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II J I
I. V. SAVELYEV
PHYSICS
Itndf~( ";iJirea~
A GENERAL COURSE
(In three volumes)
tJ
H. B. CABEJIbEB VOLUME II
RYPC OBIJJ;EH <DM3IffUI
ELECTRICITY
TOM II
AND MAGNETISM
8JIERTPMllECTBO M
MArHETM3M WAVES
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BOJIHbI
OPTICS
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Translated from RussianbyG.l.eib
PREFACE
Firstpublished 1980
Revised from the 1978Russian edition
The main content of the present volume is the science of electro
Second printing 1985
magnetism and the science of waves (elastic, electromagnetic, and
Thirdprinting 1989
light).
The International System of Units (81) has been used throughout
the book, although the reader is simultaneously acquainted with
the Gaussian system. In addition to a list of symbols, the appendices
at the end of the book give the units of electrical and magnetic
quantities in the 81 and in the Gaussian system of units, and also
compare the form of the basic formulas of electromagnetism in both
systems.
The course is the result of twenty five year's work in the Depart
ment of General Physics of the Moscow Institute of Engineering
Physics. I am grateful to my colleagues and friends for their helpful
discussions, criticism and advice in the course of the preparation
of the book.
The present course is intended above all for higher technical
schools with an extended syllabus in physics. The material has
been arranged, however, so that the book can be used as a teaching
/)I/.'~At'J lj aid for higher technical schools with an ordinary syllabus simply
by omitting some sections.
•...-:r -...,.,..._.;~.::r:_~.~~
~ .~"....~:..~...;.,..:.J
Igor Savelyeu
Moscow, November, 1979
HaaHZAUUCKOMJl3blKe '
Printedin the Union ofSovietSocialistRepublics
ISBN 5-03-000902-7 ©Ib.naTenbcTBo«HayKa», 1978
ISBN 5-03-000900-0 © Englishtranslation,MirPublishers, 1980
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Content.
CONTENTS
CHAPTER 5. STEADY ELECTRIC CURRENT
5.1. Electric Current 98
5.2. Continuity Equation 100
5.3. Electromotive Force 102 <t
5.4. Ohm's Law. Resistance of Conductors 1M
5.5. Ohm's Law for an Inhomogeneous Circuit Section 106
5.6. Multiloop Circuits. Kirchhoff's Rules 108
5.7. Power of a Current 110
5.8. The Ioule-Lenz Law 112
Preface 5
CHAPTER 6. MAGNETIC FIELD IN A VACUUH
6.1. Intersction of Currents 113
+
PART I. 6.2. Magnetic Field 115
ELECTRICITY AND MAGNETISM 11 6.3. Field of a Moving Charge 116
6.4. The Biot-Savart Law 120
CHAPTER f. ELECTRIC FIELD IN A VACUUM 6.5. The Lorentz Force 122
6.6. Ampere's Law 125
1.1. Electric Charge 11
