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ELECTRICITY AND

MAGNETISM BLEANEY AND

IJLEANKY

SECOND EDITION

Electricity

and Magnetism SECOND EDITION

B. B.

OXFORD

BLEANEY BLEANEY I.

and

ELECTRICITY AND

MAGNETISM BY B.

I.

BLEANEY

Fellow of St. Hugh's College, Oxford

AND B.

BLEANEY

Dr. Lee's Professor of Experimental Philosophy University of Oxford

SECOND EDITION

OXFORD AT THE CLARENDON PRESS 1965

£l°)S5l Oxford University Press,

I

t

Amen

« House, London E.C.4

GLASGOW NEW YORK TORONTO MELBOURNE WELLINGTON BOMBAY CALCUTTA MADRAS KARACHI LAHORE DACCA CAPETOWN SALISBURY NAIROBI IBADAN ACCRA KUALA LUMPUR HONG KONG

i

Oxford University Press, 1965

PRESTOM POLYTECHNIC

7 557

17601 fei-e-

FIRST EDITION 1957 REPRINTED LITHOGRAPHICALLY IN GREAT BRITAIN AT THE UNIVERSITY PRESS, OXFORD BY VIVIAN RIDLER, PRINTER TO THE UNIVERSITY FROM CORRECTED SHEETS 01 THE FIRST EDITION 1959 SECOND EDITION 1965

PREFACE TO THE FIRST EDITION I

In

Honour School of Physics in Oxford the authors have long felt the need for an up-to-date text on Electricity and Magnetism which would cover the whole field, both the theory and the practice. This book is an attempt to supply this need, and to make it as comprehensive as possible chapters have been included which may form part of a graduate course rather than an undergraduate course. The plan ofthe book is as follows the first eight chapters cover the fundamentals of the theory and include accounts of electrical conductors and magnetism at an elementary level chapters 9 to 11 deal with the theory of alternating currents and waves the next five chapters cover the experimental aspects of generators, radio, and alternating current measurements the teaching for the Final

:

;

;

;

devoted to fuller accounts of noise, dielectrics, conduct ors, and magnetism, and a chapter on magnetic resonance, with particular final section is

reference to measurements of

some fundamental constants. The writing of any book on electricity and magnetism is bedeviled by the question of units. The authors were brought up on the two centimetre-gramme-second systems, and the practical system. To t iese is now added the metre-kilogramme-second system, making four systems at present in use. General adoption of the m.k.s. system w^uld reduce this to one system, which is such an obvious advantage that the rationalized m.k.s. system has been adopted in this book. Unfortunately no general agreement on the definition of magnetization obtained

while this book was in preparation the authors have therefore adopted the definition which is closest to the c.g.s. systems, which has the advantage that magnetostatics is closely parallel to electrostatics. tThis ;

made to simplify as far as possible the transition from systems to the m.k.s. system, since many students who may wish to use this book will have been brought up on the former. For these students, a chapter on units has been included where methods are choice has been c.g.s.

numbered equations into their equivaThe choice of the rationalized m.k.s. system makes this translation more cumbersome, and the rather mlinor advantages of 'rationalization' are outweighed by the disadvantages of changes in the defining equations of a number of the fundamental quantities. These will disappear when the c.g.s. systems fall out of use, and the authors have therefore adopted the rationalized m.k.s. system detailed for translating all the

lents in

one of the

c.g.s.

systems.

to conform with present practice.

vi

PREFACE TO THE FIBST EDITION

—

The authors are much indebted to their colleagues in particular M. S. Bagguley, A. H. Cooke, J. H. E. Griffiths, H. G. Kuhn, and G. W. Series who have criticized parts of the manuscript and made many helpful suggestions to M. H. W. Gall, Esq., of Messrs. H. Tinsley & Co. Ltd., Professor L. F. Bates, F.R.S., and A. Hart, Esq., of Nottingham University, and to Messrs. L. J. Arundel and R. A. Kamper of the Drs. D.

—

;

Clarendon Laboratory, for the considerable trouble they took in obtaining the photographs for Figs. 7.3, 21.6, 17.3, and 22.7 respectively; and to a number of pupils who have read various chapters and eliminated

numerous errors. The authors are not so sanguine as to believe that no errors remain in other parts of the book, and they will be grateful to readers who inform them of any errors. B.

I.

B.

B. B. Clarendon Laboratory Oxford

April 1955

PREFACE TO THE SECOND EDITION Since

the

first

edition of this

book appeared in 1957 there has been and theoretical, in under-

considerable progress, both experimental

standing the electrical and magnetic properties of materials.

Much

of

advanced in approach for a book intended primarily for undergraduates or graduates starting research, but the authors have attempted to distil a suitable fraction of appropriate density for this is too

presentation at this level in the second edition. This is not always and the authors apologize both to those who find sections lacking in the simple clarity which the ideal textbook should possess, and to easy,

those who find that over-simplification has resulted in a lack of accuracy.

The plan of the book is substantially the same as in the first edition. The first eight chapters cover the fundamentals of the theory and include accounts of electrical conductors and magnetism at an elementary level; Chapters 9-11 deal with the theory of alternating currents

and waves the next four chapters cover the experimental aspects of radio and alternating current measurements the final part is devoted ;

;

to fuller accounts of noise, dielectrics, conductors, and magnetism, ending with a chapter on magnetic resonance. This part, which overlaps

with solid state physics, has been considerably rewritten; the section on semiconductors has been expanded into a separate chapter, with some account of the principles of junctions but stopping short of transistor circuitry ; that

on anti-ferromagnetism has been incorporated

new chapter which also includes ferrimagnetism and the rare-earth metals. The discussion of conduction in metals, paramagnetism and

in a

ferromagnetism, and magnetic resonance, has been considerably revised and somewhat enlarged. Material elsewhere has been pruned wherever possible to minimize the increase in overall size in particular the chapter ;

on electrical machines has been omitted, except for low frequency transformers, whose equivalent circuit is discussed at the end of Chapter 9 on alternating current theory. In the first edition the electromagnetic dipole moment of an elementary current circuit was defined in such a way as to retain H as the force (couple) vector on a magnetic dipole, while B is the force vector on a current. This was a compromise, intended to reduce the gap between the older c.g.s. system and the new m.k.s. system. However, the definition ju,/* 1 dS had obvious difficulties in ferromagnetic media, and led to inconsistencies between dia- and paramagnetism,

m=

PBEFACE TO THE SECOND EDITION

viii

in the numerator in one case where the atomic formulae contained and in the denominator in the other. In common with other books using the m.k.s. system, the authors have therefore adopted the == I dS in the second edition, which gives a more logical definition treatment in which B is the force vector both for currents and magnetic dipoles. This has necessitated considerable changes in Chapter 5, and the opportunity has been taken to revise the treatment to give a more rigorous approach. Amongst other minor changes a fuller treatment of spherical harmonics is included in Chapter 2, together with the ju,

m

multipole expansion.

