158 11. FIELDS
theory give absurd results. And so there was not much for a cosmologist to do but assume the
cosmological constant to be zero, unless new evidence were to arise that suggests a different
value.
is all changed in the late 1990s when observations indicated that the expansion of the
universe is accelerating, rather than decelerating. is observed effect is well fit by Einstein’s
cosmological constant.
e term “dark energy” is more popular than “cosmological constant” to describe this
observed acceleration. We have no coherent theoretical underpinning for this effect on the ex-
pansion of the universe, and so the “dark” part of the term emphasizes the mystery. And it
suggests a linguistic kinship with the not-quite-as, but still very mysterious dark matter. Still,
its historical origins in Einstein’s cosmological constant are preserved in the symbol ƒ, which
cosmologists use to represent the observed effect of this mysterious property of the universe.
11.3 THE ELECTRIC AND MAGNETIC FIELDS
e phenomena of electricity and magnetism have been known since antiquity. Magnetic rocks
occur naturally, and the effects of so-called static electricity are easily observed. But it was not until
the late 19th century that a complete understanding developed of the wide variety of electrical
and magnetic effects. e culmination of these insights is the first true field theory, formalized
in the 1860s and 1870s by James Clerk Maxwell, building on the work of many others.
Maxwell’s field theory explains all known electrical and magnetic phenomena with four
equations—known as Maxwell’s equations. We will neither solve nor use Maxwell’s equations
in this book! But even absent an understanding of the mathematics, there is much one can learn
simply by looking at them; they are the four Equations (11.14)–(11.17):
r ¨
E
E D
0
(11.14)
r ¨
E
B D 0 (11.15)
r ˆ
E
E D ´
B
E
B
Bt
(11.16)
r ˆ
E
B D
0
E
J
C
0
0
B
E
E
Bt
: (11.17)
Each of the four Maxwell equations refers to one—or both—of the electric field (symbol-
ized by
E
E) and the magnetic field (
E
B). ere are some other symbols as well.
• e symbols
0
and
0
are constants that, roughly speaking, scale the measured strengths
of electricity and magnetism, respectively.
• e symbol (called charge density) is a measure of the concentration of electric charge at
any given point in space.
11.3. THE ELECTRIC AND MAGNETIC FIELDS 159
• e symbol
E
J (called current density) represents the rate and direction of flow of charge
through any give region of space.
• e symbol t represents time.
All of the other symbols in Equations (11.14)–(11.17) are purely mathematical in nature, and
do not refer to any physical phenomena per se.
ere are some other things to note. First, the electric and magnetic fields (as well as
current density) are given symbols in boldface type, topped by an arrow. is is to signify that
they are vector quantities—their directions matter as well as their strengths. And so
E
E and
E
B
represent vector fields.
In Maxwell’s theory, these fields permeate all of space; in fact, we consider them to be
part of the very properties of space itself. Accordingly, every point in space has a particular value—
consisting of both a magnitude and a direction—of both
E
E and
E
B. How does one then determine
the values of
E
E and
E
B for some region of space? Solve Equations (11.14)–(11.17)!
We will not solve Maxwell’s equations here, but we will consider further what they mean.
But even so, this is clearly only part of the story. Maxwell’s equations allow one to calculate the
electric and magnetic fields at any point in space. But so what? We also need to say what these
fields do. What are the physical consequences if the electric and magnetic fields have one value
instead of another?
ese questions bring to the fore one of the hallmarks of any field theory. Rather than
describing directly how one thing affects another, a field theory breaks the interaction into two
separate questions.
1. What are the physical causes of the field? What is it that affects the value of the field at
some point in space, and how do we calculate it?
2. What are the physical consequences of the field? If there is a field at some point in space,
what happens?
