Electricity and magnetism

447 Pages·39.802 MB·English

by  Matveev A.N.

Electricity
and
Magnetism
A. N. Matveev
Mir Publishers Moscow
A. N. Matveev
Electricity and MagnetiSm
Translated from the Russian
by
Ram Wadhwa and Natalia Deineko
Mir Publishers Mo~~()w
First published f986
Revised from the f983 Russian edition
© H3AaTeJIbCTBO eBucmaa mKOJIa&, 1.983
© English translation, Mir Publishers, 1.986
Preface
This course reflects the present level of advancement in science and takes into
account the changes in the general physics curriculum.
Since the basic concepts of the theory of relativity are known from the course
on mechanics, we can base the description of electric and magnetic phenomena
on the relativistic nature of a magnetic field and present the mutual correspon
dence and unity of electric and magnetic fields. Hence we start this book not
with electrostatics but with an analysis of basic concepts associated with charge,
force, and electromagnetic field. With such an approach, the information about
the laws of electromagnetism,accumulatedby studentsfrom school-level physics,
is transformed into modern scientificknowledge, and the theory is substantiated
in the lightofthecurrentstate ofexperimental foundationsofelectromagnetism,
taking into account the limits of applicability of the concepts involved. Some
times, this necessitates a transgression beyond the theory of electromagnetism
in the strict sense of this word. For example, the experimental substantiation of
Coulomb's law for large distances is impossible without mentioning its con
nection with the zero rest mass of photons. Althoughthisquestion is discussed
fully and rigorously in quantum electrodynamics, it is expedient to describe its
main features in the classical theory of electromagnetism. This helps the stu
dent to acquire a general idea of the problem and of the connection of the mate
rial of this book with that of the future courses. The latter circumstance is quite
significant from the methodological point ofview.
This course mainly aims at the description of the experimental substantiation
of the theory of electromagnetism and the formulation of the theory in the local
form, i.e. in the form of relations between physical quantities at the same point
in space and time. In most cases, these relations are expressed in the form of
differential equations. However, it is not the differential form but the local
nature which is important. Consequently, the end product of the course are Max
we~l's equations obtained as a result of generalization and mathematical formu
lat~on of experimentally established regularities. Consequently, the analysis is
mainly based on induction. This, however, does not exclude the application of
the deductive method but rather presumes the combination of the two methods
of analysis in accordance with the principles of scientific perception of physical
Jaws. Hence, Maxwell's equations appear in this book not only as a result of
6 prefece
mathematical formulation of experimentally establishedregularitiesbutalso as •
an instrument for investigating these laws.
The choice of experimental facts which can be used to substantiate the theory
is not unique. Thus, the theory of electromagnetism is substantiated here with
and without taking the theory of relativity into account.The former approach
ispreferable, sincein thiscasethetheoryofrelativityappears as ageneralspace
time theoryon which all physical theories must be based. Such a substantiation
has become possible only within the framework of the new general physics cur
riculum.
An essential part of the theory is the determination of the limits of its ap
plicability and the ranges of concepts and models employed in it. These ques
tions, which are described in this book, are of vital importance. In particular,
the analysis of the force of interaction between charges in the framework ofthe
classical theory (i.e. without employing any quantum concepts) shows that the
classical theory of electricity and magnetism cannot be applied for analyzing
the interaction between isolated charged particles.
The author is grateful to his colleagues at Moscow State University as well
as other universities and institutes for a fruitful discussion ofthetopicscovered
in this book. He is also indebted to Acad. A. I. Akhiezer of the Academy of
Sciences of the Ukrainian SSR, Prof. N. I. Kaliteevskii and the staff of the
Department of General Physics at the Leningrad State University who care
fully reviewed the manuscript and made valuable comments.
A. Matveev
Contents
Preface 5
Introduction 13
Chapter t. Charge. Field. Force
Sec. t. Microscopic Charge Carriers . . • • . . • . . • • • • • • • . . . . .. 15
Classification. Electron. Proton. Neutron. What does the continuous distribution
of an elementary electric charge mean? Spin and magnetic moment.
Sec. 2. Charged Bodies. Electrostatic Charging •••••••....•.... f9
Thermionic work function. Energy spectrum of electrons. Fermi energy. Contact po
tential difference. Electrostatic charging.
Sec. 3. Elementary Charge and Its Invariance . . . . • • • . • • . . . . . .. 26
Millikan oil-drop experiment. Resonance method formeasurement ofcharge. Nonexis-
tence of fractional charges. Equality of positive and negative elementary charges.