1.2. Coulomb's Law 12 6.7. Magnetism as a Relativistic Effect 127
1.3. Systems of Units 14 6.8. Current Loop in a Magnetic Field 133
6.9. Magnetic Field of a Current Loop 138
-r- 11..54.. ERlaetciotrniacliFzeiedld.FoFrimeldofStrWenrgittihng16Formulas 16 t 6.10. Work Done When a Current Moves in a Magnetic Field 140
1.6. Potential 20 6.11. Divergence and Curl of a Magnetic l"ield 144
6.12. Field of a Solenoid and Toroid 148
1.7. Interaction Energy ofa System of Charges 24
1.8. Relation Between Electric Field Strength and Potential 25
1.9. Dipole 28 CHAPTER 7. MAGNETIC FIELD IN A SUBSTANCE
1.10. Field of a System of Charges at Great Distances 33 7.1. Magnetization of a Magnetic 153
1.11. A Description of the Properties of Vector Fields 36 7.2. Magnetic Field Strength 154
1.12. Circulation and Curl of an Electrostatic Field 51 7.3. Calculation of the Field in Magnetics 160
1.13. Gauss's Theorem 53 7.4. COllditions at the Interface of Two Magnetics 162
1.14. Calculating Fields with the Aid of Gauss's Theorem 55 7.5. Kinds of Magnetics 166
7.6. Gyromagnetic Phenomena 166
7.7. Diamagnetism 171
CHAPTER 2. ELECTRIC FIELD IN DIELECTRICS
+- 7.8. Paramagnetism 174
2.1. Polar and Non-Polar Molecules 61 7.9. Ferromagnetism 177
2.2. Polarization of Dielectrics 63
2.3. The Field Inside a Dielectric 64 CHAPTER 8. ELECTROMAGNETIC INDUCTION
2.4. Space and Surface Bound Charges 65
2.5. Electric Displacement Vector 70 8.1. The Phenomenon of Electromagnetic Induction 182
8.2. Induced E.M.F. 183
2.6. Examples of Calculating the Field in Dielectrics 73
8.3. Ways of Measuring the Magnetic Induction 187
2.7. Conditions on the Interface Between Two Dielectrics 77 8.4. Eddy Currents 188
2.8. Forces Acting on a Charge in a Dielectric 81
8.5. Self-Induction 18\}
2.9. Ferroelectrics 82
8.6. Current When a Circuit Is Opened or Closed 192
8.7. Mutual Induction 194
CHAPTF.R 3. CONDUCTORS IN AN ELECTRIC FIELD 8.8. Energy of a Magnetic Field 196
8.9. Work in Magnetic Reversal of a Ferromagnetic 198
3.1. Equilibrium of Charges on a Conductor 84
3.2. A Conductor in an External Electric Field 86 CHAPTER 9. MAXWELL'S EQUATIONS
3.3. Capacitance 87
3.4. Capacitors 89 --l 9.1. Vortex Electric Field 200
-+ 9.2. Displacement Current 202
9.3. Maxwell's Equations 206
CHAPTER 4. ENERGY OF AN ELECTRIC FIELD
4.f. Energy of a Charged Conductor 92 CHAPTER 10. MOTION OF CHARGED PARTICLES IN ELECTRIC AND MAGNETIC
FIELDS
4.2. Energy of a Charged Capacitor 92
4.3. Energy of an Electric Field 95 to.1. Motion of a Charged Particle in a Homogeneous Mag
netic Field 209
'It
',.,--_. -----.~ ft~__~.·,_.·, .~__ ~ ..••_.~~ ._'•._'
