<<

. 3
( 5)



>>

poking a bar magnet through a loop of wire generates an electric ¬eld
within that wire, but holding the magnet in a ¬xed position with respect
to the loop induces no electric ¬eld.
And what does the negative sign in Faraday™s law tell you? Simply that
the induced emf opposes the change in ¬‚ux “ that is, it tends to maintain
the existing ¬‚ux. This is called Lenz™s law and is discussed later in this
chapter.
60 A student™s guide to Maxwell™s Equations


Here™s an expanded view of the standard form of Faraday™s law:

Reminder that the Dot product tells you to find The magnetic flux
electric field is a the part of E parallel to dl through any surface
vector (along path C) bounded by C

An incremental segment
of path C


d
«
« E dl = “ ˆ
B n da
dt S
C
The electric
The rate of change
field in V/m
with time
Tells you to sum up the
contributions from each Reminder that this is a line
portion of the closed path C integral (not a surface or a
in a direction given by the volume integral)
right-hand rule


Note that ˜ in this expression is the induced electric ¬eld at each segment
E
d˜ of the path C measured in the reference frame in which that segment is
l
stationary.
And here is an expanded view of the alternative form of Faraday™s law:

Dot product tells you to find
the part of E parallel to dl
(along path C)
Reminder that the The flux of the time
An incremental segment
electric field is a rate of change of
of path C
vector the magnetic field




«E «
dl = “ ˆ
n da
S
C
The electric The rate of change of the
field in V/m magnetic field with time



Tells you to sum up the
Reminder that this is a line
contributions from each
integral (not a surface or a
portion of the closed path C
volume integral)
61
Faraday™s law


In this case, ˜ represents the electric ¬eld in the laboratory frame of
E
reference (the same frame in which ˜ is measured).
B
Faraday™s law and the ¬‚ux rule can be used to solve a variety of
problems involving changing magnetic ¬‚ux and induced electric ¬elds, in
particular problems of two types:
(1) Given information about the changing magnetic ¬‚ux, ¬nd the
induced emf.
(2) Given the induced emf on a speci¬ed path, determine the rate of
change of the magnetic ¬eld magnitude or direction or the area
bounded by the path.
In situations of high symmetry, in addition to ¬nding the induced emf, it
is also possible to ¬nd the induced electric ¬eld when the rate of change of
the magnetic ¬eld is known.
62 A student™s guide to Maxwell™s Equations


˜ The induced electric ¬eld
E
The electric ¬eld in Faraday™s law is similar to the electrostatic ¬eld in its
effect on electric charges, but quite different in its structure. Both types of
electric ¬eld accelerate electric charges, both have units of N/C or V/m,
and both can be represented by ¬eld lines. But charge-based electric ¬elds
have ¬eld lines that originate on positive charge and terminate on
negative charge (and thus have non-zero divergence at those points),
while induced electric ¬elds produced by changing magnetic ¬elds have
¬eld lines that loop back on themselves, with no points of origination or
termination (and thus have zero divergence).
It is important to understand that the electric ¬eld that appears in the
common form of Faraday™s law (the one with the full derivative of the
magnetic ¬‚ux on the right side) is the electric ¬eld measured in the ref-
erence frame of each segment d˜ of the path over which the circulation is
l
calculated. The reason for making this distinction is that it is only in this
frame that the electric ¬eld lines actually circulate back on themselves.


(a) +
Electric field lines
orginate on positive
charges and terminate
on negative charges
E






(b)
Electric field lines
B form complete loops
around boundary
E

As magnet moves to
N S
right, magnetic flux
through surface
decreases
Magnet motion

Surface may be real
or purely imaginary

Figure 3.1 Charge-based and induced electric ¬elds. As always, you should
remember that these ¬elds exist in three dimensions, and you can see full 3-D
visualizations on the book™s website.
63
Faraday™s law


Examples of a charge-based and an induced electric ¬eld are shown in
Figure 3.1.
Note that the induced electric ¬eld in Figure 3.1(b) is directed so as to
drive an electric current that produces magnetic ¬‚ux that opposes the
change in ¬‚ux due to the changing magnetic ¬eld. In this case, the motion of
the magnet to the right means that the leftward magnetic ¬‚ux is decreasing,
so the induced current produces additional leftward magnetic ¬‚ux.
Here are a few rules of thumb that will help you visualize and sketch
the electric ¬elds produced by changing magnetic ¬elds:

 Induced electric ¬eld lines produced by changing magnetic ¬elds must
form complete loops.
 The net electric ¬eld at any point is the vector sum of all electric ¬elds
present at that point.
 Electric ¬eld lines can never cross, since that would indicate that the
¬eld points in two different directions at the same location.
In summary, the ˜ in Faraday™s law represents the induced electric ¬eld at
E
each point along path C, a boundary of the surface through which the
magnetic ¬‚ux is changing over time. The path may be through empty
space or through a physical material “ the induced electric ¬eld exists in
either case.
64 A student™s guide to Maxwell™s Equations

H
C °Þdl The line integral
To understand Faraday™s law, it is essential that you comprehend the
meaning of the line integral. This type of integral is common in physics
and engineering, and you have probably come across it before, perhaps
when confronted with a problem such as this: ¬nd the total mass of a wire
for which the density varies along its length. This problem serves as a
good review of line integrals.
Consider the variable-density wire shown in Figure 3.2(a). To deter-
mine the total mass of the wire, imagine dividing the wire into a series of
short segments over each of which the linear density k (mass per unit
length) is approximately constant, as shown in Figure 3.2(b). The mass of
each segment is the product of the linear density of that segment times the
segment length dxi, and the mass of the entire wire is the sum of the
segment masses.
For N segments, this is

X
N
Mass ¼ ki dxi : °3:1Þ
i¼1

Allowing the segment length to approach zero turns the summation of
the segment masses into a line integral:

ZL
Mass ¼ k°xÞ dx: °3:2Þ
0

This is the line integral of the scalar function k(x). To fully comprehend
the left side of Faraday™s law, you™ll have to understand how to extend
this concept to the path integral of a vector ¬eld, which you can read
about in the next section.