The authors are much indebted to Drs. B. V. Rollin, R. J. Elliott, and R. A. Stradling, who read part of the manuscript and made suggestions for improvements also to many colleagues in Oxford and readers elsewhere who took the trouble to send comments on the first edition. With their help the authors have endeavoured to eliminate the errors that remained, but no doubt a fresh crop has been sown in the second edition, and the authors will be grateful to readers who inform them of such errors. ;

B.

I.

B.

B.B. Clarendon Laboratory

Oxford April 1964

;

ACKNOWLEDGEMENTS The

authors are indebted to the following for permission to use pub-

lished diagrams as a basis for figures in the text: the late Sir

K.

S.

Krishnan; G. Benedek; R. Berman; D. F. Cochran; S. Dresselhaus; G. Duyckaerts; R. D. Frauenfelder S. A. Friedberg; M. P. Garfunkel; ;

W.

E. Henry; A. F. Kip; C. Kittel; J. F. Koch; B. T. Matthias; K. A. G. Mendelssohn; D. E. Nagle; H. M. Rosenberg; C. G. Shull; J. S. Smart; J. W. Stout; R. A. Stradling; W. Sucksmith; R. W. Taylor P. Vigoureux W. E. Willshaw W. P. Wolf; American Institute ;

;

;

American Physical Society; Institution of Electrical Institute of Physics and the Physical Society (London)

of Physics;

Engineers

;

Royal Society (London) Bell Telephone Laboratories A.E.I. Research Laboratories G.E.C. Research Laboratories North Holland Publish;

;

ing Co.

;

;

CONTENTS

xii

5.

THE MAGNETIC EFFECTS OF CURRENTS AND MOVING CHARGES, AND MAGNETOSTATICS 5.1

126

Forces between currents

130

5.2 Magnetic shells

and magnetic media

135

problems

139

Steady currents in magnetic media

142

5.3 Magnetostatics

5.4 Solution of magnetostatic 5.5

5.6 Calculation of the magnetic fields of simple circuits 5.7

6.

Moving charges

in electric

and magnetic

fields

148 151

ELECTROMAGNETIC INDUCTION AND VARYING CURRENTS 6.1

Faraday's laws of electromagnetic induction

6.2 Self-inductance

and mutual inductance

158 161

6.3 Transient currents in circuits containing inductance, resistance,

165

and capacitance

7.

6.4

Magnetic energy and mechanical

6.5

Magnetic energy in magnetic media

175

7.1

Galvanometers, ammeters, and voltmeters; the wattmeter

179

7.2

Galvanometer damping

184

7.3

The

ballistic

galvanometer and fluxmeter

186

190

measurements

MAGNETIC MATERIALS AND MAGNETIC MEASUREMENTS 195

magnetism Diamagnetism Paramagnetism Ferromagnetism

8.1 Origins of

8.2 8.3

8.4

8.5 Production of

magnetic

198 201

204 fields

207

214

8.7

Measurement of magnetic fields Measurement of susceptibility

216

8.8

Experimental investigation of the hysteresis curve

221

8.6

8.9 Terrestrial

9.

172

DIRECT CURRENT MEASUREMENTS

7.4 Absolute

8.

forces in inductive circuits

magnetism

223

ALTERNATING CURRENT THEORY 227

9.1

Forced

9.2

Use of vectors and complex numbers

231

9.3

Tuned

236

9.4

Coupled resonant

9.5

Low-frequency transformers

oscillations

circuits circuits

243 247

CONTENTS 10.

ELECTROMAGNETIC WAVES 10.1 Maxwell's equations of the electromagnetic field 10.2.

10.3

10.4 10.5 10.6

Plane waves in isotropic dielectrics

dielectrics

269

265 267

277

278 281

FILTERS, TRANSMISSION LINES, AND WAVEGUIDES 11.1

Elements of filter theory

291

11.2

Some simple types

297

of

filter

waves on transmission lines Terminated loss-free lines Attenuation on lossy lines, and resonant lines Guided waves propagation between two parallel conducting

11.3 Travelling

302

11.4

306

11.5

11.6

11.7

311

—

planes

315

Waveguides

321

THERMIONIC VACUUM TUBES 12.1 Construction of the thermionic

12.2 12.3

vacuum tube

The diode The three-halves power law

12.4 Uses of the diode 12.5

The

triode

329 331

332 335 339

12.6 Characteristics of the triode

340

12.7 Equivalent circuit of the triode

343

12.8

Input impedance of the triode

The screen-grid tetrode 12.10 The pentode 12.9

13.

260 263

from the surface of a metal 10.8 The pressure due to radiation 10.9 Radiation from an oscillating dipole

12.

256

The Poynting vector of energy flow Plane waves in conducting media The skin effect Reflection and refraction of plane waves at the boundary of two

10.7 Reflection

11.

xiii

344 347

349

APPLICATIONS OF THERMIONIC VACUUM TUBES 13.1

Audio-frequency voltage amplifiers

351

13.2 Negative feed-back amplifiers

354

Audio -frequency power amplifiers 13.4 Radio -frequency amplifiers

359'

13.3

13.5 13.6

13.7 13.8

Tuned anode and tuned grid oscillators Power oscillators The Kipp relay and the multivibrator Amplitude modulation and detection

Frequency changing 13.10 Frequency modulation 13.9

13.11

Radio receivers

355 364 368

370 375

380 383

386

xiv

CONTENTS

14.

