
2. Choose a direction for the area vector S or dS, and then
use it consistently. Remember the sign rules for the
positive directions of magnetic flux and emf, and use them
consistently when you implement Eq.(30-3) or (30-4). If
your conductor has N turns in a coil, don't forget to
multiply by N.
3. Use Faraday's law to obtain the induced eml Use the sign
rules to determine the direction of the induced emf and
induced current. If the circuit resistance is known, you can
then calculate the magnitude of the current.
30-4 LENZ'S LAW
Lenz's law is a convenient alternative method for
determining the direction of an induced current or emf.
Lenz's law is not an independent principle; it can be derived
from Faraday's law. It always gives the same results as the
sign rules we introduced in connection with Faraday's law,
but it is often easier to use. Lenz's law also helps us gain
intuitive understanding of various induction effects and of
the role of energy conservation. H. F. E. Lenz (1804-1865) was
a German scientist who duplicated independently many of
the discoveries of Faraday and Henry. Lenz's law states:
The direction of any magnetic induction effect is such as to
oppose the cause of the effect. Lenz's law is also directly
related to energy conservation. If the induced current in
Example 30-7 were in the direction opposite to that given by
Lenz's law, the magnetic force on the rod would accelerate it
to ever-increasing speed with no external energy source,
even though electrical energy is being dissipated in the
circuit. This would be a clear violation of energy conservation
and doesn't happen in nature.
LENZ'S LAW ANDTHE RESPONSE TO FLUX CHANGES
Since an induced current always opposes any change in
magnetic flux through a circuit, how is it possible for the flux
to change at all? The answer is that Lenz's law gives only the
direction of an induced current; the magnitude of the current
depends on the resistance of the circuit. The greater the
circuit resistance, the less the induced current that appears
to oppose any change in flux and the easier it is for a flux
change to take effect. If the loop in Fig. 30-12 were made out
of wood (an insulator), there would be almost no induced
current in response to changes in the flux through the loop.
30-5 MOTIONAL ELECTROMOTIVE FORCE
In situations in which a conductor moves in a magnetic field,
as in the generators discussed in Examples 30-4 through 30-7,
we can gain additional insight into the origin of the induced
emf by considering the magnetic forces on mobile charges in
the conductor. Figure 30-13a shows the same moving rod that
we discussed in Example 30-6, separated for the moment from
the U-shaped conductor. The magnetic field B is uniform and
directed into the page, and we move the rod to the right at a
constant velocity v . A charged panicle q in the rod then
experiences a magnetic force



F  q v B
with magnitude F  q vB
We'll assume in the following discussion that q is positive; in
that case the direction of this force is upward along the rod,
from b toward a.
Now suppose the moving rod slides along a stationary Ushaped conductor, forming a complete circuit (Fig. 30-13b).
No magnetic force acts on the charges in the stationary Ushaped conductors, but the charge that was near points a
and b redistributes itself along the stationary conductors,
creating an electric field within them. This field establishes a
current in the direction shown. The moving rod has become a
source of electromotive force; within it, charge moves from
lower to higher potential, and in the remainder of the circuit,
charge moves from higher to lower potential. We call this emf
a motional electromotive force, denoted by  . From the above
discussion, the magnitude of this emf is
  vBL
(30-6)
(motional emf; length and velocity perpendicular to uniform B)
We can generalize the concept of motional emf for a
conductor with any shape, moving in any magnetic field,
uniform or not (assuming that the magnetic field at each
point does not vary with time). For an element dl of conductor,
the contribution to the emf is the magnitude dl multiplied by
the component of v  B (the magnetic force per unit charge)
parallel to dl; that is,



d  ( v  B)  d l
For any closed conducting loop, the total emf is



   ( v  B)  d l
(30-7)
30-6 INDUCED ELECTRIC FIELDS
As an example, let's consider the situation shown in Fig.
30-15. A long, thin solenoid with cross-section area A and n
turns per unit length is encircled at its center by a circular
conducting loop. The galvanometer G measures the current
in the loop.
I
G
I'
I
B
G
I
B
(a)
E
r
(b)
E
As an example, let's consider the situation shown in Fig. 3015. A long, thin solenoid with cross-section area A and n
turns per unit length is encircled at its center by a circular
conducting loop. The galvanometer G measures the current
in the loop. A current I in the winding of the solenoid sets up
a magnetic field B along the solenoid axis, as shown, with
magnitude B as calculated in Example 29-10 (Section 29-8): B
= 0 n I, along the solenoid axis, as shown, with magnitude B
as calculated in Example 29-10 (Section 29-8):
 B  BA   0 nIA
When the solenoid current I changes with time, the magnetic
flux Balso changes, and according to Faraday's law the
induced emf in the loop is given by
d B
dI
 
   0 nA
dt
dt
(30-8)
But what force makes the charges move around the loop? It
can't be a magnetic force because the conductor isn't moving
in a magnetic field and in fact isn't even in a magnetic field.
We are forced to conclude that there has to be an induced
electric field in the conductor caused by the changing
magnetic flux. This may be a little jarring; we are accustomed
to thinking about electric field as being caused by electric
charges, and now we are saying that a changing magnetic
field somehow acts as a source of electric field. Furthermore,
it's a strange sort of electric field. When a charge q goes once
around the loop, the total work done on it by the electric field
must be equal to q times the emf .