8 Contents Contents 9
10.2. Deflection of Moving Charged Particles by an Electric 15.4. Energy of Electromagnetic Waves 310
and a Magnetic Field 210 15.5. Momentum of Electromagnetic Field 312
10.3. Determination of the Charge and Mass ofan Electron 214 " 15.6. Dipole Emission 314
10.4. Determination of the Specific Charge of Ions. Mass
Spectrographs 219
10.5. Charged Particle Accelerators 223 PART III. OPTICS 318
CHAPTER fl. THE CLASSICAL THEORY OF ELECTRICAL CONDUCTANCE OF CHAPTER 16. GENERAL
METALS
16.1. The Light Wave 318 .
11.1. The Nature of Current Carriers in Metals 228 ~. Representation of Harmonic Functions Using Expo
11.2. The Elementary Classical Theory of Metals 230 nents 321
11.3. The Hall Effect 234 «h3: Reflection and Refraction of a Plane Wave at the
Interface Between Two Dielectrics 323
CHAPTER 12. ELECTRIC CURRENT IN GASES 16.4. Luminous Flux 328
16:5'. Photometric Quantities and Units 330
12.1. Semi-Self-Sustained and Self-Sustained Conduction 237 ~ Geometrical Optics 333
12.2. Semi-Self-Sustained Gas Discharge 237 . 16.7. Centered Optical System 337
12.3. Ionization Chambers and Counters 240 16.8. Thin Lens 344
12.4. Processes Leading to the Appearance of Current Carriers 16.9. Huygens' Principle 346
in a Self-Sustained Discharge 245
12.5. Gas-Discharge Plasma 249
CHAP1'ER 17. INTERFERENCE OF LIGHT
12.6. Glow Discharge 251
12.7. Arc Discharge 254 17.1. Interference of Light Waves 348
12.8. Spark and Corona Discharges 255 17.2. Coherence 353
4.7;3: Ways of Observing the Interference of Light 361
CHAPTER 13. ELECTRICAL OSCILLATIONS 17.4. Interference of Light Reflected from Thin Plates 363
11.S~ The Michelson Interferometer 373
13.1. Quasistationary Currents 259 -41.6.-Multibeam Interference 376
13.2. Free Oscillations in a Circuit Without a Resistance 259
13.3. Free Damped Oscillations 263
CHAPTER 18. DIFFRACTION OF LIGHT
13.4. Forced Electrical Oscillations 266
13.5. Alternating Current 271 -1.8;1. Introduction 384
18.2. Huygens-Fresnel Principle 385
18.3. Fresnel Zones 387
18.4. Fresnel Diffraction from Simple Barriers 392
PART II. WAVES 275 18.5. Fratinhofer Diffraction from a Slit 403
18.6. Diffraction Grating 411
1:83", Diffraction of X-Rays 419
CHAPTER U. ELASTIC WAVES
'18.8. Resolving Power of an Objective 425
14.1. Propagation of Waves in an Elastic Medium 275 18.9. Holography 427
14.2. Equations of a Plane and a Spherical Wave 278
14.3. Equation of a Plane Wave Propagating in an Arbitrary CHAPTER 19. POLARIZATION OF LIGHT
Direction 281
14.4. The Wave Equation 283 19.1. Natural and Polarized Light 432
14.5. Velocity of Elastic Waves in a Solid Medium 284 19:2;- Polarization in Reflection and Refraction 436
14.6. Energy of an Elastic Wave 286 19.3. Polarization in Double Refraction 440
14.7. Standing Waves 290 19.4. Interference of Polarized Rays 443
14.8. Oscillations of a String 293 -19.5. Passing of Plane-Polarized Light Through a Crystal
14.9. Sound 294 Plate 445
14.10. The Velocity of Sound in Gases 297 t9.6. A Crystal Plate Between Two Polarizers 447
14.11. The Doppler Effect for Sound Waves.302 ·1.9.7. Artificial Double Refraction 451
19.8. Rotation of Polarization Plane 453
CHAPTER 15. ELECTROMAGNETIC WAVES
CHAPTER 20. INTERACTION OF ELECTROMAGNETIC WAVES WITH A SUBSTANCE
15.1. The Wave Equation for an Electromagnetic Field 304
I 15.2. Plane Electromagnetic Wave 305 20.1. Dispersion of Light 456
15.3. Experimental Investigation of Electromagnetic Waves 309 ~ Group Velocity 456
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Content.
PART I ELECTRICITY
20.3. Elementary Theory of Dispersion 462
20.4. Absorption of Light 466
AND MAGNETISM
20.5. Scattering of Light 467
20.6. The Vavilov-Cerenkov Effect 470
--cHA1'TER-~ MOVING-MEDIA OPTICS
21.1. The Speed of Light 472
21.2. Fizeau's Experiment 474
21.3. Michelson's Experiment 477
21.4. The Doppler Effect 481 CHAPTER 1 ELECTRIC FIELD
APPENDICES 485
IN A VACUUM
A.1. List of Symbols 485
A.2. Units of Electrical and Magnetic Quantities in the Inter
national System (SI) and in the Gaussian System 487
A.3. Basic Formulas of Electricity and Magnetism in the SI
and in the Gaussian System 489
Name Index 495 1.1. Electric Charge
Subject Index 497
All bodies in nature are capable of becoming electrified, Le. acquir
ing an electric charge. The presehce of such a charge manifests
itself in that a charged body interacts with other charged bodies.
Two kinds of electric charges exist. They are conventionally called
positive and negative. Like charges ~pel each other, and unlike
charges attract each other.
An electric chargeisan integral part of certain elementary parti
cles*. The charge of all elementary particles (if it is not absent) is
identical in magnitude. It can be called an elementary charge.