THERMIONIC VACUUM TUBES AT VERY HIGH FREQUENCIES 14.1 Effects of electrode

390

impedance

on input conductance Modified circuits and tubes for metre and decimetre wavelengths The klystron The magnetron

14.2 Effect of transit time

393

14.3

395

14.4 14.5

15.

wave tubes

15.1

of voltage, current, and power

15.2

of impedance at low frequencies

423

of impedance at radio frequencies

429

of frequency

434

and wavelength

of dielectric constant

442

of the velocity of radio waves

445

Brownian motion and fluctuations

452

16.2 Fluctuations in galvanometers

454

16.3 The relation between resistance noise and thermal radiation 16.4 Shot noise

458

16.5 Design of receivers for

16.6

optimum performance (minimum

462

noise

figure)

466

Measurement of receiver noise

470

THEORY OF THE DIELECTRIC CONSTANT 17.1 Molecular structure

and the

dielectric constant

475

17.2 Dielectric constant of non-polar gases

477

17.3 Static dielectric constant of polar gases

480 483

17.4 Dispersion in gases 17.5 Static dielectric constants of liquids

18.

414

FLUCTUATIONS AND NOISE 16.1

17.

412

ALTERNATING CURRENT MEASUREMENTS Measurement Measurement 15.3 Measurement 15.4 Measurement 15.5 Measurement 15.6 Measurement

16.

405 411

14.6 Crystal diodes 14.7 Travelling

400

and

solids

488

17.6 Static dielectric constants of polar liquids

491

17.7 Radio-frequency dispersion in polar liquids

493

17.8 Scattering

498

ELECTRONS IN METALS 18.1 Kinetics of free electrons in metals

18.2 18.3

504

The energy band approximation Conductors and insulators on the band theory

18.4 Specific heat of the conduction electrons 18.5 Electrical 18.6

and thermal conductivity of metals

The Hall effect and paramagnetism of conduction

18.7 Dia-

506 515 517 521

528 electrons

529

CONTENTS 19.

xv

SEMICONDUCTORS and extrinsic conductivity Elementary and compound semiconductors

19.1 Intrinsic

19.2

19.3 Electron distribution

538 543

level

19.4 Optical properties

548

19.5 Transport properties

553

19.6 Metal-semiconductor junctions

560

19.7 19.8

20.

and the Fermi

536

The p-n junction The junction transistor

565

569

THE ATOMIC THEORY OF PARAMAGNETISM 20.1

A general precession theorem

574

20.2

The vector model of the atom Magnetic moments of free atoms The measurement of atomic magnetic moments

576

20.3 20.4

585

—the

Stern-

Gerlach experiment 20.5 Curie's law

590

and the approach to saturation

591

—the 4/ group —the 3d group 20.8 Susceptibility of paramagnetic solids—strongly bonded compounds 20.9 Electronic paramagnetism—a summary 20.6 Susceptibility of paramagnetic solids

593

20.7 Susceptibility of paramagnetic solids

598

20.10 Nuclear

21.

moments and hyperfine

609

610

structure

FERROMAGNETISM 21.1

21.2

Exchange interaction between paramagnetic ions The Weiss theory of spontaneous magnetization

618

622

21.3 Ferromagnetic domains

626

21.4

The gyromagnetic

634

21.5

Thermal effects in ferromagnetism Measurement of the spontaneous magnetization

21.6

effect

637

M as a func-

tion of temperature

640

21.7 Foundations of the theory of ferromagnetism

644

waves 21.9 Mechanisms of exchange interaction

648

21.8 Spin

22.

606

650

ANTI-FERROMAGNETISM AND FERRIMAGNETISM 22.1 Anti-ferromagnetism

22.2

The molecular field

657

—two sub-lattice model

22.3 Ferrimagnetism

22.4

The lanthanide

659 664

('rare earth')

22.5 Neutron diffraction

metals

670 673

CONTENTS

Xvi

MAGNETIC RESONANCE

23. *'

23.1

The magnetic resonance phenomenon beams and nuclear magnetic resonance

23.2 Molecular

677 681

23.3 Nuclear magnetic resonance in bulk material

685

23.4 Relaxation effects in nuclear magnetic resonance

689

23.5 Applications of nuclear resonance

692

23.6 Electron magnetic resonance in atomic

beams

697

23.7 Electron magnetic resonance in solids

703

23.8 Cyclotron resonance with free charged particles

710

23.9 Cyclotron resonance of charge carriers in semiconductors

717

23.10 Azbel-Kaner resonance in metals

722

UNITS

24i

24.1 Unrationalized c.g.s. systems

729

24.2 Practical units

733

24.3

The

24.4 Conversion factors from

rationalized m.k.s. system

24.5 Equivalent equations in unrationalized

Appendix A.

733

rationalized m.k.s. system

c.g.s.

systems

737

VECTORS

A.l Definition of scalar and vector quantities

744

A.2 Vector addition and subtraction

744

A.3 Multiplication of vectors

745

A.4 Differentiation and A.5 The divergence of a vector

747

A.6 The curl of a vector

750

A.7 Laplace's operator

751

A.8 Stokes's theorem

751

A.9 The divergence theorem

752

A. 10 Transformation from a rotating coordinate system

753

A. 11 Larmor's theorem

753

integration of vectors

;

736

THE UNIQUENESS THEOREM Appendix C. NUMERICAL VALUES OF THE FUNDAMENTAL CONSTANTS Appendix D. SOME ATOMIC FORMULAE IN M.K.S. UNITS INDEX

Appendix B.

748

755

756 757

759

ELECTROSTATICS 1.1.

The

electrical nature of

I

matter

The fundamental laws of electricity and magnetism were discovered by experimenters who had little or no knowledge of the modern theory of the atomic nature of matter. It should therefore be possible to present these

laws in a textbook

by dealing

at

first

and then introducing gradually the

purely in macroscopic phenomena

details of atomic theory as required.

In this way the subject might be developed almost in the historical order of discovery, and these opening sentences would talk of amber and cat's fur. It is more interesting, however, to discuss here and there throughout this book the interpretation of the macroscopic laws in terms of present atomic theory. In the later chapters a considerable knowledge of such theory will be assumed, since to give an adequate account of it would greatly increase the size of the book. This will not be attempted, but in the following paragraphs a summary is presented of what may be regarded almost as common knowledge of the nature of the atom. On modern theory the atom consists of a central core, or nucleus, of diameter about 10 -12 cm, surrounded by a number of electrons. These electrons move round the nucleus in orbits whose diameter is about 10~ 8 cm, and these determine the size of the atom. The nucleus contains two kinds of particles: protons, which are particles roughly 1836 times as heavy as electrons, but with a positive electric charge +e, and neutrons, of very nearly the same mass as protons, but with no electric charge. The number of electrons surrounding the nucleus is equal to the number of protons, and each electron has a negative charge — e, so that the atom as a whole is electrically neutral. The physical and chemical properties of the atoms are determined by the number of electrons they contain, and hence the number of protons in the nucleus is characteristic of a particular element. The number of neutrons is roughly equal to the number of protons in light elements but is over 1-5 times as great in the heaviest elements.

