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

FÞ

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

F˜

˜ 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

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 Þ

¼ 9:4 · 10À10 A:

I¼

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 xtÞ ^

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˜

˜ 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