 E d l  
(30-9)
From Faraday's law the emf  is also the negative of the rate
of change of magnetic flux through the loop. Thus for this
case we can restate Faraday's law as
d B
 E d l   dt


(stationary integration path).
(30-10)
30-7 Eddy Currents
In the examples of induction effects that we have studied, the
induced currents have been confined to well-defined paths in
conductors and other components forming a circuit. However,
many pieces of electrical equipment contain masses of metal
moving in magnetic fields or located in changing magnetic
fields. In situations like these we can have induced currents
that circulate throughout the volume of a material. Because
their flow patterns resemble swirling eddies in a river. We call
these eddy currents.
 B
0
t
E
30-8 MAXWELL’S EQUATIONS
We are now in a position to wrap up in a wonderfully neat
package all the relationships between electric and magnetic
fields and their sources that we have studied in the past
several chapters. The package consists of four equations,
called Maxwell’s equations. You may remember Maxwell as
the discoverer of the concept of displacement current, which
we studied in Section 29-10. Maxwell didn't discover all of
these equations single-handedly, but he put them together
and recognized their significance, particularly in predicting
the existence of electromagnetic waves.
For now we'll state Maxwell's equations in their simplest form,
for the case in which we have charges and currents in
otherwise empty space. In Chapter 33 we'll discuss how to
modify these equations if a dielectric or a magnetic material
is present.


 E d A 

Qencl
0
( Gauss's law for E)
(30-12)
( Gauss's law for B)
(30-13)

 B d A  0
d E
 B d l   0 (iC   0 dt ) encl ( Ampere's law).


d B
(Faraday's law).
 E d l   dt


(30-14)
(30-15)
31 Inductance
31-1 INTRODUCTION
How can a 12-volt car battery provide the thousands of volts
needed to produce sparks across the gaps of the spark plugs
in the engine? If you turn off a vacuum cleaner by yanking its
plug out of the wall socket, what's the source of the high
voltage that causes breakdown in the air so that sparks fly in
the socket? The answers to these questions and many others
concerned with varying currents in circuits involve the
induction effects that we studied in Chapter 30. A changing
current in a coil induces an emf in an adjacent coil. The
coupling between the coils is described by their mutual
inductance. A changing current in a coil also induces an emf
in that same coil. Such a coil is called an inductor; the
relationship of current to emf is described by the inductance
(also called self-inductance) of the coil. If a coil is initially
carrying a current, energy is released when the current
decreases; this principle is used in automotive ignition
systems. We'll find that this released energy was stored in
the magnetic field caused by the current that was initially in
the coil, and we'll look at some of the practical applications of
magnetic-field energy.
31-2 MUTUAL INDUCTANCE
In Section 29-5 we considered the magnetic interaction
between two wires carrying steady currents; the current in
one wire causes a magnetic field, which exerts a force on the
current in the second wire. But an additional interaction
arises between two circuits when there is a changing current
in one of the circuits. Consider two neighboring coils of wire,
as in Fig.31-1. A current flowing in coil I produces a magnetic
field B and hence a magnetic flux through coil 2. If the current
in coil 1 changes, the flux through coil 2 changes as well;
according to Faraday's law, this induces an emf in coil 2. In
this way, a change in the
current in one circuit can
induce a current in a
second circuit. Let's
analyze the situation shown
in Fig. 31-1 in more detail.
We will use lowercase
letters to represent
quantities that vary with
time; for example, a timevarying current is i, often
with a subscript to identify
the circuit. In Fig. 31–1 a
current i1 in coil 1 sets up a
magnetic field (as indicated
by the blue lines), and some
of these field lines pass
through coil 2. We denote
the magnetic flux through each turn of coil 2, caused by the
current i1 in coil l, as B2 . (If the flux is different through
different turns of the coil, then B2 denotes the average flux.)
The magnetic field is proportional to i1, so B2 is also
proportional to i1 . When i1 changes, B2 changes; this
changing flux induces an emf 2 in coil 2, given by
d B2
(31-1)
 2  N 2
dt
We could represent the proportionality of B2 and i1 in the
form  B 2  (cons tan t )i1 , but instead it is more convenient to
include the number of turns N2 in the relation. Introducing a
proportionality constant M21, called the mutual inductance of
the two coils, we write
N 2  B 2  M 21i1
(31-2)
Where B2 is the flux through a single turn of coil 2. From this,
d B 2
di1
N2
 M 21
dt
dt
and we can rewrite Eq.(31 - 1) as
di1
(31-3)
dt
That is, a change in the current i1 in coil I induces an emf in
coil 2 that is directly proportional to the rate of change of i1 .
We may also write the definition of mutual inductance,
Eq.(31-2), as
N 2 B2
M 21 
i1
We can repeat our discussion for the opposite case in which
a changing current i2 in coil 2 causes a changing flux B1 and
an emf 1 in Coil 1. We might expect that the corresponding
constant M12 would be different from M12 because in general
the two coils are not identical and the flux through them is
not the same. It turns out, however, that M12 is always equal
 2   M 21
to M12 , even when the two coils are not symmetric. We call
this common value simply the mutual inductance, denoted by
the symbol M without subscripts; it characterizes completely
the induced-emf interaction of two coils. Then we can write
di1
 2  M
dt
and
 1  M
di2
dt
(mutually induced emf’s),
(31-4)
where the mutual inductance M is
N 2  B 2 N1 B1
M