The mass of the nucleus

is

determined by the total number of protons and neutrons, and a given element may have several stable forms of different nuclear mass, corresponding to nuclei with different numbers of neutrons, but the same number of protons. These are called isotopes. Thus the oxygen nucleus 851110 B

ELECTKOSTATICS

2

I

[1.1

has 8 protons, and there are three stable isotopes, oxygen 16, 17, and 18, with 8, 9, and 10 neutrons respectively, although the percentage of isotopes 17 It

is

now

and

very small. established that the electronic charge is the fundamental 18 occurring in nature

is

and all charges are integral multiples of -f e or — e. It is assumed that the electron is indivisible, and is a fundamental particle of matter so also is the proton. We may summarize the properties of electron, proton, and neutron as follows: unit of charge,

therefore

;

Particle

Electron

Proton Neutron e

=

1-602

X

Charge

—e +e

KT1B coulomb; m =

Mass

m 1836m 1838m 0-911

x

10

-2 '

g.

Since charges of opposite sign attract one another, the electrons are to the atom by the electrical attraction of the protons in the

bound

nucleus.

The forces which hold the nucleus together are of different and operate only at very short ranges, of the order of the

character,

nuclear diameter.

Conductors and insulators

For the purpose of electrostatic theory all substances can be divided into two fairly distinct classes: conductors, in which electrical charge can flow easily from one place to another; and insulators, in which it cannot. In the case of solids, all metals and a few other substances such as carbon are conductors, and their electrical properties can be explained by assuming that a number of electrons (roughly one per atom) are free to wander about the whole volume of the solid instead of being rigidly attached to one atom. Atoms which have lost one or more electrons in this way have a positive charge, and are called ions. They remain fixed in position in the solid lattice. In solid substances of the second class, insulators, each electron is firmly bound to the lattice of positive ions, and cannot move from point to point. Typical solid insulators are sulphur, paraffin wax, and mica. When a substance has no net electrical charge, the total numbers of positive and negative charges within it must just be equal. Charge may be given to or removed from a substance, and a positively-charged substance has an excess of positive ions, while a negatively -charged substance has an excess of electrons. Since the electrons can move so much more easily in a conductor than the positive ions, a net positive charge

ELECTKOSTATICS

1.1]

is

usually produced

I

3

by the removal of electrons. In a charged conductor

move to positions of equilibrium under the influence of the forces of mutual repulsion between them, while in an insulator they are fixed in position and any initial distribution of charge will the electrons will

remain almost

indefinitely. In a good conductor the movement of almost instantaneous, while in a good insulator it is extremely slow. While there is no such thing as a perfect conductor or perfect insulator, such concepts are useful in developing electrostatic theory; metals form a good approximation to the former, and substances such

charge

is

as sulphur to the latter.

Coulomb's law and fundamental definitions The force of attraction between charges of opposite sign, and of repul-

1.2.

sion between charges of like sign,

found to be inversely proportional them to be located at points), and proportional to the product of the magnitudes of the two charges. This law was discovered experimentally by Coulomb in 1785. In his apparatus the charges were carried on pith balls, and the force between them was measured with a torsion balance. The experiment was not very accurate, and a modern method of verifying the inverse square law with high precision will be given later (§ 1.3). From here on we shall assume it to be exact. If the charges are q x and q 2 and r is the distance between them, then the force F on q2 is along r. If the charges are of the same sign, the force is one of repulsion, whose magnitude is is

to the square of the distance between the charges (assuming

,

r2

The vector equation

for the force is

F==Cf £i|2 r 3

/

L1 )

r

Here F, r are counted as positive when directed from qx to q2 Equation (1.1) is the mathematical expression of Coulomb's law. The units of F and r are those already familiar from mechanics; it remains to determine the units of C and q. Here there are two alternatives: either C is arbitrarily given some fixed numerical value, when .

equation

may be

used to determine the unit of charge, or the unit of charge may be taken as some arbitrary value, when the constant C is to be determined by experiment. The electrostatic system of units (e.s.u.) makes the use of the first method. The force F is in dynes, and (1.1)

ELECTROSTATICS

4

the distance r in centimetres

I

[1.2

both are measured in the centimetre-

(i.e.

gramme-second system), and the constant G is set equal to unity. Then q v q 2 are measured in e.s.u. of charge, the unit being denned as that charge which repels an equal charge at a distance of 1 cm in vacuo with a force of 1 dyne. In the metre-kilogramme-second-coulomb system (m.k.s.), which will be used throughout this book, the unit of charge is the coulomb, the standard practical unit of charge (equal to one-tenth of the unit of charge in the electromagnetic system of units). For the present purpose

may be

it

regarded as denned by the charge required

to deposit a certain mass of silver in a silver voltameter, being thus

denned in an arbitrary manner in the same way as the standard metre and standard kilogramme. Equation (1.1) for Coulomb's law is then analogous to that for gravitational attraction, except that it deals with electrical charges instead of masses, and the unknown constant of pro-

must be determined by experiment. In the 'rationalized' metre-kilogramme-second-coulomb system, the constant G is written as l/47re the factor in being introduced to simplify certain equations

portionality

,

which appear

later in the theory.

F where €

is

F

is

Equation

= -LM?r, 47re r3

in newtons, r in metres,

known

(1.1) therefore

and q

becomes (1.2)

in coulombs.

The quantity

as the 'permittivity of free space' (see § 1.5); its experi-

mental value is found to be (see § 7.4) 8-85 X 10~ 12 coulomb 2 newton -1 metre -2 (this unit can be more conveniently called farad metre -1 (see 10 5 dyne, and 1 metre 10 2 cm, it may § 1.6)). Since 1 newton 9 2-998 X 10 e.s.u. readily be shown that 1 coulomb

=

=

=

Electric field

and

electric potential

The force which a charge q 2 experiences when in the neighbourhood of another charge q x may be ascribed to the presence of an 'electric field' E produced by the charge qx Since the force on a charge g 2 is proportional to the magnitude of q%, we define the field E by the equation .

F From

Eg,.

