`
`PANASONIC EX1014, page 001
` IPR2021-01115
`
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`
`
`INTERNATIONAL SERIES IN PURE AND APPLIED PHYSICS
`G. P. HARNWELL, CONSULTING EDITOR
`
`ADVISORY EDITORIAL COMMITTEE; E. U. Condon, George R. Harrison,
`Elmer Hutchisson, K. K. Darrow
`
`STATIC AND DYNAMIC
`
`ELECTRICITY
`
`PANASONIC EX1014, page 002
` IPR2021-01115
`
`

`

`
`
`INTERNATIONAL SERIES IN
`PURE AND APPLIED PHYSICS
`G. P. HARNWELL, Consulting Editor
`
`______________.——-——-—
`
`BRILLoUIN—WAVE PROPAGATION IN PERIODIC STRUCTURES
`CADY—PIEZOELECTRICITY
`CLARK!APPLIED X-RAYS
`CURTIS—ELECTRICAL MEASUREMENTS
`EDW’ARDS—ANALYTIC AND VECTOR MECHANICS
`FINKELN'BCRG—ATOMIC PHYSICS
`GURNEY—INTRODUCTION TO STATISTICAL MECHANICS
`HARDY AND PERRIN—THE PRINCIPLES OF OPTICS
`HARNWELL—ELECTRICITY AND ELECTROMAGNETISM
`HARNWELL AND LIVINGoon—EXPERIMENTAL ATOMIC PHYSICS
`HOUSTON—PRINCIPLES OF MATHEMATICAL PHYSICS
`HUGHES 'AND DUBRIDGE—PHOTOELECTRIC PHENOMENA
`HUND—HIGH—FREQUENCY MEASUREMENTS
`INGERSOLL, ZOBEL, AND INGERsOLL——HEAT CONDUCTION
`KmmLE—THE FUNDAMENTAL PRINCIPLES OF QUANTUM
`MECHANICS
`KENNARD—KINETIC THEORY OF GASES
`KOLLER—THE PHYSICS OF ELECTRON TUBES
`MORSE—VIBRATION AND SOUND
`MUSKAT—PHYSICAL PRINCIPLES OF OIL PRODUCTION
`PAULI'NG AND GOUDSMIT—THE STRUCTURE OF LINE SPECTRA
`RICHTMYER AND KENNARD—INTRODUCTION T0 MODERN PHYSICS
`RUARK AND UREY—ATOMS, MOLECULES, AND QUANTA
`SCHIFF—QUANTUM MECHANICS
`SEITZ—THE MODERN THEORY OF SOLIDS
`SLATER——INTRODUCTION TO CHEMICAL PHYSICS
`MICROWAVE TRANSMISSION
`SLATER AND FRANK—-ELECTROI\IAGNETISNI
`INTRODUCTION TO THEORETICAL PHYSICS
`MECHANICS
`
`SMYTI—IE—STATIC AND DYNAMIC ELECTRICITY
`STRATToN—ELECTROMAGNETIC THEORY
`WHITE—INTRODUCTION TO ATOMIC SPECTRA
`___________._————-—
`
`Dr. Lee A. DuBridgo was consulting udiuu' uf Hus Huritm from 193‘.) to 1946.
`
`STATIC AND
`
`DYNAMIC ELECTRICITY
`
`BY
`
`WILLIAM Rt“ SMYTHE
`Prrgfesmr of Physics
`Caléforrria Insliiulc of Technology
`
`SECOND EDITION
`
`’3’
`
`McGRAW-HILL BOOK COMPANY, INC.
`
`PANASONIC EX1014, page 003
` IPR2021-01115
`
`