i1
i2
(mutual inductance).
(31-5)
The negative signs in Eq.(31-4) are a reflection of Lenz's law.
The first equation says that a change in current in coil I
causes a change in flux through coil 2, inducing an emf in
coil 2 that opposes the flux change; in the second equation
the roles of the two coils are interchanged.
31-3 SELF-INDUCTANCE AND INDUCTORS
An important related effect occurs even if we consider only a
single isolated circuit. When a current is present in a circuit,
it sets up a magnetic field that causes a magnetic flux
through the same circuit; this flux changes when the current
changes. Thus any circuit that carries a varying current will
have an emf induced in it by the variation in its own magnetic
field. Such an emf is called a self-induced emf. By Lenz's law
a self-induced emf always opposes the change in the current
that caused the emf and so tends to make it more difficult for
variations in current to occur. For this reason, self-induced
emfs can be of great importance whenever there is a varying
current. Self-induced emf's can occur in any circuit, since
there will always be some magnetic flux through the closed
loop of a current-carrying circuit. But the effect is greatly
enhanced if the circuit includes a coil with N turns of wire
(Fig. 31-3). As a result of the current i, there is an average
magnetic flux B through each turn of the coil.
In analogy to Eq. (31-5) we define the self-inductance L of the
circuit as follows:
N B
(self-inductance).
(31-6)
L
i
When there is no danger of
confusion with mutual inductance,
the self-inductance is also called
simply the inductance.
Comparing Eqs.(31-5) and (31-6),
we see that the units of selfinductance are the same as those
of mutual inductance; the SI unit
of self-inductance is one henry. If
the current i in the circuit
changes, so does the flux B ;
from rearranging Eq.(31-6) and
taking the derivative with respect
to time, the rates of change are
related by
d B
di
N
L
dt
dt
From Faraday's law for a coil with N turns. Eq.(30-4), the selfinduced emf is
   Nd B / dt
di
  L
dt
, so it follows that
(self-induced emf).
(31-7)
The minus sign in Eq. (31-7) is a reflection of Lenz's law; it
says that the self-induced emf in a circuit opposes any
change in the current in that circuit. (Later in this section
we'll explore the significance of this minus sign in greater
depth.) Equation (31-7) also states that the self-inductance of
a circuit is the magnitude of the self-induced emf per unit rate
of change of current.
A circuit device that is designed to have a particular inductance is
called an inductor, or a choke. The usual circuit symbol for an
inductor is


di
 En  d l   L dt


Ec  E n  0

b 

di
En  d l  L
dt
a

b 
a

di
Ec  d l  L
dt
di
Vab  Va  Vb  L
dt
(31-8)
31-4 MAGNETIC-FIELD ENERGY
Establishing a current in an inductor requires an input of
energy, and an inductor carrying a current has energy stored
in it. Let's see how this comes about. In Fig. 31-4, an
increasing current in the inductor causes an emf  between
its terminals, and a corresponding potential difference Vab
between the terminals of the source, with point a at higher
potential than point b. Thus the source must be adding
energy to the inductor, and the instantaneous power P (rate
of transfer of energy into the inductor) is P  Vab i
We can calculate the total energy input U needed to establish
a final current I in an inductor with inductance L if the initial
current is zero. We assume that the inductor has zero
resistance, so no energy is dissipated within the inductor. Let
the current at some instant be i , and let its rate of change be
di/dt; the current is increasing, so di/dt  0. The voltage
between the terminals a and b of the inductor at this instant
is Vab = L di/dt , and the rate P at which energy is being
delivered to the inductor (equal to the instantaneous power
supplied by the external source) is
di
P  Vab i  Li
dt
The energy dU supplied to the inductor during an
infinitesimal time interval dt is dU = Pdt, so The total energy
U supplied while the current increases from zero to a final
value I is
1 2
U  L  idi  LI (energy stored in an inductor).
0
2
I
(31-9)
B2
u
2 0
(magnetic energy density in a vacuum). (31-10)
B2
u
2
( magnetic energy density in a material). (31-11)