Electricity and Magnetism

322 Pages·1965·6.736 MB·English

by  Edward M. Purcell

Edward M. Purcell
Electricity and Magnetism
Electricity and Magnetism
Copyright (c) 1963, 1964, 1965 by Education Development
Center, Inc., edc.org.
The following notices are reproduced from the 1965 edition of
the book:
“The preparation of this course was supported by a grant from
the National Science Foundation to Educational Services
Incorporated.”
“In accordance with the National Science Foundation’s policies
concerning curriculum revision material developed under their
auspices, McGraw-Hill Book Company, a division of
McGraw-Hill, Inc., announces that the material in the
Berkeley Physics Course, Vol. II ELECTRICITY AND
MAGNETISM, which is copyrighted by Education
Development Center (a successor by merger to Educational
Services, Inc.) and published by McGaw-Hill Book Company
in 1965, will be available for use by authors and publishers on
a royalty-free basis on or after April 30, 1970. Interested
parties should address inquiries to the Managing Director,
Educational Development Center, 55 Chapel Street, Newton,
Massachusetts, 02160.”
Copyright (c) 2013 by Benjamin Crowell and others as named
in log files. All contributors have agreed that the work involved
in producing a verbatim reproduction of the 1965 edition is to
be released under the Creative Commons CC0 license,
http://creativecommons.org/publicdomain/zero/1.0/.
i
Contents
1 Electrostatics: charges and fields 7
1.1 Electric charge . . . . . . . . . . . . . . . . . 7
1.2 Conservation of charge . . . . . . . . . . . . . 9
1.3 Quantization of charge. . . . . . . . . . . . . . 10
1.4 Coulomb’s law . . . . . . . . . . . . . . . . . 12
1.5 Energy of a system of charges . . . . . . . . . . 15
1.6 Electrical energy in a crystal lattice. . . . . . . . . 18
1.7 The electric field . . . . . . . . . . . . . . . . 20
1.8 Charge distributions . . . . . . . . . . . . . . . 24
1.9 Flux. . . . . . . . . . . . . . . . . . . . . . 26
1.10 Gauss’s law. . . . . . . . . . . . . . . . . . 28
1.11 Field of a spherical charge distribution. . . . . . . 30
1.12 Field of a line charge . . . . . . . . . . . . . . 32
1.13 Field of an infinite flat sheet of charge . . . . . . . 33
2 The electric potential 35
2.1 Line integral of the electric field . . . . . . . . . . 35
2.2 Potential difference and the potential function . . . . 36
2.3 Gradient of a scalar function . . . . . . . . . . . 38
2.4 Derivation of the field from the potential. . . . . . . 39
2.5 Potential of a charge distribution. . . . . . . . . . 40
Potentialoftwopointcharges,40.—Potentialofalongcharged
wire, 41.