`

`
`
`STATIC AND DYNAMIC ELECTRICITY
`
`Copyright, 1939, 1950, by the McGraW—Hill Book Company, Inc. Printed in
`the Umted States of America. All rights reserved. This book, or parts thereof,
`may not be reproduced in any form Without permission of the publishers.
`
`PREFACE TO THE SECOND EDITION
`
`The wide use of rationalized mks units and the increased importance of
`microwaves made this radical revision of the first edition imperative.
`The units are changed throughout. The resultant extensive resetting
`of the text permits a modernization of nomenclature through such changes
`as “capacitor” for "condenser” ant “electromotancc” for “electro—
`motive force.” The original wording has been preserved only in the
`Cambridge problems.
`In static-field chapters, forty problems of above—
`average ditficulty have been added, usually covering boundary conditions
`omitted in the first edition. The expanded treatment of electromagnetic
`waves made necessary the rewriting of the parts of Chapter V dealing
`with Bessel functions and led to the introduction of vector surface har—
`monics, which greatly simplify some calculations.
`LVIuch of Chapter X1
`on eddy currents has been rewritten, and two of the three electromag—
`net-ic-wave. chapters are entirely new. Both the text and the 150 prob
`lems include methods and results not found in the literature. Two groups
`of advanced PhD. students Worked over this material to get practice
`in attacking every type of wavewfield problem. Many are too difficult
`for first-year graduate students, but every problem was solved by at
`least one of the advanced students. They can be worked either directly
`from the text or by fairly obvious extensions of it. Some useful results
`appear in the problems and are listed in the Index, which should be con—
`sulted by engineers with boundar}r value problems to solve. Chapter
`XV of the first edition is omitted because none of the remaining theory is
`based on it and because to bring it up to date would require an excessive
`amount of Space.
`None of the new topics appears to lie outside the scope of the mathe»
`matical preparation assumed for readers of the first edition. That the
`successful solution of electrical problems depends on physical rather than
`mathematical insight is borne out by the author’s experience with the
`first editionJ which shows that graduate students in electrical engineering
`and physics greatly excel those in mathematics.
`It is believed that very few of the errors and obscure or ambiguous
`statements in the first edition escaped the scrutiny of the 375 students
`at the California Institute of Technology who worked it through. No
`infallible system for locating errors caused by the transposition of units
`has been found, and the author will appreciate letters from readers point—
`,
`
`RECEIVED
`
`SEP261950
`
`
`
`
`
`
`COPYRIGHT OFFICE
`
`PANASONIC EX1014, page 004
` IPR2021-01115
`
`