(a)
L
0 x


Density varies with x: » = » (x)
(b)
»1 » 2 » 3 »N

dx1 dx2 dx3 dxN

Figure 3.2 Line integral for a scalar function.
65
Faraday™s law

H
˜  d˜ The path integral of a vector ¬eld
Al
C
The line integral of a vector ¬eld around a closed path is called the
˜˜circulation™™ of the ¬eld. A good way to understand the meaning of this
operation is to consider the work done by a force as it moves an object
along a path.
As you may recall, work is done when an object is displaced under the
in¬‚uence of a force. If the force °˜ is constant and in the same direction

as the displacement °d˜ the amount of work (W) done by the force is
lÞ,
simply the product of the magnitudes of the force and the displacement:

W ¼ j˜ jd˜ °3:3Þ
Fj lj:
This situation is illustrated in Figure 3.3(a). In many cases, the dis-
placement is not in the same direction as the force, and it then becomes
necessary to determine the component of the force in the direction of the
displacement, as shown in Figure 3.3(b).
In this case, the amount of work done by the force is equal to the
component of the force in the direction of the displacement multiplied by
the amount of displacement. This is most easily signi¬ed using the dot
product notation described in Chapter 1:

W ¼ ˜  d˜ ¼ j˜ ˜ cos°hÞ; °3:4Þ
F l Fjjd lj

where h is the angle between the force and the displacement.
In the most general case, the force ˜ and the angle between the force
F
and the displacement may not be constant, which means that the pro-
jection of the force on each segment may be different (it is also possible
that the magnitude of the force may change along the path). The general
case is illustrated in Figure 3.4. Note that as the path meanders from the
starting point to the end, the component of the force in the direction of
the displacement varies considerably.


(a) (b)
F
F
u
dl
dl

Work = F ° dl = |F | |dl| cos u
Work = |F| |dl|

Figure 3.3 Object moving under the in¬‚uence of a force.
66 A student™s guide to Maxwell™s Equations


Force F

u8
dl 8
Start F
End u1 F
dl1

Path of object

dl8
Path divided into
u8
N segments
F
8
1 Component of F
N
2
in direction of dl8
3

Figure 3.4 Component of force along object path.




To ¬nd the work in this case, the path may be thought of as a series of
short segments over each of which the component of the force is constant.
The incremental work (dWi) done over each segment is simply the com-
ponent of the force along the path at that segment times the segment
length (dli) “ and that™s exactly what the dot product does. Thus,

dWi ¼ ˜  d˜ ; °3:5Þ
F li

and the work done along the entire path is then just the summation of the
incremental work done at each segment, which is

X X
N N
˜  d˜ :
W¼ dWi ¼ °3:6Þ
F li
i¼1 i¼1


As you™ve probably guessed, you can now allow the segment length to
shrink toward zero, converting the sum to an integral over the path:
Z
˜  d˜
W¼ °3:7Þ
F l:
P


Thus, the work in this case is the path integral of the vector ˜ over path
F
P. This integral is similar to the line integral you used to ¬nd the mass of
a variable-density wire, but in this case the integrand is the dot product
between two vectors rather than the scalar function k.
67
Faraday™s law


Although the force in this example is uniform, the same analysis
pertains to a vector ¬eld of force that varies in magnitude and direction
along the path. The integral on the right side of Equation 3.7 may be
de¬ned for any vector ¬eld ˜ and any path C. If the path is closed, this
A
integral represents the circulation of the vector ¬eld around that path:
I
Circulation  ˜  d˜ °3:8Þ
A l:
C

The circulation of the electric ¬eld is an important part of Faraday™s law,
as described in the next section.
68 A student™s guide to Maxwell™s Equations
H
˜  d˜ The electric ¬eld circulation
El
C
Since the ¬eld lines of induced electric ¬elds form closed loops, these
¬elds are capable of driving charged particles around continuous circuits.
Charge moving through a circuit is the very de¬nition of electric current,
so the induced electric ¬eld may act as a generator of electric current. It is
therefore understandable that the circulation of the electric ¬eld around a
circuit has come to be known as an ˜˜electromotive force™™:
I
˜  d˜
electromotive force °emf Þ ¼ °3:9Þ
E l:
C

Of course, the path integral of an electric ¬eld is not a force (which must
have SI units of newtons), but rather a force per unit charge integrated
over a distance (with units of newtons per coulomb times meters, which
are the same as volts). Nonetheless, the terminology is now standard, and
˜˜source of emf™™ is often applied to induced electric ¬elds as well as to
batteries and other sources of electrical energy.
So, exactly what is the circulation of the induced electric ¬eld around a
path? It is just the work done by the electric ¬eld in moving a unit charge
around that path, as you can see by substituting ˜ for ˜ in the circu-
F=q E
lation integral:
H
I I˜ ˜˜W
C F  dl

˜  d˜ ¼  dl ¼ ¼: °3:10Þ
El
Cq q q
C

Thus, the circulation of the induced electric ¬eld is the energy given to
each coulomb of charge as it moves around the circuit.
69
Faraday™s law
R
˜ ^ da
dt S B 
d
n The rate of change of ¬‚ux
The right side of the common form of Faraday™s law may look intimi-
dating at ¬rst glance, but a careful inspection of the terms reveals that the
largest portion of this expression is simply the magnetic ¬‚ux °UB Þ:
Z
˜  ^ da:
UB ¼ Bn
S


If you™re tempted to think that this quantity must be zero according to
Gauss™s law for magnetic ¬elds, look more carefully. The integral in this
expression is over any surface S, whereas the integral in Gauss™s law is
speci¬cally over a closed surface. The magnetic ¬‚ux (proportional to the
number of magnetic ¬eld lines) through an open surface may indeed be
nonzero “ it is only when the surface is closed that the number of mag-
netic ¬eld lines passing through the surface in one direction must equal
the number passing through in the other direction.
So the right side of this form of Faraday™s law involves the magnetic
¬‚ux through any surface S “ more speci¬cally, the rate of change with
time (d/dt) of that ¬‚ux. If you™re wondering how the magnetic ¬‚ux
through a surface might change, just look at the equation and ask
yourself what might vary with time in this expression. Here are three
possibilities, each of which is illustrated in Figure 3.5:

 The magnitude of ˜ might change: the strength of the magnetic ¬eld
B
may be increasing or decreasing, causing the number of ¬eld lines
penetrating the surface to change.
 The angle between ˜ and the surface normal might change: varying the
B
direction of either ˜ or the surface normal causes ˜  ^ to change.
B Bn
 The area of the surface might change: even if the magnitude of ˜ and
B
the direction of both ˜ and ^ remain the same, varying the area of
B n
surface S will change the value of the ¬‚ux through the surface.
Each of these changes, or a combination of them, causes the right side of
Faraday™s law to become nonzero. And since the left side of Faraday™s
law is the induced emf, you should now understand the relationship
between induced emf and changing magnetic ¬‚ux.
To connect the mathematical statement of Faraday™s law to physical
effects, consider the magnetic ¬elds and conducting loops shown in
Figure 3.5. As Faraday discovered, the mere presence of magnetic ¬‚ux
70 A student™s guide to Maxwell™s Equations


(a) (b) (c)
Loop of decreasing radius
Rotating loop
Magnet motion
B

N B

B
Induced current
Induced current
Induced current

Figure 3.5 Magnetic ¬‚ux and induced current.



through a circuit does not produce an electric current within that circuit.
Thus, holding a stationary magnet near a stationary conducting loop
induces no current (in this case, the magnetic ¬‚ux is not a function of
time, so its time derivative is zero and the induced emf must also be zero).
Of course, Faraday™s law tells you that changing the magnetic ¬‚ux
through a surface does induce an emf in any circuit that is a boundary to
that surface. So, moving a magnet toward or away from the loop, as in
Figure 3.5(a), causes the magnetic ¬‚ux through the surface bounded by
the loop to change, resulting in an induced emf around the circuit.4
In Figure 3.5(b), the change in magnetic ¬‚ux is produced not by
moving the magnet, but by rotating the loop. This changes the angle
between the magnetic ¬eld and the surface normal, which changes ˜  ^. In
Bn
Figure 3.5(c), the area enclosed by the loop is changing over time, which
changes the ¬‚ux through the surface. In each of these cases, you should
note that the magnitude of the induced emf does not depend on the total
amount of magnetic ¬‚ux through the loop “ it depends only on how fast
the ¬‚ux changes.
Before looking at some examples of how to use Faraday™s law to solve
problems, you should consider the direction of the induced electric ¬eld,
which is provided by Lenz™s law.




4
For simplicity, you can imagine a planar surface stretched across the loop, but Faraday™s
law holds for any surface bounded by the loop.
71
Faraday™s law


À Lenz™s law
There™s a great deal of physics wrapped up in the minus sign on the right
side of Faraday™s law, so it is ¬tting that it has a name: Lenz™s law. The
name comes from Heinrich Lenz, a German physicist who had an
important insight concerning the direction of the current induced by
changing magnetic ¬‚ux.
Lenz™s insight was this: currents induced by changing magnetic ¬‚ux
always ¬‚ow in the direction so as to oppose the change in ¬‚ux. That is, if
the magnetic ¬‚ux through the circuit is increasing, the induced current
produces its own magnetic ¬‚ux in the opposite direction to offset the
increase. This situation is shown in Figure 3.6(a), in which the magnet
is moving toward the loop. As the leftward ¬‚ux due to the magnet
increases, the induced current ¬‚ows in the direction shown, which
produces rightward magnetic ¬‚ux that opposes the increased ¬‚ux from
the magnet.
The alternative situation is shown in Figure 3.6(b), in which the magnet
is moving away from the loop and the leftward ¬‚ux through the circuit is
decreasing. In this case, the induced current ¬‚ows in the opposite direc-
tion, contributing leftward ¬‚ux to make up for the decreasing ¬‚ux from
the magnet.
It is important for you to understand that changing magnetic ¬‚ux
induces an electric ¬eld whether or not a conducting path exists in which
a current may ¬‚ow. Thus, Lenz™s law tells you the direction of the cir-
culation of the induced electric ¬eld around a speci¬ed path even if no
conduction current actually ¬‚ows along that path.


(a) (b)

Leftward flux
Leftward flux
decreases as
increases as
magnet recedes
magnet approaches
B
B


N S
N S

Magnet motion Magnet motion

Current produces
Current produces
more leftward flux
rightward flux

Figure 3.6 Direction of induced current.
72 A student™s guide to Maxwell™s Equations
H R
˜  d˜ ¼ À d ˜  ^ da
CE l Bn Applying Faraday™s
S
dt law (integral form)

The following examples show you how to use Faraday™s law to solve
problems involving changing magnetic ¬‚ux and induced emf.

Example 3.1: Given an expression for the magnetic ¬eld as a function of
time, determine the emf induced in a loop of speci¬ed size.
Problem: For a magnetic ¬eld given by

˜ tÞ ¼ B0 t y
^:
B°y; z
t 0 y0
Find the emf induced in a square loop of side L lying in the xy-plane
with one corner at the origin. Also, ¬nd the direction of current ¬‚ow in
the loop.
Solution: Using Faraday™s ¬‚ux rule,
Z
d ˜  ^ da:
emf ¼ À Bn
dt S

For a loop in the xy-plane, ^ ¼ ^ and da = dx dy, so
nz

Z Z
L L
d ty
emf ¼ À ^  ^ dx dy,
B0 zz
dt t 0 y0
y¼0 x¼0


and
 !   3!
Z L
d ty d tL
emf ¼ À dy ¼ À :
L B0 B0
t0 2y0
dt t 0 y0 dt
y¼0

Taking the time derivative gives

L3
emf ¼ ÀB0 :
2t0 y0

Since upward magnetic ¬‚ux is increasing with time, the current will
¬‚ow in a direction that produces ¬‚ux in the downward °À^Þ direction.
z
This means the current will ¬‚ow in the clockwise direction as seen
from above.
73
Faraday™s law


Example 3.2: Given an expression for the change in orientation of a
conducting loop in a ¬xed magnetic ¬eld, ¬nd the emf induced in the loop.
Problem: A circular loop of radius r0 rotates with angular speed x in a
¬xed magnetic ¬eld as shown in the ¬gure.