(1.3)

and Coulomb's law it follows that E does not a vector quantity, like F. From equation (1.2) we

this definition

depend on q2 and ,

is

find that

is

=

the electric

E = -^r field

due to the charge qx

.

(1.4)

ELECTROSTATICS

1.2]

5

I

is moved an infinitesimal distance ds in a work done by the field is E.ds, and the work done against the field is —E.ds. This follows from the fact that the force on unit charge is equal to the electric field E. The work done against the field in moving a unit positive charge from a point A to a point B

If a unit positive charge

E, then the

field

will therefore

be

B

= -J E.ds. A

V This is

is

a scalar quantity

due to a

Fig.

1.1.

known

as the electric potential. If the field

single charge q at 0, as in

Kg.

1.1,

E

then the force on unit

Calculation of the potential difference between points of a point charge q at O.

A

and B due to the

field

charge at an arbitrary point P is along OP, and ds is the vector element PX PZ Now E.ds = E cos 6ds = Edr, and hence .

= - f**= --*- f * = JL(I_I). 47re J r 47re rj J

VB -YA

5

\r-

2

ri

Thus the

The

between A and B depends only on the A and B, and is independent of the path taken between them.

difference of potential

positions of

potential at a point distance r from a charge q is the work done up unit charge to the point in question from a point at zero

in bringing potential.

By

distance from

convention, the potential all charges,

that

is,

is

7=0

taken as zero at an for r

potential at a point distance r from a charge q

7 The difference ds apart is

in potential

dy

= _E

ds

=

=

A falls off

and we can

P

4weo cos0LrJ- 2

as r~ n

.

But the

solid

therefore write

LI r^- 2J"




sphere can be divided into elementary areas in this way, and the vector will not be zero unless all the individual dE are zero, since there will be a resultant towards the nearer portion of resultant of the fields at




an(i

and hence

so also

€=l+ x D = ee E,

we have

is

D.

# we write (1.21)

(1-22)

where e is the 'dielectric constant' and x is the susceptibility; for vacuum (or air, for most purposes), x °> an(i e = 1. The ratio of D to E is

=

known

D/E = e and thus e is the When a medium is present D/E is increased

as the 'permittivity'; in free space

'permittivity of free space'.

,

by the factor e, known also as the 'relative permittivity'. If we have a single point charge q in a uniform dielectric of constant e, we may apply Gauss's theorem over a sphere of radius r with centre at q. Then the surface integral reduces to 4irr zD = 2a/3. 1.12.

A sphere carrying a charge density

dielectric

medium. Verify equation

and allowing the radius to change

tj

(1.37)

per unit area is immersed in an infinite by using the principle of virtual work

infinitesimally.

1.13. Assuming that the total charge Ze of an atomic nucleus is uniformly distributed within a sphere of radius a, show that the potential at a distance r from the

centre

(r

Show


/ee

no

ee

E. Hence (2.1)

.

free charge present,

0.

(2.2)

which has its simplest form in Cartesian coordinates, where Poisson's equation becomes scalar operator,

ew 8W 8*V_ 8x 2+ 8y 2± 8z*~ Two

p

^^

'

ee

other coordinate systems will be considered. These are spherical

polar coordinates, where Poisson's equation becomes

r 2 8r\

and

8r )

Tr

2

sin Odd\

86 J

cylindrical polar coordinates,

\d_(8V r 8r\

8r )r)

^

2

sin 2 df*

~

ee

'

K

'

where we have

+ r*de + 8z* ~ z

'

(2.5)

ee

i

In principle, equation (2.1) enables us to calculate the potential distribution due to any given set of charges and conductors. A formal solution of Poisson's equation can be found,

p dr

J

(2.6)

but this holds only in a vacuum or an infinite dielectric medium. If there are conductors present we should have to allow for the effect of the charge distribution on their surfaces, but we do not in general know what this distribution is. We have therefore to resort to a number of special methods, but we must be sure that any solution we obtain which 851110

D

ELECTROSTATICS

34 satisfies

the boundary conditions

is

II

[2.1

the correct and only answer. That

shown by an important theorem, known as the Uniqueif two different potential distributions are assumed to satisfy Laplace's equation and the boundary conditions, their difference is zero. We now discuss a number of methods for the case of no free charges, where the solutions needed this is the case is

ness Theorem. This theorem (see Appendix B) shows that

are of Laplace's rather than Poisson's equation.

The required

solution

may be

a

sum

of a

number of functions, each

of which satisfies Laplace's equation; for, if the functions

Vx

,

V%,...,

Vn

are each individual solutions of Laplace's equation, then

V where a 1; a 2 ,.„, an are a

A series

= a V +a V +...+an Vn 1

1

a

1

,

set of numerical coefficients,

of functions, each of which

is

is

also

a solution.

a solution of Laplace's equation,

may sometimes be found by making use of the fact that if Vx is a solution, any differentials of T^ with respect to the space coordinates. Thus in Cartesian coordinates the functions dVJdx, dVJdy, dVJdz, dWJdx* BWjBxdy, etc., all satisfy equation (2.2) if Vx does. The proof of this can

so also are

be seen from a single example. On partial differentiation of equation (2.3) with respect to x, we have (setting p = 0)

= 1/^4.^1 -t-S r ~

8z\dx*~

dx*\dx )

dy 2

"*"

t

~dz*j

8y 2 \8x J

"*"

8z 2 \8x )

\8x}'

is immaterial when x, y, z are independent coordinates. The value of this method lies in the fact that once a series of functions which satisfy equation (2.2) is established, any linear

since the order of differentiation

combination of these functions may be taken, and if they can be chosen in such a way as to satisfy the boundary conditions by adjustment of the coefficients, they give the unique solution to the problem.

In theory, any problem involving electrostatic fields may be solved finding a solution which satisfies equation (2.2) and gives the right boundary conditions. In practice the problem is almost insoluble by ordinary mathematical methods except in cases where there is a high degree of symmetry. These may be handled by the use of a series of known functions, and some examples of this method are given below. We shall consider also another special method which can be applied to the case of one or two point charges near to a conducting surface of simple shape. Though a number of other problems may be handled by

by

ELECTKOSTATICS

2.1]

II

35

mathematical methods which are beyond the scope of this volume (see the general references at the end of Chapter 1), most of the problems met with in practice, such as the design of electron guns to give a focused beam in a cathode ray tube, are dealt with either by use of approximate solutions, or by plotting the lines of equal potential using a scale model as described in Chapter 3.