`

`
`
`' CONTENTS
`
`PREFACE TO THE SECOND EDITION.
`
`PREFACE TO THE FIRST EDITION.
`
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`A
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`r
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`4
`
`v
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`vii
`
`TABLEOFSYNEBOLS...........................XVil.
`CHAPTER I
`.
`.
`.
`.
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`.
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`.
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`.
`.
`.
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`BASIC IlJEASOFE1.ECTEOSTA1'ICS ‘
`_
`.
`.
`.
`.
`Electrification, conductors, and insulators—Positive and negative olectricity
`——Coulomb's law, unit charge, dialectriass—Limitations of the inverse—square
`law—Electrical induction—The clement-my electric nlmrgeaiElectric field
`intensity—Electrostatic pni‘-(‘.[1t-ial—Eler-.tric dipoles and mult-ipoies—Intnr—
`action (if dipoles—Lines of {DTCEiEquPOtE'DLial surfacesiGmrss’s electric
`flux theorem—lines of force from collinear charges—Lines of force at
`hlfirfityil’otentiul maxima and minjma. Earnshaw‘s theorem—Potential
`of electric double layer—Electric Cliflpllll’lcmc‘llt- and tubers of .forI-w-Strcaqcs
`in an electric fieldiGuuss’s electric flux thcorcm [or nonlmmOgcnc-ous
`medituns—Bniindury conditions and start-sacs [m the surface of conductors—
`Buundary conditions and stresses on the surface of a die-Ir'ctriciflisplacc-
`ment. and intensity in solid dielectric—Crystalline dielectrics—Problems—
`References.
`
`1
`
`CHAPTER II
`
`.
`.
`.
`.
`.
`.
`.
`.
`.
`.
`.
`.
`CAPACITORS, DrELEE'rmcs, SYSTEMS or Coxmrcrons.
`Unjqucncsfi
`theorcmiCapacitance—Capacitors in series and parallel—
`Spht-Jical capacitors—Cylindrical
`caps.(Sitar:—Pa.1's,llEl-plfl.Ln capacitors—
`Guard, ringsiEncrgy of a charged capacitor—Energy in an electric field—-
`Parallel-plate capacitor with crystalline dielectric—Stresses when the
`capncitivity is a; function of density—Elcctrostriction in liquid dielectrics—
`Force on conductor in dicluutric—Green‘s reciprocation theoremisupcr-
`
`pasition of fields—hi'duced charges on earthed conductors—Self— and mutual
`eiastnllcerSelf- and mutual capacitance—Electric screening Elastances
`and capacitance-s for two distant conductorsiEncrgy of a charged system—
`Forces and torques on charged cmuiuctors—Problemaéfiefcrcnces.
`CHAPTERIII
`GENERALTHEOREMS...........................
`Gauss’s theoremistokcs’s theorem—Equations of Poissan and Laplace—
`Orthngonal curvilinear coerdinates—Curl in orthogonal curvilinear coordi-
`nates—V ‘ (JV) in other Ccordlnata systems—Green's theorems—Green’s
`reciprocation theorem for dielectrics—Green’s function—Solution of Polar
`
`25
`
`48
`
`PANASONIC EX1014, page 005
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`
`
`CONTENTS
`
`63
`
`CHAPTER IV
`T\V0-DIMENSIONAL POTENTIAL DISTRIBUTIONS.
`Field and potential
`in two dimensions—Circular harmonics—Harmonic
`expansmn of line charge potential—Conducting or dielectric cylinder in
`uniform field#Dielectric cylinder. Method of images—Image in conduct-
`ing cylinder—Image in plane face of dielectric or conductor.
`Intersectin
`conducting planes—Dielectric wedge—Complex quantitiesiconju at:
`functions—The stream function—Electric field intensity. Electric flEX#
`Functions for a line charge—Capacitance between two circular cylinders—
`Capscrtance between cylinder and plane and between two similar cylinders
`—Con_formal transformations—Given equations of boundary in parametric
`formilDetermmation of required conjugate functions—The Schwarz trans—
`formation—IPolygons with one positive angle—Polygon with angle zero—
`Poiygons With one negative angle. Doublet.
`Inversion—Images by two—
`dunensrona]inversion—Polygon with two angles—Slotted planeiRiemann
`surfaces—-Circular cylinder
`into elliptic cylmdcr4Dielectric boundar
`conditions—Elliptic dielectric cylinder—Torque on dielectric c linder—y
`Polygon with rounded corner—Plane grating of large cylindrical wires—
`Angles not integral multiples of I,5;:r—-1°1-oblemsifleferences.
`CHAPTER V
`
`. 111
`
`.
`THREE-DIMENSIONAL POTENTIAL DISTRIBUTIONS.
`When-can a set of surfaces be equipotentials?—Potentials for confocal
`conicmds—Charged conducting ellipsoid4Elli-ptic and circular disksi
`Method of images. Conducting planes—flame boundary between dielec-
`trics—Image in spherical conductoriExample of images of point charge—
`Infinite set of images. Two spheres—Difference equations. Two spheres—
`Sphere and plane and two equal spheres—Inversion in three dimensions