We shall use the symbol e to denote a positive elementary charge.
The elementar¥ particles include, in particular, the electron
(carrying the negative charge -e), the proton (carrying the positive
charge +e), and the neutron (carrying no chllrge). These particles
are the bricks which the atoms and molecules of any substance are
built of, therefore all bodies contain electric charges. The particles
carrying charges of different signs are usually present in a body in
equal numbers and are distributed over it with the same density.
The algebraic sum of the charges in any elementary volume of the
body equals zero in this case, and each such volume (as well as the
body as a whole) will be neutral. If in some way or other we create
a surplus of particles of one sign in a body (and, correspondingly,
a shortage of particles of the opposite sign), the body will be charged.
It is also possible, without changing the total number of positive
and negative particles, to cause them to be redistributed in a body
so that one part of it has a surplus of charges of one sign and the
• Elementary particles are defined as such microparticles whose internal
,-\ structure at the present level of development of physics cannot be conceived as
a combination of other particles.
"
12 Electricity and Magnetism Electric Field ina·Vacuum 13
other part a surplus of charges of the opposite sign. This can be p. 174), Coulomb measured the force of interaction of two charged
done by bringing a charged body close to an uncharged metal one. spheres depending on the magnitude of the charges on them and
Since a charge q is formed by a plurality of elementary charges, on the distance between them. He proceeded from the fact that
it is an integral multiple of e: when a charged metal sphere was touched by an identical uncharged
sphere, the charge would be distributed equally be
q = ±Ne (1.1)
tween the two spheres.
An elementary charge is so small, however, that macroscopic charges As a resultofhisexperiments, Coulomb arrivedat
may be considered to have continuously changing magnitudes. the conclusion that the force of interaction between
~ 'iO If a physical quantity can take on only definite discrete values. ,. two stationary point charges is proportional to the
)it is said to be quantized. The fact expressed by Eq. (1.1) signifies magnitude ofeach of them and inversely proportional
that an electric charge is quantized. to the square of the distance between them. The direc
The magnitude of a charge measured in different inertial reference tion of the force coincides with the straight line
frames will be found to be the same. Hence, an electric charge is connecting the charges.
relativistically invariant. It thus follows that the magnitude of It must be noted that the direction of the force
a charge does not depend on whether the charge is moving or at rest. of interaction along the straight line.connecting the
Electric charges can vanish and appear again. Two elementary point charges follows from considerations of sym
charges of opposite signs always appear or vanish simultaneously, metry. Anemptyspace isassumedto be homogeneous
however. For example, an electron and a positron (a positive elec and isotropic. Consequently, the only direction dis
tron) meeting each other annihilate, Le. transform into neutral tinguished in the space by stationary point charges
gamma-photons. This is attended by vanishing of the charges -e
introduced into it is that from one charge to the .
and +e. In the course of the process called the birth of a pair, a gam other. Assume that the force F acting on the charge Fig. 1.1
ma-photon getting into the field of an atomic nucleus transforms
qi (Fig. 1.2) makes the angle ex with the direction
into a pair of particles-an electron and a positron. This process
causes the charges -e and +e to appear. from ql to q2, and that ex differs from 0 or n. Btlt owing to axial
symmetry, there are no grounds to set the force F aside from the
Thus, the total charge of an electrically isolated system* cannot
multitude of forces of other directions making the same angle ex with
" change. This statement forms the law of electric charge conser
vation.
\•
We must note that the law of electric charge conservation is asso
I
cIniadteeedd,viefrytheclmosaeglynitwuditehofthaecrhealragteivdisetpiecndinevdaorinanitcsevoeflocaitych,atrhgeen. 1n1~..0-4 " ! I,I onJ2 F. 'I ~---J f
by bringing charges of one sign into motion we would change the 1% 21
r
total charge of the relevant isolated system.