Solutions of Laplace's equation in spherical coordinates

2.2.

We shall consider first the pendent of

where

V

=

If

we

by

i*,

case of spherical coordinates,

we have symmetry about the

initially that

Then Laplace's equation reduces

.

has been written for cos#. where P is a function of /*

/z.

r'Pj,

t

V is inde-

to

This has solutions of the form

=

cos 6 only,

substitute such a function in equation (2.7),

we

and assume

polar axis so that

and I is an integer. and divide through

obtain Le^endre's differential equation for

P

t

|((i-rtf}+W+Di!=o. It

is

readily seen that replacing

= P_

altered, so that Pj

ft+1) ;

I

that

by

is,

V

— (£+1) leaves this equation un= t^Pj and V = r-f+^Pj are both

solutions of equation (2.7).

Solutions of Legendre's equation may be obtained by standard methods, but a quick alternative method is as follows. We know that

V

=

1/r is

a solution, and hence so

is

any partial derivative of this such

as (BVjdz) under the conditions x, y constant.

we have

2r{8rj8z)

=

2z

when

x,

Since r 2

= x +y +z 2

2

2 ,

y are kept constant, so that

f-1

-|^)

Hence

(-|=^|^1

= _| = r -2 CO s0

along the polar axis, so that z = r cos 9. The two and r~ 2 cos 6 are the first two types of solution in the inverse powers r~fJ +v> and correspond to values of I = and 1 respectively. Thus P = 1, and Px — cos0. Further functions may be

if

we take

functions

z to lie _1 r

V

=

,

generated

by

successive differentiation; thus (a/a2)(z/r3 )

= (1 — 3z /r>- = (1 — 3cos 0)»2

3

2

3

ELECTROSTATICS

36

gives the next function,

*°rP

?

which

is

II

proportional to

m

is

1

[2.2

Pz A general formula .

,

s-mkjv-

1 *'

(2 8) -

where the numerical coefficients are such that P = 1 at /x = 1, i.e. at 0. The first few functions are given in Table 2.1, together with the radial functions r~(Z+1) and r1 with which they combine to give solu}

=

tions of Laplace's equation.

Table Some

2.1

spherical harmonic functions

Legendre function

Function of r

Po-l P1 = cos P = i(3cos 0-l) P = £(5cos 0-3cos0) P4 = $(35 cos 0-3O cos 0+3) P5 = |(63 cos 0-7O cos 0+15) P» = *(231 cos 0-315 cos 0+lO5 cos

r-i r-z -3

a

r y—4

2

s

3

2

4

5

3

6

4

8 0-

-5)

1

r r2

rS

p~ 5

^.4

j.-6

r5

r-7

r

Associated Legendre functions

Table

2.1 clearly

equation, since x, y.

does not contain

we can

find others

all possible solutions

by

of Laplace's

differentiation with respect to

For example,

also be a solution. The fact that it contains shows that it is not a solution of equation (2.7), but of the more general equation which includes the dependence on This is

must

.

dW =

m+ihO8r)

x

'

3/4

{l—tj?)d

fy\

2

o,

(2.9)

where we have again written for cos 9. As before, we assume that there exists a solution of the form V = r ®®, where 0,

—T~ Ij

I

t,

—t— A

1 1

\"/V!^

I

—+—

is

(@O)

J+^'Wi+r^-*

—

I

differential equation for

.

PyPz as * ne components of the electric dipole moment of the charge distribution. This gives a general definition of the dipole moment of

a charge distribution, the components being

Px If

= J*

P x dr>

Pv

= j PV dr,

we have a number

pz

= j pzdr,

or

p

=

r Jp

dr.

(2.25)

of point charges rather than a continuous distri-

by a summation (a dipole consistand opposite charges separated by a small distance, as defined in Chapter 1, is thus a special case). Note that in equation (2.24) we have implicitly assumed that the first differentials of V are bution, the integrals can be replaced

ing of two equal

constant over the region occupied by charge, since only then can take them outside the integration. If the charge distribution has reflection symmetry in the plane z

we

= 0,

the charge density p at the point (x, y, z) is the same as that at the point {x,y, —z), the component pz of the dipole moment will be that

is, if

zero, since in the integral

j pz dr the contributions from the points and (x,y, —z) will be equal and opposite. Thus in a diatomic molecule, an electric dipole moment can exist parallel to the fine joining the two nuclei if they are different (i.e. if the molecule is heteronuclear, such as HC1), but not if they are identical, as in a homonuclear molecule (H 2 ,C12 ). (x,y,z)

Similar considerations apply to p x p of course, and show that a y diatomic molecule can have no electric dipole moment perpendicular to the internuclear axis. The method can be extended to more compli,

cated molecules, such as

CH3 C1, C 2H C6 H6 6,

.

—

r

ELECTKOSTATICS

40

II

[2.3

In an atom or nucleus the charge distribution is expected to have reflection symmetry in three mutually perpendicular planes, so that there will be no permanent electric dipole moment in any direction. Note that three such reflections change the point (x,y, z) into ( x, — y, — z) and are equivalent to inversion through the origin. The assumption we have made about the charge distribution is equivalent to assuming that the system is 'invariant under the parity operation', i.e.

that

properties are unaltered

its

by

inversion.

B=

(S,

Ojj, 4,IS )

^1b

(a)

Eia. 2.1. Expansion of the potential at A due to a point charge q B bAB (or of the potential at due to charge q^ at A) using spherical harmonics. In (b), the points A, B are not necessarily in the plane 0.

B

=

Expansion in spherical harmonics Similar considerations can be applied to the higher terms in equation (2.23), but

obvious that the large number of terms makes

it is

the expansion in Cartesian coordinates clumsy to handle. fore turn to another

We

there-

method of expanding the potential of a point charge

which makes use of spherical harmonics. Assume that we have a charge at the point B and wish to know how its potential varies in the neighbourhood of the point O, e.g. at the point

A

(Fig.

2.1(a)).

This requires evaluation of the quantity

(R 2 —2Brcosd-\-r i )~ i which is the inverse of the distance AB. If r this function can be expanded in powers of (rjB): ,

1

IR—

=


+---+7^ p -

ELECTKOSTATICS

2-3]

Here the functions (2.8),

A

P

t

41

are the Legendre functions defined

as can readily be verified

by

by equation

direct expansion for the first terms.

more general formula can be found where the points A, B in (r, 9 A A ) and (R, 9B> B ) respectively. In 2.1 (b), Oz is the polar axis, and the lines OA, OB (which are not

spherical coordinates are at Fig.