B
v




(a) Find an expression for the emf induced in the loop.
(b) If the magnitude of the magnetic ¬eld is 25 lT, the radius of the loop
is 1 cm, the resistance of the loop is 25 W, and the rotation rate x is 3
rad/s, what is the maximum current in the loop?
Solution: (a) By Faraday™s ¬‚ux rule, the emf is
Z
d ˜  ^ da
emf ¼ À Bn
dt S
Since the magnetic ¬eld and the area of the loop are constant, this becomes
Z Z
 d

°B  ^Þ da ¼ À ˜ °cos hÞ da:
emf ¼ À n B
S dt dt
S

Using h = xt, this is
Z Z
 d 
˜ °cos xtÞ da ¼ À˜ d °cos xtÞ da:
emf ¼ À B B
dt dt
S S

Taking the time derivative and performing the integration gives

emf ¼ ˜x°sin xtÞ°pr0 Þ:
2
B

(b) By Ohm™s law, the current is the emf divided by the resistance of the
circuit, which is

emf ˜x°sin xtÞ°pr0 Þ2
B
I¼ ¼ :
R R
For maximum current, sin(xt) = 1, so the current is
°25 · 10À6 Þ°3Þ½p°0:012 ފ
¼ 9:4 · 10À10 A:

25
74 A student™s guide to Maxwell™s Equations


Example 3.3: Given an expression for the change in size of a conducting
loop in a ¬xed magnetic ¬eld, ¬nd the emf induced in the loop.
Problem: A circular loop lying perpendicular to a ¬xed magnetic ¬eld
decreases in size over time. If the radius of the loop is given by r(t) =
r0(1Àt/t0), ¬nd the emf induced in the loop.

Solution: Since the loop is perpendicular to the magnetic ¬eld, the loop
normal is parallel to ˜ and Faraday™s ¬‚ux rule is
B,
Z Z
 d  d
d ˜  ^ da ¼ À˜ da ¼ À˜ °pr 2 Þ:
emf ¼ À Bn B B
dt S dt S dt
Inserting r(t) and taking the time derivative gives
 !   !
 d  2
t2 1
t
emf ¼ À˜ ¼ À˜ pr0 °2Þ 1 À
pr0 1 À À ;
2
B B
dt t0 t0 t0
or
  2 
2˜pr0
B t
emf ¼ 1À :
t0 t0
75
Faraday™s law


3.2 The differential form of Faraday™s law
The differential form of Faraday™s law is generally written as

B
˜E
r·˜ ¼ À Faraday™s law:
@t
The left side of this equation is a mathematical description of the curl of
the electric ¬eld “ the tendency of the ¬eld lines to circulate around a
point. The right side represents the rate of change of the magnetic ¬eld
over time.
The curl of the electric ¬eld is discussed in detail in the following
section. For now, make sure you grasp the main idea of Faraday™s law in
differential form:


A circulating electric ¬eld is produced by a magnetic ¬eld that changes
with time.

To help you understand the meaning of each symbol in the differential
form of Faraday™s law, here™s an expanded view:

Reminder that the Reminder that the electric
del operator is a vector field is a vector


*B

—E = “
The rate of change
of the magnetic field
*t with time

The differential The electric
operator called field in V/m
“del” or “nabla”
The cross-product turns
the del operator into the
curl
76 A student™s guide to Maxwell™s Equations

˜
r· Del cross “ the curl
The curl of a vector ¬eld is a measure of the ¬eld™s tendency to circulate
about a point “ much like the divergence is a measure of the tendency of
the ¬eld to ¬‚ow away from a point. Once again we have Maxwell to
thank for the terminology; he settled on ˜˜curl™™ after considering several
alternatives, including ˜˜turn™™ and ˜˜twirl™™ (which he thought was some-
what racy).
Just as the divergence is found by considering the ¬‚ux through an
in¬nitesimal surface surrounding the point of interest, the curl at a spe-
ci¬ed point may be found by considering the circulation per unit area
over an in¬nitesimal path around that point. The mathematical de¬nition
of the curl of a vector ¬eld ˜ is
A
I
1
˜A ˜  d˜
curl°˜ ¼ r· ˜  lim °3:11Þ
AÞ A l;
DS!0 DS
C

where C is a path around the point of interest and DS is the surface area
enclosed by that path. In this de¬nition, the direction of the curl is the
normal direction of the surface for which the circulation is a maximum.
This expression is useful in de¬ning the curl, but it doesn™t offer much
help in actually calculating the curl of a speci¬ed ¬eld. You™ll ¬nd an
alternative expression for curl later in this section, but ¬rst you should
consider the vector ¬elds shown in Figure 3.7.
To ¬nd the locations of large curl in each of these ¬elds, imagine that
the ¬eld lines represent the ¬‚ow lines of a ¬‚uid. Then look for points at

(a) (b) (c)


1
7
6

5
2

4


3



Figure 3.7 Vector ¬elds with various values of curl.
77
Faraday™s law


which the ¬‚ow vectors on one side of the point are signi¬cantly different
(in magnitude, direction, or both) from the ¬‚ow vectors on the opposite
side of the point.
To aid this thought experiment, imagine holding a tiny paddlewheel at
each point in the ¬‚ow. If the ¬‚ow would cause the paddlewheel to rotate,
the center of the wheel marks a point of nonzero curl. The direction of the
curl is along the axis of the paddlewheel (as a vector, curl must have both
magnitude and direction). By convention, the positive-curl direction is
determined by the right-hand rule: if you curl the ¬ngers of your right hand
along the circulation, your thumb points in the direction of positive curl.
Using the paddlewheel test, you can see that points 1, 2, and 3 in
Figure 3.7(a) and points 4 and 5 in Figure 3.7(b) are high-curl locations.
The uniform ¬‚ow around point 6 in Figure 3.7(b) and the diverging ¬‚ow
lines around point 7 in Figure 3.7(b) would not cause a tiny paddlewheel
to rotate, meaning that these are points of low or zero curl.
To make this quantitative, you can use the differential form of the curl
˜
or ˜˜del cross™™ °r·Þ operator in Cartesian coordinates:
 