Fig. 1.2 Fig. 1.3
1.2. Coulomb's Law the axis QCq2 (the directions.of these forces form a cone with a cone
angle of 2ex). The difficulty appearing as a result of this vanishes
The law obeyed by the force of interaction of point charges was when ex equals 0 or n.
established experimentally in 1785 by the French physicist Charles Coulomb's law can be expressed by the formula
A. de Coulomb (1736-1806). A point charge is defined as a charged
'" body whose dimensions may be disregarded in comparison with F12= - k ·ql~q2eI2 (1.2) @})
the distances from this body to other bodies carrying an electric
Here k = proportionality constant assumed to be positive
charge.
ql and q, = magnitudes of the interacting charges
Using a torsion balance (Fig. 1.1) similar to that employed by r = distance between the charges ~
H. Cavendish to determine the gravitational constant (see Vol. I.
eu = unit vector directed from the charge ql to Q2
• A system is referred to as electrically isolated if no charged particles can Fu = force acting on the charge Ql (Fig. 1.3; the figure cor
penetrate through the surface confining it. responds to the case of like charges).
L~ ~-{~~~--~-f~~~~..r-~..(~···~~~~
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J J l._.1 J .1 .1 J .1 l~. ..I J L ..I "i ..,I ..I ..1 ..i "I \ .."J ..I
1.4 Electricity and Magnetbm Electric Field in a Vacuum 15
with a force of 1 dyne in a vacuum with an equal charge at a distance
The force F21 differs from F12 in its sign:
of 1 em from it.
(gj
F -'k qlql e (1.3) Careful measurements (they are described in Sec. to.3) showed
2f- "_.-1r - f2 that an elementary charge is
The magnitude of the interaction force, which is the same for
e= 4.80 X10-fo cgse (1.6)
q
both charges, can be written in the form
Adopting the units of length, mass, time, and charge as the basic
F=k Iqlq.1 (1.4) ones, we can construct a system of units of electrical and magnetic
rl
quantities. The system based on the centimetre, gramme, second, ••
Experiments show that the force of interaction between two given and the cgse unit is called the absolute electrostatic system of units
q
chargesdoesnotchangeifother charges are placednear them. Assume (the cgse system). It is founded on Coulomb's law, i.e. the law of
IJ" •
that we have the charge qa and, in addition, N other charges interaction between charges at rest. On a later page,weshall become
Ql, Q2, .••, qN' It can be seen from the above that the resultant acquainted with the absolute electromagnetic system of units (the
force F with which all the N charges qi act on qa is cgsm system) based on the law of interaction between conductors
carryingan electric current. The Gaussian system in which the units
N
of electrical quantities coincide with those~of the cgse system, and
F= ~ Fa•t (1,5) of magnetic quantities with those of the cgsm system, is also an
i-f
absolute system.
where Fa.i is the force with which the charge q, acts on qa in the Equation (1.4) in the cgse system becomes
absence of the other N - 1 charges.
The fact expressed by Eq. (1.5) permits us to calculate the force F= Iq~21 (1.7)
of interaction between charges concentrated on bodies of finite
dimensions, knowing the law of interaction between point charges. This equation is correct if the charges are in a vacuum. It has to be
For this purpo~e,we must divide each charge into so small charges dq determined more accurately for charges in a medium (see Sec. 2.8).
that they can be considered as point ones, use Eq. (1.2) to calculate USSR State Standard GOST 9867-61, which came into force
,
the force of interaction between the charges dq taken i~rs, and on January 1, 1963. prescribes the preferable use of the Internation ••
then perform vector summation of these forces. Mathematically, al System of Units (SI). The basic units of this system are the
this procedure coincides completely with the calculation of the metre, kilogramme, second, ampere, kelvin, candela, and mole.
force of gravitational attraction between bodies of finite dimensions The SI unit of force is the newton (N) equal to 105 dynes.
(see Vol. I, Sec. 6.1). In establishing the units of electrical and magnetic quantities,
All experimental facts available lead to the conclusion that the SI system, like the cgsm one, proceeds from the law of interac
Coulomb's law holds for distances from 10-15 m to at least several tion of current-carrying conductors instead of charges. Consequently,
kilometres. There are grounds to presume that for distances smaller the proportionality constant in the equation of Coulomb's law is
than 10-16 m the law stops being correct. For very great distances, a qnantity with a dimension and rliffering from unity.
there are no experimental confirmations of Coulomb's law. But The SI unit of charge is the coulomb (C). It has been found experi
there are also no reawns to expect that this law stops being obeyed mentally that
with very great distances between charges.