`Geometrical properties—Inverse of potential and image systems—Example
`of inversion of imageS#Invcrsion of charged conducting surface—Capaci-
`tance by 1nversion—fThree-dimensional harmonics—Surface of revolution
`and orthogonal wedge—Spherical harmonics!General property of surface
`harmonlcs—Potential of harmonic charge distribution—Differential e ua—
`trons for surface harmonics—Surface zonal harmonics. Legendre’s equaiion
`
`
`
`fSeriles solution of Legendre’s equation—Legendre polynomials. Rod-
`rigucs :-
`formula—Legendre
`coe"icicnts.
`Inverse distance—Recurrence
`formulas for Legendre polynomials—Integral of product of Legendre
`oiv-
`nomrals—IExpansionof functioninLegendre polynomin-lsfiTnhle ofLe, :ndre
`polynomials;Legendre polynomial with imaginary variableil’oiential
`of charged ring—Charged ring in conducting sphere—Dielectric shell in uni-
`form field-Ofi—center spherical capacitor—Simple conical boundary—Zonal
`harmomcs of the second kind—Recurrence formulas for Legendre functions
`of the second kind—Legendre functions of the second kind in terms of
`Legendre polynomials—Special values of Legendre functions of the second
`kmd—Icgeudre function of the. second kind with imaginary variable—Use of
`Legendre function of the second kind in potential problems—Nonintegral
`linrruonica—Assnciuteri Legendre functions—Integrals of products
`
`CONTENTS
`
`xiii
`
`function for a EOHPGI'EEDIB function for a conical boxwhlate spheroidal
`coordinat-eerblnte spheroidal harmonics—Conducting sheet with circular
`hole—Torque on disk in uniform field‘Potc-ntial of charge. distribution
`on spherold4Pnt-Plltial of point charge in oblate spheroidal harmonics—-
`Prolatc spheroidal harmonies—~Prolntn spheroid in unilorm field—Laplaee’s
`oordi'nar-es—rBessel’s equation and Bessel functions
`equation in cylindrical c
`—[\-lodjfic<1 Besecl equation and functions—Solution of liesscl’s equation—
`Recurrence formulas for Bessel functions—Values of Bessel functions at
`infinity—integrals of Bessel
`imitations—Expansion in series of Bessel
`functirms—rGreeu’s
`function
`for
`cylinder.
`Inverse
`distance—Green’s
`function for cylindrical bOXrBessel functions of Hero ordewltoots and
`ifunctious of zero order—Derivatives and integrals
`numerical values of Hesse
`of Bessel
`functions of zero order—Point charge and dielectric platet
`Potential inside hollow cylindrical ring—Nonintegral order and spherical
`Bessel
`llmctions—Modified Bessel
`functions—liecurrence formulas
`for
`modified Bessel. functionerValues of modified Bessel functions at infinity
`-[ntcgrnl of
`a. product of complex modified Bessel functions—Green‘s
`fizurction for a. hollow cylindrical ringglloditicd Bessel functions of zero
`order—Definite integrals for the modified lit-3.5M function of
`the second
`kind.
`anur- at infinity—-TJef‘ulite integrals for Bessel functions of zero
`orderilnvcrsc distance. in terms of nwdilied Bessel functions-Cylindricul
`dielectric l“)lllILlul'lC‘rliPfltclltlfld inside hollow cylindrical ring—Modified
`Bessel
`functions of nunintegrnl ordnr—Appmximntc solutions. Electro-
`stutic lens—Wedge fun(-tionsw—l’mhletins;lleferenccs.
`CHAPTER VI
`
`. 218
`
`E1.Eo'r]nc (“.‘Lrnunnr.
`.
`.
`.
`.
`.
`.
`Electric current density. Equationcl’ continuity—-Electromotnnce—Ohm‘s
`law.
`lumistivity—Heating effect of electric current—Linear conductors.
`Kirchheii’e laws. Conductors in series and parallelelution of networks.
`Circulating currents. Wheatstonc bridge—Network with repeating mem-
`berSrLine with continuous leak-General network#Conjugate conductors.
`Kelvin double bridgk—Steady currents in extended meilium54Genei-al
`theorcmsFCurrcnt How in two dimensions-Long strip with abrupt change
`in widthfilurrent {low in three {iinmnsiorm—Syst-ems of electrodes. Two
`spheres. Distant electrodes—Solid conducting sphereffiolid conducting
`i-ylinderflEarth resistant-churreuts in thin Curved sheets—Current dii‘r
`t-ributiou on spherical shell-Surface of revolutiouiliimits of resistance
`—Currents in nonisoLi-opic mediums Earth straitaflSpuce—chnrgc current.
`Child’s equation4Pruhlcin5#lleferences.
`CHAPTER VII
`
`. 2‘50
`
`MAGNETIC INTERACTION or CURRENTs.
`.
`.
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`.
`.
`.
`.
`.
`.
`.
`.
`Definition of
`the ampere in terms of the magnetic tuomcnt—-l\-Iagnetic
`induction and permeabilityflingnetic vector potential. Uniform field!
`Uniqueness theorems for magnetostaticerrthogon expansions for vector
`potent-i:il—-—Vectur potential
`in cylindrical coordinates—Vector potential
`in spherical coordiimtce—Vcctor potential in terms of magnetic induction
`
`
`
`PANASONIC EX1014, page 006
` IPR2021-01115
`
`