Invariance of charge.
Sec. 4. Electric Current ••.••••••••...••.•.•..••... 30
Motion ofcharges. Continuous distribution ofcharges.Volume charge density. Charge
concentration. Surface charge density. Current density. Current through a surface.
Sec. 5. Law 01Charge Conservation • • • • • • • • • . • • . • • • . • . . .. 35
Twoaspectsoftheconceptofchargeconservation. Integralform ofchargeconservation
law. Divergence/Gauss' theorem. Differential form of charge conservation law•
."~ec. 6. Coulomb's Law •...•••.••..•..•.•.....•... 42
Experimental verification of Coulomb's law. The Cavendish method. Verification of
Coulomb's law for large distances. Verification of Coulomb's law for small distances.
Field form ofCoulomb's law. Electricfield. On the limits of applicability of the clas-
sical concept of field.
Sec. 7. SuperposltlonPrinciple •.•••.•.••.............. 50
Superpositionprincipleforinteractionofpointcharges. Fieldform ofthe superposition
principle. Test charges. Limits of applicability of the principle of superposition.
" Sec. M. Magnetic Field .•••••..•.•••••.•.•••••... 53
Inevitabilityofmagneticfieldgenerationdueto motionofchar~es. Interactionbetween
a pointcharge and an infinitelylong chargedfilament. Helativlstic nature of magne-
tic field. Forces of interaction between parallel current-earrying conductors. Unit of
current. Magnetic field.
", Sec. s. Lorentz Force. Ampere Force • • • . . . . • • • . • • . . . • • . .. 58
Transformation offorces. Lorentz force. Magnetic induction. Ampere force. Transfor
mation from steady volume currents to linear currents. Magnetic field of a rectilinear
current.
to.
/' Sec. Biot-Savart Law ...••••....•••.•.....•.... 63
Interaction between current elements. Onexperimental verificationofthe law ofinter-
8 Contents
action. Fieldform of interaction. Biot-Savartlaw. Force of interactionbetweenrecti
linear currents.
Sec. 11. Field Transformation ......................•. 69
Invariance of the expression for ferce in an electromagnetic field. Transformation of
fields. Application of formulas (11.15). Field of a point charge moving uniformly in a
straight line. Problems.
Chapter 2. Constant Electric Field
Sec. 12. Constant Electric Field 76
Fixed charge. The essence of the model. Limits of applicability of the model.
Sec. 13. Differential Form of Coulomb's Law . . . . . . . . . . . . . • . . .. 77
Gauss' theorem. Measurement ofcharge. Physical foundation of the validity ofGauss'
theorem. Differential form of Coulomb's law. Maxwell's equation for div E. Lines
of force. Sources and sinks of field E. Charge invariance.
Sec. 14. Potential Nature of Electrostatic Field . . . . . . . . . . . . • . . .. 82
Work in an electric field. Potentialnatureofa Coulomb field. Curl ofa vector. Stokes'
integral theorem. Differential form of the potential nature of the field. Gradient.
Scalarpotential. Ambiguityofscalar potential. Normalization. Expression ofwork in
terms of potential. Field potential of a point charge. Field potential of a system
of point charges. Field potential of continuously distributed charges. Field potential
of surface charges. Infinite value of the field potential of a pointcharge.Finitevalue
ofthe potential for a continuous charge distribution with a finite density.Continuity
of potential. Earnshaw's theorem.
-- Sec. 15. Electrostatic Field in Vacuum ...........••..••.•. 94
Formulation of the problem. Direct application of Coulomb's law. Calculation of
potential. Application of Gauss' theorem. Laplace's equation and Poisson's equation.
A very long uniformly charged circular cylinder.
Sec. 16. Electrostatic Fieldin the Presence of Conductors . . . . . . . . . . . .• 100
Differential form of Ohm's law. Classification of materials according to conductivity.
Absence ofelectricfield inside a conductor. Absenceofvolumechargesinside a conduc-
tor. Electrostatic induction. The field near the surface of a conductor. Mechanism of
creation of the field near the surface of a conductor. Dependence of the surface charge
density on the curvature of the surface. Charge leakage from a tip. Electroscopes and
electrometers. Metallic screen. Potential of a conductor. Capacitance of an isolated
conductor. Asystem ofconductors. Capacitors. Aconductingsphere in a uniformfield.
The field of a dipole. Method of image charges.