@ ^@ ^@
˜A · °^ x þ ^ y þ ^ z Þ:
r· ˜ ¼ ^ þ j þ k °3:12Þ
i iA jA kA
@x @y @z

The vector cross-product may be written as a determinant:
 
^ ^ ^
i j k

r· ˜ ¼  @x @y @z ;
˜A @@@ °3:13Þ
 
 Ax A y Az 

which expands to
     
@Az @Ay ^ @Ax @Az ^ @Ay @Ax ^
˜A
r· ˜ ¼ À iþ À jþ À °3:14Þ
k:
@y @z @z @x @x @y

Note that each component of the curl of ˜ indicates the tendency of the
A
¬eld to rotate in one of the coordinate planes. If the curl of the ¬eld at a
point has a large x-component, it means that the ¬eld has signi¬cant
circulation about that point in the y“z plane. The overall direction of the
curl represents the axis about which the rotation is greatest, with the
sense of the rotation given by the right-hand rule.
If you™re wondering how the terms in this equation measure rotation,
consider the vector ¬elds shown in Figure 3.8. Look ¬rst at the ¬eld in
78 A student™s guide to Maxwell™s Equations


(a) (b)
z
z




y
y



x x

Figure 3.8 Effect of @Ay/@z and @Az/@Y on value of curl.
Figure 3.8(a) and the x-component of the curl in the equation: this term
involves the change in Az with y and the change in Ay with z. Proceeding
along the y-axis from the left side of the point of interest to the right, Az is
clearly increasing (it is negative on the left side of the point of interest and
positive on the right side), so the term @Az/@y must be positive. Looking
now at Ay, you can see that it is positive below the point of interest and
negative above, so it is decreasing along the z axis. Thus, @Ay/@z is
negative, which means that it increases the value of the curl when it is
subtracted from @Az/@y. Thus the curl has a large value at the point of
interest, as expected in light of the circulation of ˜ about this point.
A
The situation in Figure 3.8(b) is quite different. In this case, both @Ay/@z
and @Az/@y are positive, and subtracting @Ay/@z from @Az/@y gives a small
result. The value of the x-component of the curl is therefore small in this
case. Vector ¬elds with zero curl at all points are called ˜˜irrotational.™™
Here are expressions for the curl in cylindrical and spherical coordinates:
   
1 @Az @A™ @Ar @Az
r·˜ ^þ ^
À À ™
A r
r @™ @z @z @r
  °3:15Þ
1 @°rA™ Þ @Ar
þ À ^ °cylindricalÞ;
z
@r @™
r

   
1 @°A™ sin hÞ @Ah 1 @Ar @°rA™ Þ ^
1
r ·˜  ^þ
À À h
A r
r sin h @h @™ r sin h @™ @r
  °3:16Þ
1 @°rAh Þ @Ar
^
þ À ™ °sphericalÞ:
@r @h
r
79
Faraday™s law

˜E
r · ˜ The curl of the electric ¬eld
Since charge-based electric ¬elds diverge away from points of positive
charge and converge toward points of negative charge, such ¬elds cannot
circulate back on themselves. You can understand that by looking at the
¬eld lines for the electric dipole shown in Figure 3.9(a). Imagine moving
along a closed path that follows one of the electric ¬eld lines diverging
from the positive charge, such as the dashed line shown in the ¬gure. To
close the loop and return to the positive charge, you™ll have to move
˜˜upstream™™ against the electric ¬eld for a portion of the path. For that
segment, ˜  d˜ is negative, and the contribution from this part of the
El
path subtracts from the positive value of ˜  d˜ for the portion of the path
El
in which ˜ and d˜ are in the same direction. Once you™ve gone all the way
E l
around the loop, the integration of ˜  d˜ yields exactly zero.
El
Thus, the ¬eld of the electric dipole, like all electrostatic ¬elds, has
no curl.
Electric ¬elds induced by changing magnetic ¬elds are very different, as
you can see in Figure 3.9(b). Wherever a changing magnetic ¬eld exists, a
circulating electric ¬eld is induced. Unlike charge-based electric ¬elds,
induced ¬elds have no origination or termination points “ they are
continuous and circulate back on themselves. Integrating ˜  d˜ around
El
any boundary path for the surface through which ˜ is changing produces
B
a nonzero result, which means that induced electric ¬elds have curl. The
faster ˜ changes, the larger the magnitude of the curl of the induced
B
electric ¬eld.



‚B/‚t
(a) (b)
E




1 2 E E
E




Figure 3.9 Closed paths in charge-based and induced electric ¬elds.
80 A student™s guide to Maxwell™s Equations

˜E ˜
r· ˜ ¼ À @ B Applying Faraday™s law (differential form)
@t
The differential form of Faraday™s law is very useful in deriving the
electromagnetic wave equation, which you can read about in Chapter 5.
You may also encounter two types of problems that can be solved using
this equation. In one type, you™re provided with an expression for the
magnetic ¬eld as a function of time and asked to ¬nd the curl of the
induced electric ¬eld. In the other type, you™re given an expression for the
induced vector electric ¬eld and asked to determine the time rate of
change of the magnetic ¬eld. Here are two examples of such problems.