II

,

(f>

make angles 6A 6B with it; the angle AOB is denoted by AB It can be shown that the Legendre function P (cos d AB ) can be expressed in terms of A) A and 6B B by the formula necessarily coplanar)

,

.

z

,

pKoob^)

= ^fj-ir%tm {eA = I {~^ m] Cl>m

where the function is

also listed in Table 2.2.

at the point

A

Oi,

,4>

A )Y _m {eB ,$B ) lt

m {SA ,A )Ch _m (eB ,B ),

= U?L\\m

By

(2.27)

(2.28)

the use of equation (2.27) the potential B may be written as

due to a charge qB at

47r«

=

|R-r|

2 ^w"2 °I=Om=-I

l

^4ri G^ e^^ ciM eB,B)-

(2-29)

if we take iasa variable point, the potential at due to the charge qB at has a series of components, and the magnitude of the component which varies as rlG (dA A ) is determined by the lm value of R- Ci -m {0B, 4>b) at the point B. We may equally well take

A

This shows that

B

,

i

B to be a variable point at which we wish to find the potential due

to

a charge qA atA; this is given by equation (2.29) on replacing qB by qA The potential at B then has a series of components, where the magnitude of the component varying as R-Q+VQ m (6B B ) is determined by the value of r*CUm (dA A ) at the point A. .

,

,

(f>

If we have a number of point charges qB at large distances, the magnitude of a given component in the potential at A near O can be found

by the summation

2 Qb B_a+1)Q,- m (^B, B •

b)>

an example of which is given

Problem 2.18. Conversely, if we have a number of point charges qA close to 0, the magnitude of a given component in the potential at a distant point such as can be found by the summation rl (> I n either case the summation is replaced by an ^,^A ^i,m(^A> l A)-

in

B

A

integration if

we have a continuous

distribution of charge.

Note that

ELECTBOSTATICS

42

II

[2.3

the series expansion for the potential does not assume that r *J ^) ^, Bi,- m = j {-l)™pB B-^C - m (eB

(2-32)

ft*

lt

The

quantities

the multipole

and

it

A l>m may

,

fa) drB

(2.33)

.

be regarded as defining the components of

moments of degree

of the charge distribution near 0, will be seen that they interact only with the conjugate comI,

Bl _m which have the same value of I, to. The monopole component (I = 0) contains only one term, while the dipole (1=1) components contain three terms which may easily be shown (Problem 2.15) to give the same interaction as equation (2.24). In general the interaction energy involving r 1 and R-Q +1 contains (2Z+1) terms, but the advantage of the quantities A lm Blm is that they can be expressed in terms of functions Clm which are tabulated. We shall go no further than the quadrupole terms (I 2), which can be written out using Table 2.2. It can then readily be verified that, for the particular case where either charge distribution is symmetrical about the axis 0, all the terms in equation (2.31) vanish except that with m 0. If pA has such symmetry (i.e. it is independent of $), its quadrupole interponents

>

,

=

=

=

ELECTROSTATICS

2.3]

II

43

action can be expressed in terms of a single component

Ao = j Pa

W

where the quantity (writing

Q

ouMa - 1) drA =

r cos 6A

frQ,

(2.34)

= z)

= -^jpA{W-r )dTA %

(2.35)

called the 'quadrupole moment' of the charge distribution, and has the dimensions of an area. It might be expected that one would take q j p A drA the total charge in the distribution, but for a nucleus,

is

=

,

by convention,

q is taken as the charge on a single proton (not the total nuclear charge), and Q is expressed in terms of the unit of a 'barn' 10~ 24 cm2 This unit is chosen because it is of the same order as the

=

.

square of the nuclear radius

be of order lO -16

(for

an atom the quadrupole moment would

cm 2).

It has already been shown that invariance under the parity operation excludes the possibility of permanent electric dipole moments in atoms nuclei. This may be expressed more generally using spherical harmonics. Inversion through the origin is equivalent to changing the point (r,9,) into (r.w— 0,tt+ ). Since

and

we have and

it

dJr-9, "+$)

=

=

Q

(- 1)^(0,

),

(2.36)

follows that

| pr*Cl>m dr

if 2 is

odd.

(2.37)

Thus invariance under the parity operation excludes the possibility of electric multipole moments of any odd degree. The form in which the interaction energy is expressed in equation (2.31) is very suitable to a case where one charge distribution (such as that of a nucleus) is confined to a small volume, but interacts with another charge distribution which is comparatively far away (such as the atomic electrons). The series then converges very rapidly, since (r/B) 1 experimentally no interactions with nuclear electric multipoles higher than the quadrupole have been detected. The convergence is


oo. In order to make V = at r = B for all values of V=

6, it

seems likely that

we can only add terms which vary with the same power we try as a solution

V It

is

that

= —E

r cos

is,

A=E B

.

= —E J

o

field

if

E B = AB~ Z

;

rcos0(l--.R3 /r3 ).

This shows that the potential outside the sphere

form

Hence (2.38)

boundary condition Hence we have

V

8.

6+Ar- 2 cos 9.

clear that this satisfies our 3

of cos

together with that of a dipole of

is

that due to the uni-

moment p

=

47ree

E B3

situated at the centre of the sphere. Inside the sphere a solution of the

B

ELECTROSTATICS

2.4]

II

45

type (2.38) is not acceptable, since it would give an infinite potential at the origin. Instead we must add a term E r cos 9, which will just cancel the potential of the external field so that

V

=

everywhere

inside the sphere.

The magnitude of the induced charge at any point on the sphere can be found from the normal component of the

Er = Hence the charge

o-

—dVjdr

(a) € X




e2 ,

Dx > D

Ex

inside the sphere

is

"

2)

parallel to

„

= -nS- E

(see Fig. 2.3),

(

o-

(

Ex < E

but

,

2 43 ) -

the reduction being due

to the reverse field of the polarization charges on the surface of the sphere; this reverse field is known as the 'depolarizing tial distribution outside 47re 2 e i? 3

E

(e 1

—

2 )/(e 1

the sphere

+2e2 ),

is

dipole

polarization

Px

=6

(ej

is

Px

just equal to the

induced by the

field.