/ Sec. 17. Electrostatic Field in the Presence ofa Dielectric . . . . . ... . . . . •• 129
Dipole moment ofa continuouscharge distribution. Polarizationof dielectrics. Molec-
ular pattern ofpolarization. Dependence of polarizationon the electric field strength.
Theeffectofpolarization onelectricfield. Volume and surfacedensityofboundcharges.
Electric displacement. Gauss' electrostatic theorem in the presence of dielectrics.
Boundaryconditions. Boundary conditions for the normal component of vector D.
Boundaryconditionsforthetangentialcomponentof vector E. Refraction of field lines
at the interface between dielectrics. Signs of bound charges at the interface between
dielectrics. Method of images. Dielectric sphere in a uniform field.
Sec. 18. Energy of Electrostatic Field . . . . . . . . . . . . . . . . . . . .. 148
Energy ofinteraction between discrete charges. Energy of interactionfor a continuous
distribution of charges. Self-energy. Energy density of a field. Energy of the field of
surfacecharges. Energy ofchargedconductors. Energy of a dipole in an external field.
Energy of a dielectric in an external field.
Sec. 19. Forces in an Electric Field .. . . . . . . . . . . . . . . . . . • .• t56
Nature of forces. Force acting on a point charge. Force acting on a continuously dis
tributed charge. Forceacting on a dipole. Moment of force acting on a dipole. Volume
forces acting on a dielectric. Forces acting on a conductor. Surface forces acting on
a dielectric. Volume forces acting on a compressible dielectric. Calculation of forces
from the expression for energy. Problems.
Co~enb 9
Chapter 3. Dielectrics
Sec. 20. Local Field •.......••••......••......... 17Z-
The difference between a local field and an external field. Calculation of local field
strength.
Sec. 21. Nonpolar Dielectrics ...••.•..•...•...••.....• 175·
Molecular dielectric susceptibility. Rarefied gases. Dense gases.
Sec. 22. Polar dielectrics 177
Temperature dependence of polarization. Saturation field. Rarefied gases. Quantum
interpretation of polarization of polar gaseous dielectrics. Dense gases. Polar liquids.
Ionic crystals.
Sec. 23. Ferroelectrics ......•.................... 183"-
Definition. Hysteresis loop. Curie point.Molecular mechanism of spontaneous polar
ization. Dielectric domains. Antiferroelectrics.
Sec. 24. Piezoelectrics' ..........•................ 187
Properties of piezoelectrics. Longitudinal andtransverse piezoelectric effects. Mecha-
nism of piezoelectric effect. Inverse piezoelectric effect. Pyroelectrics. Problems.
Chapter 4. Direct Current
Sec.25.ElectricField in theCaseofDirectCurrents . . . . . . . . . . . . . . .. 191
The field in aconductor. The sources ofa field. Field outside a conductor. Surface
charges. Volumecharges.Mechanismofgeneratingdirectcurrents. Change in potential
along a current-earrying conductor.
Sec. 26. Extraneous Electromotive Forces . . . . . . . . . . . . . . . . . . .. 195-
The originofextraneous e.m.I, s. Mechanicalextraneous e.m.f. Galvaniccells. Voltaic
cell. Range of action of extraneous e.m.f, s. Law of conservation of energy. Polar
ization of a cell. Methods ofJdepolarization. Accumulator,
Sec. 27. Differential Form of Joule's Law. Work Done During the Passage of Current
and Power Developed 202:
Workperformedduringpassageofcurrent. Power. Differentialform ofJoule'slaw. The
sourceofenergyfor theworkdoneby current. DerivationofOhm'slawfromtheelectron
patternofelectricalconductivity. DerivationofJoule'slaw from theelectrontheoryof
electrical conductivity. Drawbacks of the classical theory of electrical conductivity.
Main features of quantum-mechanical interpretation of electrical conductivity.
.....~. 28. Linear Circuits. Kirchhoff's Laws ...............•... 206·
An isolated closed loop. Branched circuits. Kirchhoff's laws.
Sec. 29. Currentsin aContinuousMedium . . . . . . . . . . . . . . . . . . .. 20~
Formulation of the problem. Derivation of the formula. Conditions of applicability
of Eq. (29.6). Coaxial electrodes. Nonhomogeneous medium.
Sec. 30. Earthing of Transmission Lines . . . . . . . . . . . . . . . . . . .. 213
Formulationofthe problem. Calculationofresistance. Experimental verification. Step
voltage. Problems.