`

`
`
`CONTENTS
`
`Field of circular loop in spherical harmonics—Riot and Savart’s law. Field
`of straight wire—Field of helical solenoid—Field in cylindrical hole in con—
`ducting rod—Field of rectilinear currents in cylindrical conducting shell—
`Force on electric circuit in magnetic field—Examples of forces between
`electric circuits—Vector potential and magnetizationfiMagnetic boundary
`conditions—Example of the use of A and A—Current images in plane face—
`Magnetic induction and permeability in crystals—Two—dimensional mag-
`netic fields—Magnetic shielding; Of bifilar circuit~Current images in two
`dimensions~Magnetomotance and. magnetic
`intensity—The magnetic
`circuit. Anchor ring—Air gaps in magnetic circuits—Field in shell-type
`transformer—Slotted pole piece. Effective air gap—Problems—References.
`CHAPTER VIII
`
`ELECTROMAGNETIC INDUCTION .
`
`. 307
`
`Faraday’s law of induction—Mutual energy of two circuits—Energy in
`a magnetic field—Mutual inductancefiBoundary conditions on A—Mutual
`inductance of simple circuits—Mutual
`inductance of circular
`loops—
`Variable mutual
`induetance—Self—inductancc—Computation
`of
`self-
`inductance. Thin wire—Self—inductance of circular loop—Self—inductance
`of solenoid—Self-inductance of bifilar lead‘—Energy of n circuits~Stresses
`in a magnetic field—Energy of a permeable body in a magnetostatic field—
`Problems~References.
`
`CHAPTER IX
`TRANSIENT PHENOMENA IN NETWORKS.
`
`
`Electrical transients—Energy relations in a network Circuit with capaci-
`inductance, and resistancefiDischarge and charge of a capacitor
`~Growth and decay of current‘ in an inductor—Circuits with mutual
`inductance—Kinetic energy and electrokinetic momentum—Equation for
`transients in general network—Solution for general network—Natural
`modes in a network—Networks containing constant electromotancesfi
`Modes of two inductively coupled eircuits~Amplitudes in two coupled
`circuits—Oscillatory solution—LOW-resistance inductively coupled circuits—-
`Low-resistance tuned inductive coupling—Circuits with repeated members
`-—-Integrated transient effects—Transients due to pulses of finite duration——
`Problems—References.
`
`. 327
`
`CHAPTER X
`
`ALTERNATING CURRENTS.
`
`.
`
`.
`
`. 355
`Harmonic electromotances. The particular integral—Circuit with resist—
`ance, capacitance, and inductanceaPower, root-mean—square quantities and
`resonance—Graphical
`representations. Phasor diagram—Impedances in
`series and parallel—Transmission of power—Impedance bridge—General
`alternating-current network—Conjugate branches in impedance network.
`Anderson bridge-—Forced oscillations in inductively coupled circuits—Induc—
`tively coupled low-resistance circuits—Tuned inductively coupled low-
`resistance circuits—Filter circuits—Terminal conditions in wave filters—
`Frequency characteristics of filters—The band-pass filter—The m—derived
`type of filter section—Termination of composite filter—Transmission lines—-
`
`CONTENTS
`
`CHAPTER XI
`
`XV
`
`.
`.
`.
`EDDY CUREENTS .
`Induced curr
`rents in attended conductors—Solution for vector potgntial of
`-
`i
`y
`"
`-
`- — '
`fleet on tubular con uctor—
`rents—bteady-state skin affect Skin e _
`_
`.
`_
`.
`$113: :i'ith on solid cylindrical conductor—Solution 1.11 spherical coordinates
`for axial symmetry—Conducting sphere in alternating. field—-l-"ower zihsorhc
`b sphere in alternating magnetic field—Transients in conducting sphereb—
`Flddy currents in plane sheets—-Eddy currents In infinite plane shegt
`[3:
`image method—Torque on small rotating current loop or magnetic. 1pc L
`li‘ddy currents from rotating diImleii-‘Ihiclding oi (alrcucllrér L‘Oll by :hin (gin-
`_ J‘
`J
`I
`I
`~.
`-
`>
`i
`"
`l-Icriealsheli—E ycurren sm .
`in
`d etm r sheet—Zonnl eddy rill-rents m sp
`‘
`_
`.
`—
`‘
`ciilindiicnl ShelliTr:msient shielding by a thlek cylindrical shell Problems
`——References.
`CHAPTER XII
`
`. 421
`-.
`MAGNETIsM.................-.....-.'._...
`Paramagnetism and dirimagnetism—MI-Ignctic
`s!lseeptlbihty—Magnctu.
`allinc sphere in uniform magnetic.- field—Ferro-
`properties of crystals—Uryst
`munent magnetism—The nature of permanent
`magnetiBIu—-Hysteresis. Per
`magnetismrUniform magnetization. Equivalent current sheflih‘IHgntL
`tized sphere and cylinder. Magnetic poles—Boundary conditions on
`permanent magnets—{ipherifial permanent magnet in. uniform field—Lifting
`power of horseshoe magnet—Field of cylindrical magnet—Magnum needles
`FProblemSFRefnrences.
`CHAPTER XIII
`
`440
`
`.
`.
`.
`.
`.
`.
`.
`.
`.
`.
`PLANE ELECTROMAGNETIC WAVES.
`Maxwell’s field equations—Propagation equation. Electrodyr}:am:3g:::::;
`—Plane waves In
`'
`‘
`om
`'
`1 Hertz vector—Paynung 3 vector
`.
`.
`_
`‘
`.
`flanged dielectric insulator—Plane wave velocity 1n anisotropic mediums
`——I?.a}r surface and polarization in anisotropic mediums—Energy, pI'BSSIIIE
`and momentum of a. plane wave-Refraction and reflection of a plane Wfivet.
`Intensity of reflected and refracted wav35#Freuuency, wave length, e 1p 1c
`olnrirution—Total
`reflectionfElectromugnetic waves
`in homogeneous
`
`inductors—Planes waves in homogeneous isotropic conductors—Rel cation
`from conducting surface—Plane waves on cylindrical perfect conclgciolrs
`'
`'
`'
`J
`'
`iIieilcution at a discontinuity.
`.' n r 1mg
`Intrinsic impedance of a medium
`.
`erfect
`'
`‘
`'
`tor—Nearly plane waves on Imp
`section;Complex Poynting vec
`.
`_
`conductors. Lccher wires—Group velocity—Problems References.
`CHAPTER XIV
`ELECTROMAGNETIC RADIATION .
`.
`.
`.
`.
`.
`.
`.
`.
`.
`.
`The radiation problem;Two useful types of vector got—egidaljslflwgirfl
`‘
`' —Retarded potentia. — iaion
`ro
`electromagnetic waves. Dipole
`‘
`.
`_
`.
`.
`.
`.
`.
`.
`na—
`linear antenna—Distant radiation from linear antenna—Radiation from
`progressive waves—Conical transmissmn lines—The broomcai anten
`s—Uniqueness of solution—Solutions of the
`Antenna arrays;Ea.rth effect
`dinatesiPolynomiai expansion for a plane
`wave equation in spherical. com
`t
`loop. Magnetic dipole—Free
`wavedllaldiution from uniform curren
`cl oscillations of dielectric or
`
`. 468
`
`PANASONIC EX1014, page 007
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`
`