Example 3.4: Given an expression for the magnetic ¬eld as a function of
time, ¬nd the curl of the electric ¬eld.
Problem: The magnetic ¬eld in a certain region is given by the expression
˜ ¼ B0 cos°kz À xtÞ ^
B°tÞ j.
(a) Find the curl of the induced electric ¬eld at that location.
(b) If the Ez is known to be zero, ¬nd Ex.
Solution: (a) By Faraday™s law, the curl of the electric ¬eld is the negative
of the derivative of the vector magnetic ¬eld with respect to time. Thus,
@˜ @ ½B0 cos°kz À xtފ ^
B j
˜E
r· ˜ ¼ À ¼À ;
@t @t
or
˜E
r· ˜ ¼ ÀxB0 sin°kz À xtÞ ^
j:
(b) Writing out the components of the curl gives
     
@Ez @Ey ^ @Ex @Ez ^ @Ey @Ex ^
k ¼ ÀxB0 sin°kz À xtÞ^
À iþ À jþ À j:
@y @z @z @x @x @y

Equating the ^ components and setting Ez to zero gives
j
 
@Ex
¼ ÀxB0 sin°kz À xtÞ:
@z
Integrating over z gives
Z
x
Ex ¼ ÀxB0 sin°kz À xtÞdz ¼ B0 cos°kz À xtÞ,
k
to within a constant of integration.
81
Faraday™s law


Example 3.5: Given an expression for the induced electric ¬eld, ¬nd the
time rate of change of the magnetic ¬eld.
Problem: Find the rate of change with time of the magnetic ¬eld at a
location at which the induced electric ¬eld is given by
"   2 #
 2
2
z^ x^ y^
˜ y; zÞ ¼ E0 iþ jþ k:
E°x;
z0 x0 y0

Solution: Faraday™s law tells you that the curl of the induced electric ¬eld
is equal to the negative of the time rate of change of the magnetic ¬eld.
Thus

B ˜ E;
¼ Àr · ˜
@t
which in this case gives
     
@˜ @Ez @Ey ^ @Ex @Ez ^ @Ey @Ex ^
B
¼À À iÀ À jÀ À k,
@t @y @z @z @x @x @y


   !
@˜ 2y ^ 2z ^ 2x
B
iþ 2 jþ 2 ^ :
¼ ÀE0 k
@t y2 z0 x0
0



Problems
You can exercise your understanding of Faraday™s law on the following
problems. Full solutions are available on the book™s website.
3.1 Find the emf induced in a square loop with sides of length a lying in
the yz-plane in a region in which the magnetic ¬eld changes over time
as ˜ ¼ B0 eÀ5t=t0^
B°tÞ i.
3.2 A square conducting loop with sides of length L rotates so that the
angle between the normal to the plane of the loop and a ¬xed
magnetic ¬eld ˜ varies as h(t) ¼ h0(t/t0); ¬nd the emf induced in the
B
loop.
3.3 A conducting bar descends with speed v down conducting rails in the
presence of a constant, uniform magnetic ¬eld pointing into the page,
as shown in the ¬gure.

(a) Write an expression for the emf induced in the loop.
(b) Determine the direction of current ¬‚ow in the loop.
82 A student™s guide to Maxwell™s Equations




3.4 A square loop of side a moves with speed v into a region in which a
magnetic ¬eld of magnitude B0 exists perpendicular to the plane of the
loop, as shown in the ¬gure. Make a plot of the emf induced in the loop
as it enters, moves through, and exits the region of the magnetic ¬eld.

a
B
a


v

2a
0


3.5 A circular loop of wire of radius 20 cm and resistance of 12 W surrounds
a 5-turn solenoid of length 38 cm and radius 10 cm, as shown in the
¬gure. If the current in the solenoid increases linearly from 80 to
300 mA in 2 s, what is the maximum current induced in the loop?




3.6 A 125-turn rectangular coil of wire with sides of 25 and 40 cm rotates
about a horizontal axis in a vertical magnetic ¬eld of 3.5 mT. How
fast must this coil rotate for the induced emf to reach 5V?
3.7 The current in a long solenoid varies as I(t) = I0 sin(xt). Use
Faraday™s law to ¬nd the induced electric ¬eld as a function of r
both inside and outside the solenoid, where r is the distance from
the axis of the solenoid.
3.8 The current in a long, straight wire decreases as I(t) ¼ I0eÀt/s. Find the
induced emf in a square loop of wire of side s lying in the plane of the
current-carrying wire at a distance d, as shown in the ¬gure.

I(t)
s
s


d
4
The Ampere“Maxwell law




For thousands of years, the only known sources of magnetic ¬elds were
certain iron ores and other materials that had been accidentally or
deliberately magnetized. Then in 1820, French physicist Andre-Marie
Ampere heard that in Denmark Hans Christian Oersted had de¬‚ected a
compass needle by passing an electric current nearby, and within one
week Ampere had begun quantifying the relationship between electric
currents and magnetic ¬elds.
˜˜Ampere™s law™™ relating a steady electric current to a circulating
magnetic ¬eld was well known by the time James Clerk Maxwell began
his work in the ¬eld in the 1850s. However, Ampere™s law was known to
apply only to static situations involving steady currents. It was Maxwell™s
addition of another source term “ a changing electric ¬‚ux “ that extended
the applicability of Ampere™s law to time-dependent conditions. More
importantly, it was the presence of this term in the equation now called
the Ampere“Maxwell law that allowed Maxwell to discern the electro-
magnetic nature of light and to develop a comprehensive theory of
electromagnetism.



4.1 The integral form of the Ampere“Maxwell law
The integral form of the Ampere“Maxwell law is generally written as
 
I Z

˜  d˜ ¼ l0 Ienc þ e0 E  ^ da The Ampere“Maxwell law:
Bl n
dt S
C

The left side of this equation is a mathematical description of the
circulation of the magnetic ¬eld around a closed path C. The right side



83
84 A student™s guide to Maxwell™s Equations


includes two sources for the magnetic ¬eld; a steady conduction current
and a changing electric ¬‚ux through any surface S bounded by path C.
In this chapter, you™ll ¬nd a discussion of the circulation of the mag-
netic ¬eld, a description of how to determine which current to include in
calculating ˜ and an explanation of why the changing electric ¬‚ux is
B,
called the ˜˜displacement current.™™ There are also examples of how to use
the Ampere“Maxwell law to solve problems involving currents and
magnetic ¬elds. As always, you should begin by reviewing the main idea
of the Ampere“Maxwell law:


An electric current or a changing electric ¬‚ux through a surface
produces a circulating magnetic ¬eld around any path that bounds
that surface.