1 C= 2.998 X t09 ~ 3 X 109 cgseq (1.8)
To form an idea of the magnitude of a charge of 1 C, let us calcu
late the force with which two point charges of 1 Ceach would interact
1.3. Systems of Units
with each other if they were 1 m apart. By Eq. (1.7)
,. We can make the proportionality constant in Eq. (1.2) equal F= 3x109.nX~n3x109 cgseF=9 X lOf4 dyn=9 X 109N~ t09 kgf (1.9)
unity by properly choosing the unit of charge (the units for F and r
were established in mechanics). The relevant unit of charge (when
An elementary charge expressed in coulombs is
F and r are measured in cgs units) is called the absolute electrostatic
unit of charge (cgseq). It is the magnitude of a charge that interacts e=1.60X 10-f9 C (1.10)
~
Electric Field in a Vacuum 17
Electricity and Magnetism
16
at a pointof it experiencesthe action of a force. Hence, to see whether
1.4. Rationalized Form
there is an electric field at a given place, we must place a charged
of Writing Formulas body (in the following we shall say simply a charge for brevity) at
it and determine whether or not it experiences the action of an
Many formulas of electrodynamics when written in the cgs systems
electric forcfl. We can evidently assess the "strength" of the field
.. (in particular, in the Gaussian one) include as factors 41t and the
according to the magnitude of the
so-called electromagnetic constant c equal to the speed of light
force exerted on the given charge.
in a vacuum. To eliminate these factors in the formulas that are
Thus, to detectand study an electric F
most important in practice, the proportionality constant in Cou
field, we must use a "test" charge. For
lomb's law is 'taken equal to 1I41t£o. The equation of the law for
the force acting on our test charge to
-charges in a vacuum will thus become
characterize the field "at the given
F=_1_ IqIq21 (1.11) point", the test charge must be a point
4n80 r2
one. Otherwise, the force acting onthe
The other formulas change accordingly. This modified way of writ charge will characterize the properties
ing formulas is called rationalized. Systems of units constructed of the field averaged over the volume fJ
with the use of rationalized formulas are also called rationalized. occupied by the body that carries the
They include the SI system. test charge.
Fig. 1.4
The quantity £0 is called the electric constant. It has the dimenSiOn Let us study the field set up by the
of capacitance divided by length. It is accordingly expressed in stationary point charge q with the aid
units called the farad per metre. To find the numerical value of £0' of the point test charge qt. We place the testcharge at a pointwhose
we shall introduce the values of the quantities corresponding to the position relative to the charge q is eterminedbythe position vector
case of two charges of 1 C each and 1 m apart into Eq. (1.11). By r (Fig. 1.4). We see that the test charge experiences the force
Eq. (1.9), the force of interaction in this case is 9 X 109 N. Using
this value of the force, and also ql = q2 = 1 C and r = 1 m in F = qt (_1_...!Le ) (1.13)
4neo r2 r
Eq. (1.11), we get
[see Eqs. (1.3) and (1.11}J. Here e, is the unit vector of the position ..
9 X 109=_1- 1xt
4neo -1-1 vector r.
x: A glance at Eq. (1.13) shows that the force actingon our test charge
whence
depends not only on the quantities determining the field (on q and r),
c' 'c £0= 4n x 109 = 0.885 X 10-11 F/m (1.12) but also on the magnitude of the test charge qt. If we take different
test charges qt, qt, etc., then the forces F', F", etc. which they "
The Gaussian system of units was widely used and is continuing
experience at the given point of the field will be different. We can .
to be used in physical puhlications. We therefore consider it essen see from Eq. (1.13), however, that the ratio F/qt for all the test ,
tial to acquaint our reader with both the SI and the Gaussian system. ~
charges will be the same and depend only on the values of q and r
We shall set out the material in the SI units showing at the same
determining the field at the given point. It is therefore natural
time how the formulas look in the Gaussian system. The fundamen
to adopt this ratio as the quantity characterizing an electric fiel~:
tal formulas of electrodynamics written in the SI and the Gaussian
system are compared in AppendiX 3. E=F- (1.14)
qt
This vector quantity is called the electric field strength (or intensity)
1.5. Electric Field. Field Strength at a given point (Le. at the point where the test charge qt experi
ences the action of the force F).