2.6 Uniformly charged disk. . . . . . . . . . . . . . 42
2.7 The force on a surface charge. . . . . . . . . . . 47
2.8 Energy associated with an electric field . . . . . . . 49
2.9 Divergence of a vector function . . . . . . . . . . 51
2.10 Gauss’s theorem and the differential form of Gauss’s
law . . . . . . . . . . . . . . . . . . . . . . . . 53
2.11 The divergence in Cartesian coordinates. . . . . . 54
2.12 The Laplacian . . . . . . . . . . . . . . . . . 57
2.13 Laplace’s equation . . . . . . . . . . . . . . . 58
2.14 Distinguishing the physics from the mathematics . . 59
2.15 The curl of a vector function . . . . . . . . . . . 60
2.16 Stokes’ theorem . . . . . . . . . . . . . . . . 62
2.17 The curl in Cartesian coordinates. . . . . . . . . 63
2.18 The physical meaning of the curl . . . . . . . . . 66
3 Electric fields around conductors 69
3.1 Conductors and insulators . . . . . . . . . . . . 69
3.2 Conductors in the electrostatic field . . . . . . . . 70
3.3 Thegeneralelectrostaticproblem;uniquenesstheorem
75
3.4 Some simple systems of conductors . . . . . . . . 77
3.5 Capacitors and capacitance. . . . . . . . . . . . 80
3.6 Potentials and charges on several conductors . . . . 83
3.7 Energy stored in a capacitor . . . . . . . . . . . 85
3.8 Other views of the boundary-value problem . . . . . 86
4 Electric currents 89
4.1 Charge transport and current density. . . . . . . . 89
4.2 Stationary currents . . . . . . . . . . . . . . . 91
4.3 Electrical conductivity and Ohm’s law. . . . . . . . 93
4.4 A model for electrical conduction. . . . . . . . . . 95
4.5 Where Ohm’s law fails . . . . . . . . . . . . . .100
4.6 Electrical conductivity of metals . . . . . . . . . .102
4.7 Resistance of conductors. . . . . . . . . . . . .103
4.8 Circuits and circuit elements . . . . . . . . . . .105
4.9 Energy dissipation in current flow . . . . . . . . .107
4.10 Electromotive force and the voltaic cell. . . . . . .108
4.11 Variable currents in capacitors and resistors . . . .112
5 The fields of moving charges 115
5.1 From Oersted to Einstein. . . . . . . . . . . . .115
5.2 Magnetic forces. . . . . . . . . . . . . . . . .116
5.3 Measurement of charge in motion . . . . . . . . .118
5.4 Invariance of charge. . . . . . . . . . . . . . .119
5.5 Electric field measured in different frames of reference122
5.6 Field of a point charge moving with constant velocity .124
5.7 Field of a charge that starts or stops . . . . . . . .127
5.8 Force on a moving charge . . . . . . . . . . . .130
5.9 Interactionbetweenamovingchargeandothermoving
charges . . . . . . . . . . . . . . . . . . . . . .134
6 The magnetic field 141
6.1 Definition of the magnetic field. . . . . . . . . . .141
6.2 Some properties of the magnetic field . . . . . . .146
6.3 Vector potential . . . . . . . . . . . . . . . . .150
6.4 Field of any current-carrying wire . . . . . . . . .154
6.5 Fields of rings and coils . . . . . . . . . . . . .155
6.6 Change in B at a current sheet . . . . . . . . . .157
6.7 How the fields transform . . . . . . . . . . . . .160
6.8 Rowland’s experiment . . . . . . . . . . . . . .165
6.9 Electric conduction in a magnetic field: the Hall effect .165
7 Electromagnetic induction 169
Contents iii
7.1 Faraday’s discovery . . . . . . . . . . . . . . .169
7.2 A conducting rod moves through a uniform magnetic
field . . . . . . . . . . . . . . . . . . . . . . .172
7.3 A loop moves through a nonuniform magnetic field . .173
7.4 A stationary loop with the field source moving . . . .179
7.5 A universal law of induction . . . . . . . . . . . .180
7.6 Mutual inductance. . . . . . . . . . . . . . . .184
7.7 A “reciprocity” theorem. . . . . . . . . . . . . .186
7.8 Self-inductance . . . . . . . . . . . . . . . . .188
7.9 A circuit containing self-inductance. . . . . . . . .189
7.10 Energy stored in the magnetic field . . . . . . . .191
7.11 “Something is missing” . . . . . . . . . . . . .193
7.12 The displacement current . . . . . . . . . . . .195
7.13 Maxwell’s equations . . . . . . . . . . . . . .197
8 Alternating-current circuits 203
8.1 A resonant circuit . . . . . . . . . . . . . . . .203
8.2 Alternating current. . . . . . . . . . . . . . . .206
8.3 Alternating-current networks . . . . . . . . . . .211
8.4 Admittance and impedance. . . . . . . . . . . .213
8.5 Power and energy in alternating-current circuits . . .216
9 Electric fields in matter 219
9.1 Dielectrics . . . . . . . . . . . . . . . . . . .219
9.2 The moments of a charge distribution. . . . . . . .222
9.3 The potential and field of a dipole . . . . . . . . .225