What makes a field theory such as Maxwell’s equations complex and subtle is that the an-
swer to each question is affected by the answer to the other. For electricity and magnetism,
Maxwell’s equations are the answer to the first question. We will consider the answer to the
second question in Section 11.3.3, and then further consider the meaning of Maxwell’s equa-
tions in Section 11.3.1. Finally, in Section 12.2, we will consider an important and revolutionary
consequence—unexpected at the time—of Maxwell’s equations.
11.3.1 WHAT THE FIELDS DO
What are the direct physical consequences of electric and magnetic fields in space? e answer
lies in the mathematical description of what is known as the Lorentz force,
E
F
L
:
E
F
L
D q
E
E C q
Ev ˆ
E
B
: (11.18)
160 11. FIELDS
e Lorentz force is exerted on any particle with electric charge, q, located in the presence of
E
E
or
E
B, or both. And so in rough terms, the answer is this: electric and magnetic fields in space
exert forces on any electric charges located there. ese electric forces and magnetic forces will then
alter the motions of those charged particles.
e right-hand side of Equation (11.18) has two parts that are added together to calculate
the total effect on the charged particle:
• q
E
E, the electric force and
• q.Ev ˆ
E
B/, the magnetic force.
Both of these forces, considered individually, are proportional to the electric charge in question.
So particles with a greater electric charge experience a greater force, all else being equal.
But the electric and magnetic forces act in quite different ways. e electric force is sim-
pler; its strength is simply proportional to—and in the same direction as—the electric field,
E
E.
6
is means, for example, that electric fields can easily be used to accelerate charged particles to
high velocities. Simply arrange for a region of space to have a uniform electric field throughout,
and a charged particle placed there will feel a constant electric force from that field. us, the
particle will accelerate faster and faster. A consequence is that electric fields can do mechanical
work on charged particles, and so increase (or decrease) their kinetic energy. us, there is an
energy associated with electric fields.
e magnetic force, on the other hand, only acts upon charged particles that are moving.
e symbol Ev in Equation (11.18) represents the velocity of the charged particle in question—its
speed, combined with its direction of travel. And so a magnetic field has no effect at all on a
motionless charged particle.
But the action of magnetic fields is even more intriguing. e magnetic force relates only
to the part of the charged particle’s motion that is perpendicular to
E
B. Only motion crosswise to
the magnetic field counts. And furthermore, the magnetic force itself is always directed perpen-
dicular to both the magnetic field and the motion of the charged particle.
Because of this cross-wise nature of the magnetic forces, they cannot be used to change the
speed of a charged particle. Instead, they change the direction. A force that is always crosswise
to one’s motion is just what is needed to cause a swirly motion. And so charged particles tend to
swirl around magnetic fields rather than move along them. is also means the magnetic force
cannot directly alter the kinetic energy of a charged particle.
11.3.2 WHAT IS CHARGE?
e existence of the Lorentz force—a force that is exerted only on particles that have the prop-
erty of electric charge—begs an important question: what exactly is electric charge? At one level,
6
If the charge is negative, the electric force acts in the direction opposite the electric field.
11.3. THE ELECTRIC AND MAGNETIC FIELDS 161
we can say simply that “it is what it is.” Charge is a property of certain fundamental particles—
the electron and proton being the most familiar examples—such that they interact with each
other according to the rules set down by Maxwell’s equations.
But is there a deeper explanation? Why is charge quantized? Why does the smallest unit
of charge have the particular value that it does? Why are there two kinds of charge? Why do,
for example, electrons and protons have charge, but neutrons do not? Why is charge conserved?
ese questions cannot be answered from within the confines of Maxwell’s 19th-century
theory of electricity and magnetism. But modern physics does provide at least some hints of
answers to some of these questions. e conservation of charge, for example, can be seen to
arise as a consequence of a particular symmetry of nature (see Section 14.2.2). And some sense
has been made—at least at the level of categorization—of the charges and other properties of the
many different fundamental particles (see Section 14.3). But many of the deep questions about
the fundamental nature of charge are beyond current understanding (see, for example, Penrose
[2004, p. 66]).