Charges at rest interact through an electric field·. A charge alters
According to Eq. (1.14), the electric field strength numerically
the properties of the space surrounding it-it sets up an electric
equals the force acting on a unit point charge at the given point
field in it. This field manifests itself in that an electric charge placed
of the field. The direction of the vector E coincides with that of the
• We shallsee inSec.6.2 that when considering moving charges, their inter force acting on a positive charge.
action in addition to an electric field is due to a magnetic field.
2--7796
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18 Electricity and MagneU6m Electric Fteld in a Vacuum 19
It must be noted that Eq. (1.14) also holds when the test charge According to Eq. (1.14), the force exerted on a test charge is
is negative (qt < 0). In this case, the vectors E and F have opposite F = qtE
directions.
It is obvious that any point charge q* at a point of a field with the
We have arrived at the concept of electric field strength when
strength E will experience the force
studying the field of a stationary point charge. Definition (1.14),
however, also covers the case of a field set up by any collection of F = qE (1.18)
stationary charges, But here the following clarification is needed.
If the charge q is positiv~,the d.irection of the force coincides with " ...
The arrangement of the charges setting up the field being studied
may change under the action of the test charge. This will happen, that of the vectorE. If q IS negahve, the vectors F and E are directed J
oppositely.
for example, when the charges producing the field are on a conduc
We mentioned in Sec. 1.2 that the force with which a system of
tor and can freely move within its limits. Therefore, to avoid appre
charges acts on a charge not belonging to the system equals the
ciable alterations in the field being studied, a sufficiently small test
vector sum of the forces which each of the charges of the system
charge must be taken.
exerts separately on the given charge [see Eq. (1.5»), Hence it fol
It follows from Eqs. (1.14) and (1.13) that the field strength of
j a point charge varies directly with the magnitude of the charge q lows that the field strength of a system of charges equals-the vector sum .Q
of the field strengths that would be produced by eac~,Pf the charges of
o"i and inversely with the square of the distance r from the charge to
the system separately: l''j
I the given point of the field:
E = ~ E, (1.19)
E=-n14eo--rq.er (1.15) This statement is called the principle of electric field superposition.
The superposition principle allows us to calculate the field strength
The vectorE is directed along the radial straight line passing through of any system of charges. By dividing extended charges into suffi
the charge and the given point of the field, from the charge if the ciently small fractions dq, we can reduce
latter is positive and toward the charge if it is negative. any system of charges to a collection of [ E
In the Gaussian system, the equation for the field strength of ~
p.oint charge•. We calculatethe oontrihu-
a point charge in a vacuum has the form hon of each of such charges to the resul- •
~~
tant field by Eq. (1.15).
q
E=rser (1.16) An electric field can be described by
indicatingthe magnitude aod direction of
The unit of electric field strength is the strength at a point where the vector E for each of its points. The "/
unit force (1 N in the SI and 1 dyn in the Gaussian system) acts combination of these vectors forms the Fig. 1.5
on unit charge (1 C in the SI and 1 cgse in the Gaussian system). field of the electric field strength vector
q
This unit has no special name in the Gaussian system. The SI unit (compare with the field of the velocity vector, Vol. I, p. 249). The
of electric field strength is called the volt per metre (VIm) [see velocity vector field can be represented very illustratively with the
Eq. (1.44)]. aid of flow lines. Similarly, an electric field can be described with
According to Eq. (1.15), a charge of 1 C produces the following the aid of strength lines, which we shall call for short E lines or
field strength in a vacuum at a distance of 1 m from this charge: field lines. These lines are drawn so that a tangent to them at every] •~
point coincides with the direction of the vector E. The density of
1 1
E=. ....~__ .~a. i2=9x 101 VIm the lines is selected so that their number passing through a unit
area at right angles to the lines equals the numerical value of the
This strength in the Gaussian system is vector E. Hence, the pattern of field lines permits us to assess the
direction and magnitude of the vector E at various points of space
E = 7q = 34xnt1\20' 3 X 106 cgseE (Fig. 1.5).