9.4 The torque and the force on a dipole in an external field226
9.5 Atomic and molecular dipoles; induced dipole moments228
9.6 The polarizability tensor . . . . . . . . . . . . .231
9.7 Permanent dipole moments. . . . . . . . . . . .233
9.8 The electric field caused by polarized matter. . . . .235
9.9 The capacitor filled with dielectric . . . . . . . . .240
9.10 The field of a polarized sphere . . . . . . . . . .242
9.11 A dielectric sphere in a uniform field. . . . . . . .245
9.12 Charge in a dielectric . . . . . . . . . . . . . .247
9.13 Susceptibility and atomic polarizability. . . . . . .250
9.14 Energy changes in polarization. . . . . . . . . .254
9.15 Dielectrics made of polar molecules. . . . . . . .255
9.16 Polarization in changing fields . . . . . . . . . .256
9.17 The bound-charge current. . . . . . . . . . . .257
10Magnetic fields in matter 263
iv Contents
10.1 How various substances respond to a magnetic field.263
10.2 The absence of magnetic “charge” . . . . . . . .268
10.3 The field of a current loop . . . . . . . . . . . .270
10.4 The force on a dipole in an external field . . . . . .273
10.5 Electric currents in atoms . . . . . . . . . . . .276
10.6 Electron spin and magnetic moment. . . . . . . .282
10.7 Magnetic susceptibility . . . . . . . . . . . . .284
10.8 The magnetic field caused by magnetized matter . .286
10.9 The field of a permanent magnet . . . . . . . . .289
10.10 Free currents, and the field H . . . . . . . . . .291
10.11 Ferromagnetism . . . . . . . . . . . . . . .295
Contents v
Chapter 1
Electrostatics: charges
and fields
1.1 Electric charge
Electricity appeared to its early investigators as an extraordi-
nary phenomenon. To draw from bodies the subtle fire, as it
was sometimes called, to bring an object into a highly electri-
fied state, to produce a steady flow of current, called for skillful
contrivance. Except for the spectacle of lightning, the ordi-
nary manifestations of nature, from the freezing of water to
the growth of a tree, seemed to have no relation to the curi-
ous behavior of electrified objects. We know now that electrical
forces largely determine the physical and chemical properties
of matter over the whole range from atom to living cell. For
this understanding we have to thank the scientists of the nine-
teenth century, Ampere, Faraday, Maxwell, and many others,
who discovered the nature of electromagnetism,, as well as the
physicists and chemist of the twentieth century who unraveled
the atomic structure of matter.
Classical electromagnetism deals with electric charges and
currents and their interactions as if all the quantities involved
could be measured independently, with unlimited precision.
Here classical means simply “non-quantum.” The quantum law
with its constant h is ignored in the classical theory of elec-
tromagnetism, just as it is in ordinary mechanics. Indeed, the
classical theory was brought very nearly to its present state
of completion before Planck’s discovery. It has survived re-
markably well. Neither the revolution of quantum physics nor
the development of special relativity dimmed the luster of the
electromagnetic field equations Maxwell wrote down a hundred
years ago.
Of course, the theory was solidly based on experiment, and
because of that was fairly secure within its original range of
7
application –to coils, capacitors, oscillating currents, eventually
radio waves and light waves. But even so great a success does
not guarantee validity in another domain, for instance, the in-
side of a molecule.
Twofactshelpexplainthecontinuingimportanceinmodern
physics of the classical description of electromagnetism. First,
special relativity required no revision of classical electromag-
netism. Historically speaking, special relativity grew out of
classical electromagnetic theory and experiments inspired by
it. Maxwell’s field equations, developed long before the work of
LorentzandEinstein, provedtobeentirelycompatiblewithrel-
ativity. Second, quantum modifications of the electromagnetic
force have turned out to be unimportant down to distances less
than 10−10 cm, a hundred times smaller than the atom. We can
describe the repulsion and attraction of particles in the atom
using the same laws that apply to the leaves of an electroscope,
although we need quantum mechanics to predict how the parti-
cles will behave under those forces. For still smaller distances,
there is a rather successful fusion of electromagnetic theory and
quantum theory, called quantum electrodynamics, which seems
to agree with experiment down to the smallest distances yet
explored.