`

`CONTENTS
`
`from apertures in plane conducting screens—Diffraction from rectangular
`aperture in conducting plane—Orthogonal functions in diffraction problems.
`Coaxial line~Problems~References
`
`CHAPTER. XV
`WAVE GUIDES AND CAVITY RESONATORS.
`
`511
`
`TABLE OF SYMBOLS
`
`Waves in hollow cylindrical tubes—Attenuation in hollow wave guides—
`The rectangular wave guide—The circular wave guide—The coaxial wave
`guide—Plane discontinuities in coaxial
`lines—Coupling to wave guides
`~Excitation of circular guide by current element~Loop coupling with
`circular guide—Orifice coupling with circular guide—Plane discontinuities
`in rectangular guides—Cavity resonators. Normal modesiIndependent
`oscillation modes Of a cavity—Inductance and capacitance of cylindrical
`cavity—Damping of normal modes. Cavity resistance—Normal modes
`of a cylindrical cavityiPropertics of a rectangular cavity~Properties
`of a right circular cylindrical cavity—Multiply connected cylindrical
`cavities—Coaxial cable resonators—The normal modes of a spherical cavity
`—Normal modes of an imperfect cavity—Complex cavities—Excitation of
`cavities,
`inductive coupling—Inductive coupling to a circular cylindrical
`cavity—Excitation of cavity by internal electrode—Cavity excitation
`through orifice—Problems—References.
`CHAPTER XVI
`SPECIAL RELATIVITY AND THE MOTION OF CHARGED PARTICLES.
`
`. 560
`
`The postulates of special relativity—The Lorentz transformation equations
`—Transformation equations for velocity and acceleration—Variation of
`mass with velocity—The transformation equations for force—Force on
`charge moving in magnetic fieldfiMotion of charge in uniform magnetic
`field—Energy of a charged moving particle—Magnetic cutoff of therm—
`ionic rectifier—Path of cosmic particle in uniform field—Magnetic field
`of moving charge—Retarded fields and potentials of moving charge—
`Radiation from linearly accelerated electron—Transformation of Maxwell’s
`equations—Ground speed of an airplane—Motion of charged particle in
`crossed electric and magnetic fields—Aberration and Doppler Effect——
`Problems—References.
`
`SYSTEMS or ELECTRICAL UNITS.
`
`APPENDIX
`
`TABLE I. RELATIONS BETWEEN cos AND MKS MECHANICAL UNITs.
`
`TABLE II. TRANsrosrTION or MKS FORMULAS INTO CGS EsU .
`
`.
`
`.
`
`.
`
`.
`
`TRANSPOSITION or MKS FORMULAS INTO ch EMU .
`
`TRANSPOSITION or was ESU on cos EMU FORMULAS INTO MKS .
`TABLE V. DIMENSIONS OF ELECTRIC AND MAGNETIC QUANTITIES.
`TABLE VI. NUMERICAL VALUES
`
`. 585
`
`. 586
`
`. 586
`
`. 587
`
`. 588
`
`. 590
`
`_
`
`. 591
`. 593 i
`
`.) are space vectors
`.
`Note: In this table. hold-face symbols (v, u, o, .
`.
`.
`.) o; conjugate
`except in Chap. X, where they are phasors (I, 6,
`phasol‘s (1*, 6*,
`.
`. J; In “subsequent chapters, phasors i. , é’,
`.
`.
`.)Ci
`phasor space vectors (E, 3,11,
`.
`.4), conjugate phasors (1,6,
`..
`I.
`.) arr-t
`conjugate phasor space vectors (E, 3, II,
`.
`.
`.) are shownI‘byi..1.i:iderecf
`(v) or an inverted (A) flat vee above the symbol.
`h-I‘agmtu‘e: o
`vectors and scalars. whether time—dependent or not, are written Wu. out
`designation.
`A, A4,, A“ etc. Vector potential.
`A“ Normalized vector potential.
`A, A., etc. Quasi—vector potential.
`B, B9, B 1, etc. Magnetic induction.
`B Susceptance.
`B“ Normalized or relative susceptance, BZk.
`C Capacitance. A constant.
`-
`(7“ Normalized or relative capacitance, CZk.
`0 Velocity of light. A length.
`cm. Self-capacitance. Operator in 9.07.
`cm Mutual capacitance. Operator in 9.07.
`D, DA, Dz, etc. Electric displacement.
`Dw Dwight integral tables.
`ds Differential element along 3.
`dr Differential change in r.
`E, E, E, E, etc. Electric field intensity.
`E(k) Complete elliptic integral.
`6 Electronic charge.
`2.71828.
`6, 6, E, 6", etc. Electromotance.
`6’. Effective or rms electromotance.
`F, Fz Force.
`v
`G’ Conductance, Y = G + 3'3.
`9 Acceleration of gravity.
`H, H, II, II, etc. Magnetic field intens1ty.
`H5,”, H5,“ (v), H$3) , H5?) (v) Hankel functions.
`it Plank’s constant.
`h1, h2, h».
`In orthogonal curvilinear coordinates.
`
`PANASONIC EX1014, page 008
` IPR2021-01115
`
`