11.3.3 MAXWELL’S EQUATIONS AND THE CAUSES OF
E
E AND
E
B
e four Maxwell equations, (11.14)–(11.17), represent two equations each for the two vector
fields,
E
E and
E
B. Notice that there are two versions of the left-hand side, repeated for each field.
• Equations (11.14) and (11.15) begin with “r¨.” is is called a divergence, and it refers to
a vector field that diverges away from—or converges toward—some point in space.
• Equations (11.16) and (11.17) begin with “rˆ.” is is called a curl, and it refers to a
vector field that swirls around a point in space.
And so Maxwell’s four equations answer two questions about each of the two fields,
E
E and
E
B.
1. What causes the field to diverge from or converge toward points in space?
2. What causes the field to swirl around points in space?
From a mathematical perspective, to answer both of these questions is to say all there is to say
about a vector field. For the electric and magnetic fields, the answers are on the right-hand sides
of Equations (11.14)–(11.17). An intriguing thickening of this plot is the fact that two of the
Equations—(11.16) and (11.17)—include
E
E and
E
B in their right-hand side; thus the answer is,
in effect, part of the very question. Furthermore,
E
E is in the right-hand side of the equation
for
E
B, and vice versa. us, the two fields are coupled; one cannot really know one, without also
knowing the other. In that sense, the electric and magnetic fields
E
E and
E
B are but two individual
sides of a complex and subtle electromagnetism.
And so let us use words to relate the right-hand sides of Maxwell’s equations to their
left-hand sides, in the same order as they appear in Equations (11.14)–(11.17).
162 11. FIELDS
Gauss’s Law
Equation (11.14) is called Gauss’s Law. It says that a diverging electric field is caused by the
very presence of electric charge in space. is is, for example, the origin of the simple force of
static electricity. Put an electric charge on the table, and electric field diverges away from it in all
directions.
7
Put another charge next to it, and Equation (11.18) for the Lorentz force says the
field will exert a force on that second charge, in the same direction as the electric field.
8
us, it
is as if the one charge puts a force of repulsion (or attraction) on the other.
Krauss’s Law
Equation (11.15) has no particular historical name, but it is clearly mathematically similar to
Gauss’s law, except that it applies to
E
B instead of
E
E. Many years ago one of my physics students—
named Jeff Krauss—gave me permission to use his name for this equation. And so “Krauss’s
Law” says that a diverging magnetic field is caused by—nothing at all! Magnetic fields never
diverge away from (or converge upon) points in space. is is sometimes taken to be equivalent
to the statement that there is no such thing as “magnetic charge.”
Faraday’s Law
Equation (11.16) is called Faraday’s Law, and it says that a swirly electric field is caused by
a magnetic field that changes with time. e magnet sitting motionless on the door of your
refrigerator has no effect on the electric field. But move the magnet back and forth in front
of the door and it will cause electric fields to swirl around in space. ese electric fields will
then, according to the Lorentz force, accelerate electrons in the conducting metal door along
those swirly
E
E fields. is causes a swirling electric current in the metal. ese are called induced
currents, and they can occur in an isolated electric circuit disconnected from any external source
of power.
e Ampère-Maxwell Law
Equation (11.17) is the most complex of the four, and the second term on the right-hand side
is Maxwell’s addition to what had been known as Ampère’s Law. is last piece of the puzzle
transformed the four equations into a mathematically coherent field theory. And so although
the rest were discovered by others (Gauss, Faraday, and Ampère in particular), we honor the set,
taken as a whole, with Maxwell’s name.
e Ampère-Maxwell law says that there are two ways to make a swirly magnetic field.
But first, “Krauss’s Law” has already told us that there is no such thing as a diverging magnetic
field. And so the only kind of magnetic field that exists is one that swirls back upon itself to
form a closed loop of magnetism.
7
If it is a negative charge, then
E
E will instead converge toward it.
8
If it is a negative charge, the force is opposite the direction of
E
E.
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