Chapter 5. Electrical Conductivity
Sec. 31. Electrical Conductivity of Metals . . . . . . . . . . . . . . . . . .. 218·
Theproofoftheabsenceofmasstransportby electriccurrentinmetals. The Tolmanand
Stewartexperiments. Onthebandtheory. Temperature dependence of resistance. Hall
effect. Magnetoresistance. Mobility of electrons. Superc.onductivity. Critical tempera-
ture. Critical field. Meissner effect. Surface current. Soft and hard superconductors.
The theory of superconductivity.
Sec. 32. Electrical Conductivi~of Liquids . . . . . . . . . . . . . . . . . .. 22&
Dissociation. Calculation ofelectrical conductivity. Dependence of electrical conduc-
tivity on concentration. Temperature dependence of electrical conductivity. Electro-
lytes.
10 Contents
.0. . •.
Sec, 33. Electrical Conductivity of Gases . . . . • . . • . . . . . . . 228
Self-sustained and non-self-sustained currents. Non-self-sustained current. Saturation
currentdensity. The characteristicofcurrent. Self-sustainedcurrent. The effect of vol-
ume charge. Mobility of charges. Comparison of results with experiment.
Sec. 34. Electric Current in Vacuum . • . • • • • . • • . . • . . . • . • . .. 232
Thermionic emission. The characteristicsof an electroncloud. Saturationcurrent den-
aity. Three-halves power law. Problems.
Chapter 6. Stationary Magnetic Field
Sec. 35. Ampere's Circuital Law . . . . . . . . . . . . . . . . . . . . . .. 240
Formulation of the problem.The integral form of Ampere's circuital law. Differential
form of Ampere's circuital law. Experimental verification of Ampere's circuital law.
The derivation of the differential form of Ampere's law by direct differentiation of
the Biot-Savart law•
.Sec. 36. Maxwell's Equations for a Stationary Magnetic Field . . . . . . . • • .• 245
Equation for div B. Maxwell's equations. The types of problems involved.
Sec. 37. Vector Potential ...•.•.................•.. 247
The possibility of introducing a vector potential. Ambiguity of vector potential. Po
tentialgauging. Equationfor vectorpotential. Biot-Savartlaw. The field ofan elemen-
tary current.
Sec. 38. Magnetic Field in the Presence of Magnetics . . . . . • • • . . . . •• 254
Definition. Mechanisms of magnetization. Magnetization. Vector potential in the
presence of magnetics. Volume density of molecular currents. Surface molecular
currents. Uniformly magnetized cylinder. Magnetic fieldstrength. Equation for the
magneticfieldstrength. Relationbetweenmagnetization and magnetic field strength.
Field in a magnetic. Permanent magnets. Boundary conditions forthefieldvectors.
Theboundaryconditionforthe normalcomponent ofvector B. The boundarycondition
forthetangentialcomponentofvectorH. Refraction of magneticfield lines.The measu
rement of magnetic induction.The fields ofa very long solenoidandauniformlymag
netizedverylongcylinder.The measurement ofpermeability, magnetic induction and
the field strength inside magnetics. A magnetic sphere in a uniform field. Magnetic
-shielding,
Sec. 39. Forces in 8 Magnetic Field • . . . . . • • . • . • • • • . • • • • •• 270
Forces acting on a current. Lorentz'sforce. The force and the torque acting on a mag-
netic dipole. Body forces acting on incompressible magnetics. Problems.
Chapter 7. Magnetics
~c.40. Diamagnetics •........•.•••...•.••••••.•• 277
Larmer precession. Diamagnetism. Diamagnetic susceptibility. Temperature indepen-
-dence of diamagnetic susceptibility. .
Sec. 41. Paramagnetics •.••.••••.••..••••...•..•• 282
Mechanism ofmagnetization. Temperature dependence of paramagneticsusceptibility.
Magnetic moments of free atoms. Magnetic moments of molecules. Magnetism due to
free electrons. Paramagnetic resonance.
Sec. 42. Ferromagnetics •.•.••••.••....•••.•.••••• 287
Definition. Magnetizationcurveandhysteresisloop. Permeabilitycurve. Classification
of ferromagnetic materials. Interaction of electrons. Basic theory of ferromagnetism.
Curie-Weiss law. Magnetization anisotropy. Domains. Domain boundaries. Magnetic
reversal. Antiferromagnetism. Ferrimagnetism. Ferromagnetic resonance.