`

`
`
`CAPACITORS, DIELECTRICS, SYSTEZLIS 0F CONDUCTORS
`
`§2.04
`
`§2.06
`
`.
`
`GUARD RINGS
`
`‘
`
`29
`
`It should be noted that in deriving (1) it is assumed that there is no
`charge on the outside of b, which requires that b be at zero potential.
`If this is not the case, the additional capacitance between the outside
`of b and infinity, computed from (2), must be considered.
`2.04. Cylindrical Capacitors—Consider a pair of concentric circular
`conducting cylinders of infinite length, the inner, of radius a, carrying a
`charge Q per unit length and the outer, of radius b, carrying a charge — Q
`per unit length,
`the space between being filled with a homogeneous
`isotropic medium of capacitivity c. From symmetry, the electric dis—
`placement must be directed radially outward from the axis and lie in a
`plane normal to the axis, and its magnitude must depend only on 7'.
`Apply Gauss’s electric flux theorem to the volume enclosed by two
`planes, normal to the axis and one centimeter apart, and the concentric
`circular cylinder of radius r when b > r > e. The plane walls contribute
`nothing to the surface integral; we have therefore
`LEE-nets = awn 2 Q
`
`so that
`
`E=—#=Ne
`
`The potential difference betWeen the cylinders is therefore
`Q<
`adr # ___Q_
`a
`#27rre
`b 7 _
`27reln3
`Thus the capacitance per unit length of a long cylindrical capacitor is
`27m
`
`n—m:
`
`,
`
`o :
`
`ln (b/a)
`
`(n
`
`(1'1)
`
`<2)
`
`if there is a finite
`If we let I) -—> oo , we see that C’ —> 0. Therefore,
`charge per unit length on a circular cylinder of finite radius and infinite
`length, the potential difference between its surface and infinity is infinite.
`Since physically we deal only with cylinders of finite length, this difficulty
`does not arise, but it indicates that the results of this article apply only
`Where the distance to the surface of the cylinder is small compared with
`the distance to the ends.
`2.05. Parallel—plate Capacitors.—When two infinite parallel con-
`ducting planes, carrying charges +Q and —Q, are a distance it apart,
`the space between being filled with a homogeneous isotropic dielectric,
`we see from symmetry that the field between them must be uniform and
`normal to the plates.
`If 0- is the charge per square meter, then there
`must be, from 1.13, a' unit tubes leaving every square meter of the plates.
`
`D = 6E = —eflf = «r
`61:
`
`The difference of potential between the plates is
`[1
`(1.1)
`V2 _ V1 : if dx = ‘11
`C
`0
`6
`Therefore the capacitance per unit area is e/a, and the capacitance of an
`area A is
`
`611
`C = a
`
`.
`ll)
`
`In practice, the field will be uniform only far from the edges of the plate,
`so that this formula is an appmximation which is good if a is small com—
`pared with all surface dimensions of the plate and still better if, in addi-
`tion, the capacitivity of the region between the plates is higher than that
`of the region beyond the edges.
`2.06. Guard Rings.—The derivation of formula 2.04 (2) for the capaci—
`tance per unit length of a cylindrical capacitor and that of 2.05 (1)
`f
`/
`for the capacitance of a parallel—
`plate capacitor both involve the
`hypothesis of conductors of infi—
`nite dimensions. To permit
`
`\
`
`\"
`
`FIG. 2.06a.
`
`FIG. 2.0(5b.
`
`the application of these formulas to actual capacitors, a device known as
`a guard ring is used. For the cylindrical capacitors, shown in Fig. 2.06m,
`the end sections of one of the members are separated from the center sec—-
`lion by narrow cracks but are maintained at the same potential. Thus the
`distorted field near the edges does not affect the center section, except
`[or a very small effect produced by the cracks, so that the charge on this
`section is related to the potential difference by 2.04 (1.1}.
`A similar arrangement
`is used for the parallel-plate capacitor by
`leaving a narrow gap between the central section of one plate and the
`area surrounding it, maintaining both at the same potential as shown in
`Fig. 2.061;. The field between the central areas is nosr uniform except
`
`
`
`PANASONIC EX1014, page 009
` IPR2021-01115
`
`