In other words, a magnetic ¬eld is produced along a path if any current is
enclosed by the path or if the electric ¬‚ux through any surface bounded
by the path changes over time.
It is important that you understand that the path may be real or purely
imaginary “ the magnetic ¬eld is produced whether the path exists or not.
Here™s an expanded view of the Ampere“Maxwell law:

Dot product tells you to find the
Reminder that the
part of B parallel to dl (along path C)
magnetic field
is a vector
An incremental The electric The rate of change
segment of path C current in amperes with time




« «
d
B dl = ˆ
E n da
Ienc +
0 0
dt S
C Reminder that
The magnetic only the enclosed
field in teslas current contributes
The electric
Tells you to sum up the contributions The magnetic The electric flux through
from each portion of closed path C in permeability permittivity a surface
a direction given by the right-hand rule of free space of free space bounded by C


Of what use is the Ampere“Maxwell law? You can use it to determine the
circulation of the magnetic ¬eld if you™re given information about the
enclosed current or the change in electric ¬‚ux. Furthermore, in highly
symmetric situations, you may be able to extract ˜ from the dot product
B
and the integral and determine the magnitude of the magnetic ¬eld.
85
The Ampere“Maxwell law

H
˜  d˜ The magnetic ¬eld circulation
Bl
C
Spend a few minutes moving a magnetic compass around a long, straight
wire carrying a steady current, and here™s what you™re likely to ¬nd: the
current in the wire produces a magnetic ¬eld that circles around the wire
and gets weaker as you get farther from the wire.
With slightly more sophisticated equipment and an in¬nitely long wire,
you™d ¬nd that the magnetic ¬eld strength decreases precisely as 1/r,
where r is the distance from the wire. So if you moved your measuring
device in a way that kept the distance to the wire constant, say by circling
around the wire as shown in Figure 4.1, the strength of the magnetic ¬eld
wouldn™t change. If you kept track of the direction of the magnetic ¬eld
as you circled around the wire, you™d ¬nd that it always pointed along
your path, perpendicular to an imaginary line from the wire to your
location.
If you followed a random path around the wire getting closer and
farther from the wire as you went around, you™d ¬nd the magnetic ¬eld
getting stronger and weaker, and no longer pointing along your path.
Now imagine keeping track of the magnitude and direction of the
magnetic ¬eld as you move around the wire in tiny increments. If, at each
incremental step, you found the component of the magnetic ¬eld ˜ along
B
that portion of your path d˜ you™d be able to ¬nd ˜  d˜ Keeping track of
l, B l.
each value of ˜  d˜ and then summing the results over your entire path,
B l
you™d have a discrete version of the left side of the Ampere“Maxwell law.
Making this process continuous by letting the path increment shrink




Magnetic field
strength depends
Field strength only on distance
Weaker field
meter and magnetic from wire
NS
compass
Stronger field
Long straight wire carrying steady current

I
Magnetic field direction
Magnetic field is along circular path
strength is constant
along this path

Figure 4.1 Exploring the magnetic ¬eld around a current-carrying wire.
86 A student™s guide to Maxwell™s Equations


toward zero would then give you the circulation of the magnetic ¬eld:
I
Magnetic field circulation ¼ ˜  d˜ °4:1Þ
B l:
C

The Ampere“Maxwell law tells you that this quantity is proportional to
the enclosed current and rate of change of electric ¬‚ux through any
surface bounded by your path of integration (C). But if you hope to use
this law to determine the value of the magnetic ¬eld, you™ll need to dig ˜
B
out of the dot product and out of the integral. That means you™ll have to
choose your path around the wire very carefully “ just as you had to
choose a ˜˜special Gaussian surface™™ to extract the electric ¬eld from
Gauss™s law, you™ll need a ˜˜special Amperian loop™™ to determine the
magnetic ¬eld.
You™ll ¬nd examples of how to do that after the next three sections,
which discuss the terms on the right side of the Ampere“Maxwell law.
87
The Ampere“Maxwell law


l0 The permeability of free space
The constant of proportionality between the magnetic circulation on the
left side of the Ampere“Maxwell law and the enclosed current and rate of
¬‚ux change on the right side is l0, the permeability of free space. Just as
the electric permittivity characterizes the response of a dielectric to an
applied electric ¬eld, the magnetic permeability determines a material™s
response to an applied magnetic ¬eld. The permeability in the Ampere“
Maxwell law is that of free space (or ˜˜vacuum permeability™™), which is
why it carries the subscript zero.
The value of the vacuum permeability in SI units is exactly 4p · 10À7
volt-seconds per ampere-meter (Vs/Am); the units are sometimes given as
newtons per square ampere (N/A2) or the fundamental units of (m kg/C2).
Therefore, when you use the Ampere“Maxwell law, remember to multiply
both terms on the right side by
l0 ¼ 4p · 10À7 Vs=Am:
As in the case of electric permittivity in Gauss™s law for electric ¬elds, the
presence of this quantity does not mean that the Ampere“Maxwell law
applies only to sources and ¬elds in a vacuum. This form of the Ampere“
Maxwell law is general, so long as you consider all currents (bound as well
as free). In the Appendix, you™ll ¬nd a version of this law that™s more useful
when dealing with currents and ¬elds in magnetic materials.
One interesting difference between the effect of dielectrics on electric ¬elds
and the effect of magnetic substances on magnetic ¬elds is that the magnetic
¬eld is actually stronger than the applied ¬eld within many magnetic
materials. The reason for this is that these materials become magnetized
when exposed to an external magnetic ¬eld, and the induced magnetic ¬eld is
in the same direction as the applied ¬eld, as shown in Figure 4.2.
The permeability of a magnetic material is often expressed as the
relative permeability, which is the factor by which the material™s per-
meability exceeds that of free space:

Relative permeability lr ¼ l=l0 : °4:2Þ

Materials are classi¬ed as diamagnetic, paramagnetic, or ferromagnetic
on the basis of relative permeability. Diamagnetic materials have lr
slightly less than 1.0 because the induced ¬eld weakly opposes the applied
¬eld. Examples of diamagnetic materials include gold and silver, which
have lr of approximately 0.99997. The induced ¬eld within paramagnetic
88 A student™s guide to Maxwell™s Equations




Applied magnetic
field produced by
current I




Magnetic dipole
I I
moments align with
applied field



Figure 4.2 Effect of magnetic core on ¬eld inside solenoid.


materials weakly reinforces the applied ¬eld, so these materials have lr
slightly greater than 1.0. One example of a paramagnetic material is
aluminum with lr of 1.00002.
The situation is more complex for ferromagnetic materials, for which
the permeability depends on the applied magnetic ¬eld. Typical max-
imum values of permeability range from several hundred for nickel and
cobalt to over 5000 for reasonably pure iron.
As you may recall, the inductance of a long solenoid is given by the
expression
lN 2 A
L¼ ; °4:3Þ
˜
where l is the magnetic permeability of the material within the solenoid,
N is the number of turns, A is the cross-sectional area, and ˜ is the length
of the coil. As this expression makes clear, adding an iron core to a
solenoid may increase the inductance by a factor of 5000 or more.
Like electrical permittivity, the magnetic permeability of any medium
is a fundamental parameter in the determination of the speed with which
an electromagnetic wave propagates through that medium. This makes it
possible to determine the speed of light in a vacuum simply by measuring
l0 and e0 using an inductor and a capacitor; an experiment for which, to
paraphrase Maxwell, the only use of light is to see the instruments.
89
The Ampere“Maxwell law


Ienc The enclosed electric current
Although the concept of ˜˜enclosed current™™ sounds simple, the question
of exactly which current to include on the right side of the Ampere“
Maxwell law requires careful consideration.
It should be clear from the ¬rst section of this chapter that the
˜˜enclosing™™ is done by the path C around which the magnetic ¬eld is
integrated (if you™re having trouble imagining a path enclosing anything,
perhaps ˜˜encircling™™ is a better word). However, consider for a moment
the paths and currents shown in Figure 4.3; which of the currents are
enclosed by paths C1, C2, and C3, and which are not?
The easiest way to answer that question is to imagine a membrane
stretched across the path, as shown in Figure 4.4. The enclosed current is
then just the net current that penetrates the membrane.
The reason for saying ˜˜net™™ current is that the direction of the current
relative to the direction of integration must be considered. By convention,
the right-hand rule determines whether a current is counted as positive or
negative: if you wrap the ¬ngers of your right hand around the path in the



(a) (b) (c)




I2 C2 I3
C1
I1
C3

Figure 4.3 Currents enclosed (and not enclosed) by paths.



(a) (b) (c)

C3
C2
C1



I2 I3
I1

Figure 4.4 Membranes stretched across paths.
90 A student™s guide to Maxwell™s Equations


(b) (c)
(a)




I2
C1
I1 C2 C3
I3

Figure 4.5 Alternative surfaces with boundaries C1, C2, and C3.


direction of integration, your thumb points in the direction of positive
current. Thus, the enclosed current in Figure 4.4(a) is þI1 if the inte-
gration around path C1 is performed in the direction indicated; it would
be ÀI1 if the integration were performed in the opposite direction.
Using the membrane approach and right-hand rule, you should be able
to see that the enclosed current is zero in both Figure 4.4(b) and 4.4(c).
No net current is enclosed in Figure 4.4(b) , since the sum of the currents is
I2 þ ÀI2 = 0, and no current penetrates the membrane in either direction
in Figure 4.4(c).
An important concept for you to understand is that the enclosed
current is exactly the same irrespective of the shape of the surface you
choose, provided that the path of integration is a boundary (edge) of that
surface. The surfaces shown in Figure 4.4 are the simplest, but you could
equally well have chosen the surfaces shown in Figure 4.5, and the
enclosed currents would be exactly the same.
Notice that in Figure 4.5(a) current I1 penetrates the surface at only
one point, so the enclosed current is þI1, just as it was for the ¬‚at
membrane of Figure 4.4(a). In Figure 4.5(b), current I2 does not penetrate
the ˜˜stocking cap™™ surface anywhere, so the enclosed current is zero, as it
was for the ¬‚at membrane of Figure 4.4(b). The surface in Figure 4.5(c) is
penetrated twice by current I3, once in the positive direction and once in
the negative direction, so the net current penetrating the surface remains
zero, as it was in Figure 4.4(c) (for which the current missed the mem-
brane entirely).
Selection of alternate surfaces and determining the enclosed current is
more than just an intellectual diversion. The need for the changing-¬‚ux
term that Maxwell added to Ampere™s law can be made clear through just
such an exercise, as you can see in the next section.
91
The Ampere“Maxwell law

R
˜ ^ da
dt S E 
d
n The rate of change of ¬‚ux


This term is the electric ¬‚ux analog of the changing magnetic ¬‚ux term in
Faraday™s law, which you can read about in Chapter 3. In that case, a
changing magnetic ¬‚ux through any surface was found to induce a cir-
culating electric ¬eld along a boundary path for that surface.
Purely by symmetry, you might suspect that a changing electric ¬‚ux
through a surface will induce a circulating magnetic ¬eld around a
boundary of that surface. After all, magnetic ¬elds are known to circulate “
Ampere™s law says that any electric current produces just such a circulating
magnetic ¬eld. So how is it that several decades went by before anyone saw
¬t to write an ˜˜electric induction™™ law to go along with Faraday™s law of
magnetic induction?
For one thing, the magnetic ¬elds induced by changing electric ¬‚ux are
extremely weak and are therefore very dif¬cult to measure, so in the
nineteenth century there was no experimental evidence on which to base
such a law. In addition, symmetry is not always a reliable predictor
between electricity and magnetism; the universe is rife with individual

<<

. 3
( 5)



>>