1

field 1

If e 2

=

1,

the size of this

volume of the sphere times the within the sphere since then

— 1)E = 3e (e — l)E l(e + 2). X

The poten-

situated at the centre of the sphere, super-

imposed on that due to the uniform

moment

field' .

that of a dipole of magnitude

Note that

as e x

->-

oo,

the solu-

tions tend to those obtained for the conducting sphere; this follows

from

ELECTROSTATICS

2.4]

II

47

the fact that the boundary condition then requires that the sphere be zero.

the

field in

—conducting cylinder in a uniform

Problems with cylindrical symmetry field

In some three-dimensional problems the potential may be independent of one coordinate, and the problem then reduces to a two-dimensional one. It is often convenient to use cylindrical coordinates in such a case, taking the z direction as that in which the potential is invariant. Then

Table Some

cylindrical

Cylindrical harmonic

putting p equation

=

logr

1

r~l (A cos 6+B sin 0) r~ l (A cos 26+B sin 26) r-*(A cos 36 sin 39)

+B

3

and

S 2 F/0z 2

=

6+B sin 6) r*(A cos 26+B sin 26) r(A cos

^(A

in equation (2.5),

cos 36

+B sin 30)

we have

for Laplace's

,

r

r

Tr[

This

harmonic functions

Corresponding solutions of Laplace's equation

Do

D

2.3

is satisfied

tion of 9 alone

W]+W =

by a function of the form (known as a

the differential equation

V

°-

= rn Dn

cylindrical harmonic)

^+

n*Dn

=


-

where Dn is a funcwhich must satisfy

0.

(2.45)

is unchanged by the substitution of — n for n, so that if a solution of Laplace's equation, so also is V r- n Dn One solution is V loge r, and other solutions may be obtained either by partial differentiation with respect to x r cos 9, or by direct solution

This equation

V

= rn Dn

=

is

.

=

—

of equation (2.45).

A

number of the simplest functions are given in Table 2.3; note that the general form of n will be A n cosnd+Bn smnd, where A and n n

D

B

are constants.

The type of problem to which the solutions may be applied is illustrated by the case of a conducting circular cylinder, initially uncharged, lying with

its

the cylinder

bution will tend to V

axis at right angles to a uniform field

E

.

If the axis of

taken as the z-axis, it is clear that the potential distribe independent of z. At large distances the potential will is

= —E

Q

r cos

9,

and we

will

assume that other terms required

E

ELECTROSTATICS

48

must

also

II

[2.4

vary as cos 0. Then the potential outside the cylinder

of the form

T7

V

=—

„

.

.

be

r cos 6 -\-Ar~ 1 cos 6.

To satisfy the boundary condition V = we must have E R = ^li?-1 so that the ,

at r

= R for all values of 0,

potential

F= — .0 rcos0(l— i? o

The first term is the an extended dipole

will

a

,

2 //- 2

potential of the external

is

(2.46)

).

field,

the second that of

consisting of two parallel lines of positive and negative charge close to the z-axis. In these problems, we have assumed a potential containing just the required number of terms. This is a matter of intelligent anticipation

rather than guesswork or knowing the answer beforehand. If we had taken any less terms, we could not have satisfied the boundary condiIf we had taken more terms, the coefficients of the additional terms would have been found to be zero. In the case just considered, terms such as cosnd or sin n9 could not satisfy the condition V at tions.

=

=R

for all values of

because the potential of the external field varies only as cos 6. We are justified in assuming that the solution we have found is the correct and only solution because of the uniqueness theorem. This theorem also justifies the use of another special method, r

which we

shall

now

9,

consider.

images we have two equal

2.5. Electrical

If

point charges of opposite sign separated by a certain distance 2a, the plane passing through the midpoint of the fine joining

them and normal

zero potential.

Therefore

if

to this line

is

an equipotential surface at

the negative charge (say)

is

replaced

by

a plane conducting sheet A B in Fig. 2.4, the field to the right of will remain unaltered. Conversely, if a point charge is placed in front of an infinite conducting plane, the resultant electric field to the right

AB

AB will be the same as that produced by the original charge plus a negative charge an equal distance from the plane on the opposite side. of

The negative charge the plane

is

the 'electrical image' of the original charge in

AB.

The method of images thus consists in replacing a conductor by a point charge such that the conducting surface is still an equipotential Then Laplace's equation is still satisfied at all points outside by the principle of uniqueness, the problem of a point charge and its image is identical with that of a point charge and an surface.

the conductor, and infinite

conducting surface as regards the region outside the conducting

ELECTROSTATICS

2.5]

II

49

surface.

Electrical images are entirely virtual; a field closed equipotential surface cannot be represented by

on one side of a an image on the

same

side of the surface, since this would give a singularity at the point occupied by the image charge. The field on the one side of the surface is identical with that which would be produced if the surface were replaced by an image charge on the other side of it.

\

/

\

^1/ ///l\\

/A\

/ /

\.

\

/

\

/ /

\

/

Fig. 2.4.

A point charge q and its image charge — q in an infinite conducting plane AB,

showing the

lines of force

from q on the right of the plane which end on the surface charge = 2o. on AB.

XT

Point charge and

infinite

conducting plane

The method of images will now be applied to a number of special cases, the simplest of which

is that of a point charge a placed a distance a from conducting plane at zero potential. In this case it is obvious that the image must be a charge q at a distance a behind the plane,

an

infinite

—

as in Fig. 2.5.

The

potential at an arbitrary point

= -£-11. HV

4Tree \r

,2

{/

P is then

*

-|-4a 2 +4arcos0} i

where e is the dielectric constant of the medium outside the conductor. In order to calculate the charge density at any point on the plane, we 851110

ELECTKOSTATICS

50

must

find the

is (see

component of the

II

electric field

[2.5

normal to the plane. This

Fig. 2.5)

8V 1 8V Ex = Er cosd-E$ sme= _ — sin# cos0+-^ " r '

8r

q

at the point P.

r 86

rcos0+2a

Tcoi cos#

47ree [ 47re€ L

'

2

r r

{>

2

+4a 2 +4arcos0)*J

At a point Q on the

Ex =

plane, roosd

—qal2ire€ r

=

—a,

so that at

Q

3

Conductor

V = Fig. 2.5.

Image of point charge

in a conducting plane.

(cf. Problem 2.6), where r is the distance of induced charge per unit area at Q is then e

0^x

Q from the

—qaj2Trr 3

.

charge +