The E linesof a point charge field are a collection of radial straight
hI
lines directed away from the charge if it is positive and toward it if
Comparing these two results, we find that
1 cgseE=3 X 10' VIm (1.17) • In Eq. (1.15), q stands for the charge setting up the field. In Eq. (1.18),
q stands for the charge experiencing the force F at a point of strength E.
2'"
20 Electricity and Magnetism Electric Field in a Vacuum 21
to another does not depend on the path. This work is
it is negative (Fig. 1..6). One end of each line is at the charge, and
lthe other extends to infinity. Indeed, the total number of lines J2
jintersecting a spherical surface of arbitrary radius r will equal the
A = F (r) e,.dl (1.21)
• • 0 0 1!. product of the density of the lines and the surface area of the sphere 12 1
4nr2 We have assumed that the density of the lines numerically
•
where dl is the elementary displacement of the charge q'. Inspection
\ equals E = (1I4n80) (q/r2). Hence, the number of lines is
c(1/4n80) (q/r2) 4nr2 = qleo. This result signifies that the number of of Fig. 1.7 shows that the scalar product e,. dl equals the increment
of the magnitude of the position vector r, Le. dr. Equation (1.21)
lines at any distance from a charge will be the same. It thus follows
can therefore be written in the form
J2
Jldl _ !
A = F (r) dr
12 J_
1
(comparewith Eq. (3.24) ofVol. I, p. 84]. In
troduction of the expression for F (r) yields
A = -ll!t.- TrZ2.!....-_1_ ( qq' _ qq' )
J
12 4neo rS - 4neo rl TI
"1
(1.22)
The work of the forces of a conservative 'I
....
field can be represented as a decrement of
the potential energy: Fig. 1.7
Fig. 1.6
A12= WP. t- WP.2 (1.23)
A comparison of Eqs. (1.22) and (1.23) leads to the following expres
fhat the lines do not begin and do not terminate anywhere except
sion for the potential energy of the charge q' in the field of the
tor the charge. Beginning at the charge, they extend to infinity (the
charge q:
charge is positive), or arriving from infinity, they terminate at thd
1 qq'
charge (the latter is negative). This property of the E lines is com WP=-4--+const
neo T
mon for all electrostatic fields, i.e. fields set up by any system of
The value of the constant in the expression for the potential energy
stationary charges: the field lines can begin or terminate only at
is usually chosen so that when the charge moves away to infinity
charges or extend to infinity.
(i.e. when r = 00), the potential energy vanishes. When this con
dition is observed, we get
W __1_ qq' \
1.6. Potential p - 4neo T (1.24)
Let us use the charge q' as a test charge for studying the field.
Let us consider the field produced by a stationary point charge q.
By Eq. (1.24), the potential energywhich the test charge has depends
At any point of this field, the point charge q' experiences the force
not only on its magnitude q', but also on the quantities q and r
1 q~' " determining the field. Thus, we can use this energy to describe the
F=-4 e,.= F(r) e,. (1.20)'1
neo r field just like we used the force acting on the test charge for this
purpose.
Here F (r) is the magnitude of the force F, and e,. is the unit vector Different test charges qi, qt, etc. will have different energies
of the position vector r determining the position of the charge q' W;, W;, etc. at the same point of a field. But the ratio Wp/qt will ~
relative to the charge q.
be the same for all the charges (see Eq. (1.24)]. The quantity
The force (1.20) is a central one (see Vol. I, p. 82). A central
W
field of forces is conservative. Consequently, the work done by the' cp=p (1.25)
forces of the field on the charge q' when it is moved from one point" qt
r
- r
~ ...r-~..r~--:.r~~..r..J
....r'
...{"'....r' :.c-~ ;..j :.c~ ~ •