We assume the reader has some acquaintance with elemen-
tary facts of electricity. We are not going to review all the ex-
periments by which the existence of electric was demonstrated
or all the evidence for the electrical constitution of matter. On
theotherhand,wedowanttolookcarefullyattheexperimental
foundations of the basic laws on which all else depends. In this
chapter we shall study the physics of stationary electric charges
— electrostatics.
Certainly one fundamental property of electric charge is its
existence in the two varieties that were long ago named positive
andnegative. Theobservedfactisthatallchargedparticlescan
be divided into two classes such that all members of one class
repel each other, while attracting members of the other class.
If two small electrically charged bodies A and B, some distance
apart, repeloneanother, andifAattractssome third electrified
body C, then we always find that B attracts C. Why this uni-
versal law prevails we cannot say for sure. But today physicists
tend to regard positive and negative charge as, fundamentally,
opposite manifestations of one quality, much as “right” and
“left” are opposite manifestations of “handedness.” Indeed, the
questionofsymmetryinvolvedinrightandleftseemstobeinti-
mately related to this duality of electric charge, and to another
fundamental symmetry, the two directions of time. Elementary
particle physics is throwing some light on these questions.
8 Chapter 1 Electrostatics: charges and fields
What we call negative charge could just as well have been
calledpositive,andviceversa.1 Thechoicewasahistoricalacci-
dent. Our universe appears to be very evenly balanced mixture
of positive and negative electric charge which, since like charges
repel one another is not surprising.
Twootherobservedpropertiesofelectricchargeareessential
in the electrical structure of matter: charge is conserved, and
chargeisquantized. Thesepropertiesinvolvequantity ofcharge,
and thus imply a measurement of charge. Presently we shall
state precisely how charge can be measured in terms of the
force between charges a certain distance apart, and so on. But
let us take this for granted, for the time being, so that we may
talk freely about these fundamental facts.
1.2 Conservation of charge
The total charge in an isolated system never changes. By iso-
lated we mean that no matter is allowed to cross the boundary
of the system. We could let light pass into or out of the system
without affecting the principle, since photons carry no charge.
For instance, a thin-walled box in vacuum, exposed to gamma
rays,mightbecomethesceneofa“pair-creation”eventinwhich
a high-energy photon ends its existence with the creation of a
negative electron and a positive electron (Fig. 1.1). Two elec-
trically charged particles have been newly created but the net
changeintotalcharge, inandonthebox, iszero. Aneventthat
would violate the law we have just stated would be the creation
of a positively charged particle without the simultaneous cre-
ation of a negatively charged particle. Such an occurrence has
never been observed.
Of course, if the electric charges of electron and positron
were not precisely equal in magnitude, pair creation would still
violate the strict law of charge conservation. As well as can be
determined from experiment, their charges are equal. An in-
teresting experimental test is provided by the structure called
positronium,astructurecomposedofanelectronandapositron,
and nothing else. This curious “atom” can live long enough-a
tenth of a microsecond or so-to be studied in detail. It behaves
as if it were quite neutral, electrically. Actually, most physicists
would be astonished, not to say incredulous, if any difference
were found in the magnitudes of these charges, for we know
that electron and positron are related to one another as par-
ticle to antiparticle. Their exact equality of charge, like their
1The charge of the ordinary electron has nothing intrinsically negative
about it. A negative integer, once multiplication has been defined, differs a/Charged particles are created in
pairs with equal and opposite charge.
essentiallyfromapositiveintegerinthatitssquareisanintegerofopposite
sign. Buttheproductoftwochargesisnotacharge;thereisnocomparison.
Section 1.2 Conservation of charge 9