Sec. 43. Gyromagnetic Effects ••••••••••••••••••••... 295
Relation between angular momentum and magnetic moment. Einstein-deHaasexper
iment. Barnett effect. Problems. .
Chapter 8. Electromagnetic Induction and Quasistationary Alternating
Currents
Sec. 44. CurrentsInduced in Moving Conductors • • • • • • • • . • . . . • • •. 300
Contents 11
Emergence of an e.m.I, in a moving conductor. Generalization to an arbitrary case.
A.c. generators. The law of conservation of energy.
Sec. 45. Faraday's Law of Electromagnetic Induction ••••.•.•••.••• 304
Definition. Physicalessenceofthe phenomenon. Aconductormovingin avaryingma
gnetic field. Application of the law of electromagnetic induction to a.c. generators.
See. 46. Differential Form ofthe Law of Electromagnetic Induction'•••..•.• 306
Differentialform ofFaraday'slaw. Nonpotential nature of induced electric field. Vec
tor and scalar potentials in a varying electromagnetic field. Ambiguity of potentials
and gauge transformation.
See. 47. Magnetic Field Energy ••••.•.•.••..••••••• .•• 309
Magnetic fieldenergyfor an isolatedcurrentloop. Magneticfieldenergyforseveralcur
rentloops. Magneticfieldenergy in the presence ofmagnetics. Magneticenergy density.
Inductance. The fieldofasolenoid. Energyofa magnetic in an external field. Calcula
tion offorcesfrom the expressionfor energy. Body forces acting on compressible mag
netics. Energy of a magnetic dipole in an external magnetic field.
See. 48. Quasistationary A.C. Circuits •••••••••••••••••••• 323
Definition. Self-inductance. Connection and disconnection ofan RL-circuit contatning
a constant e.m.I, Generation of rectangularcurrent pulses. RC-circuit. Connection
anddisconnectionofan RC-circuitcontaining a constant e.m.I, LCR-circui contain
ing a source ofextraneouse.m.I.s.Alternating current. Vector diagrams. Kirchhoff's
laws. Parallel and series connections of a.c, circuit elements. Mesh-current method.
Sec. 49. Work and PowerofAlternatingCurrent • • • • • • • • • • • • • • • • • 334
Instantaneouspower.Meanvalue of power. Effective (r.m.s.) values of current and
voltage. Power factor. Electric motors. Synchronous motors. Asynchronous motors.
Generation ofrotating magnetic field. Load matching with a generator. Foucaul
(eddy) currents.
Sec. 50. Resonances in A.C.Circuits ••••.••••••• 343
Voltage resonance. Current resonance. Oscillatory circuit.
Sec. 51. Mutual Inductance Circuit .••.•.•••••••..•••.• .• 347
Mutual Inductance. Equation for a system ofconductors taking into account the self
inductance and mutual inductance. The case of two loops. Transformer. Vector dia
gram ofatransformerat no-load. Vectordiagramofatransformerunderload.Autotrans
former. Transformer as a circuit element. Real transformer.
Sec. 52. Three-Phase Current .• . • • • • • • • • • • • • . . • • • • . . • • 353
Definition. Generation of three-phase current. Star connection of generator windings.
Delta connection of generator windings. Load connection. Generation of a rotating
magnetic field.
Sec. 53. Skin Effect ••• • . • . • . • • • • . • • . • • • . . . . • • . • • 357
Essence of the phenomenon. Physical pattern of the emergence of skin effect. Basic
theory.Skindepth. Frequencydependenceofohmic resistance'ofaconductor.Frequency
dependence of inductance of a conductor. . ~ .
Sec. ~4. Four-Terminal Networks •••.•••••.•••.•.••• ••• 361
Definition. Equations. Reciprocitytheorem. Impedance of a four-terminal network.
Simple four-terminal networks. Input and output impedances. Gain factor.
I. . . . . . . . . . -. . . . .
Sec. 55. Filters ••• • • • . • • • • • • • 366
Definition. Low-pass filter. High-pass filter. Iterative filter. Band filter.
Sec. 56. Betatron ..•••••••••••••••••••.••.•..•• 369
Function. Operating principle. The betatron condition. Radial stability. Vertical sta
bility. Betatron oscillations. Energy limit attainable in betatron. Problems.
Chapter 9. Electromagnetic Waves
Sec. 57. Displacement Current ••••.•••.••••.•..••..••• 3;6
The nature of displacement current. Why do wecall the rate of variation of displace
ment the displacement current density? Maxwell's equations including displacement
current, Relativistic nature of displacement current.