`

`CAPACITORS, DIELECTRICS, SYSTEMS 0F CONDUCTORS
`
`§2.07
`
`2.07. Energy of :1 Charged Capacitor.—We can compute the mutual
`energy of any system of charges directly from the definition of potential.
`The work in joules to put the jt.h charge in place will be, from 1.06 (3),
`
`W,- : q;V,- = i {11"
`47m
`7'1,
`i=1
`
`Wherei ¢ j
`
`The total work to put all charges in place is
`n
`1L
`
`if?"
`1]
`
`W = 81—“
`
`i=1 i=1
`
`wherei 751'
`
`(1)
`
`The factor % is necessary because the summation includes not only the
`work done in bringing the with charge into its position in the field of the
`jth charge but also that done in bringing the jth charge into the field of
`the ith charge, which is the same.
`If V. is the potential at the spot Where
`the ith charge is situated, this may be written, from 1.06 (3),
`
`W : ezqiv.
`i=1
`
`(2)
`
`When all charges lie on the same conductor a, they are at the same poten-
`tial'and if their sum is Qa, We may Write
`
`, _ 1
`, g V.
`_ 1
`,
`H’a — EEQJ 1' — 729: — QQaI/a
`over a,
`over a
`
`(3)
`
`If the capacitance of a conductor is C, we have, from 2.01 (1), the follow-
`ing equivalent expressions for the energy of a charged conductor:
`W = em = tQQ/C = eat/'2
`
`(4)
`
`For a capacitor, the two members of which carry charges Q and —Q at
`potentials V1 and Vg, respectively, the energy becomes
`W = tQVr — tQVz = tQWr — V2)
`
`(5)
`
`From 2.01 (2), (5) has the same equivalent forms as (4).
`2.08. Energy in an Electric Field—We have seen that visualizing
`electric forces as transmitted by stresses in the region occupied by an
`electric field gives results consistent with the observed laws of electro—
`statics. Where stresses exist, potential energy must be stored. We
`shall now compute this energy density. Consider an infinitesimal disk-
`shaped element of volume oriented in such a way that the two faces are
`equipotential surfaces.
`If this element is taken sufficiently smallhthe
`
`§2.081
`
`PARALLEL—PLATE CAPACITOR '
`
`31
`
`faces is (aV/Os) ds = —E (is. Let n be the unit vector normal to the
`faces so that E = En. The charge on an area dS of the face is
`D-E ,
`D-ndS—Tdb
`
`and the capacitor volume is dV = ds d8, so that 2.07 (4) gives
`
`,_D-E
`dW —Tdv
`
`This gives, for the energy density in the field,
`
`dW_D-E
`W‘
`2
`
`In an isotropic dielectric, D - E = DE, giving
`
`dW d2 _ 1E _ D2
`dv
`2
`2 T 2—5
`
`(1)
`
`(2)
`
`In a crystalline dielectric, we have, from 1.19 (5),
`MW
`(3)
`1].; : %(611E% + engg + e33E§ + 2€12E1E2 + 2613E1E3 + 2623E2E3)
`nl‘ if the coordinate axes are chosen to coincide with the electric axes of
`Hm crystal, we have, from 1.19 (6),
`(WV
`air = £61152 + ezEfi + 63E?)
`
`(4)
`
`
`
`2.081. Parallel—plate Capacitor with Crystalline Dielectric.—Let us
`now calculate the capacitance per square meter of a parallel-plate
`capacitor where the dielectric consists of a slab of crystal of thickness d.
`lml, the capacitivities along the crystal axes r, y, and 2 be 51, 52, and e3,
`I't'h|)(¥Cth€iY,
`and let the direction cosines of the angle made by the
`normal to the capacitor plates with these axes be l,m, and 1».
`Since,
`cilwtl'lcaily, one square meter section is like any other, the equipotentials
`must be parallel to the plates and equally spaced and the electric intensity
`must lie along the normal. Thus we have
`
`If
`E=<X2+Y2+ZZ>%=;,—
`wlwm V is the potential across the capacitor. Thus
`11/,
`mV
`nV
`(1)
`d1
`Y _ d I
`Z _ d
`substituting in 2.08 (4), multiplying by the volume (1 of a square meter
`mmliun, and using 2.07 (4), we have
`
`Xm
`
`PANASONIC EX1014, page 010
` IPR2021-01115
`
`

`

`
`
`CAPACITORS, DIELECTRICS, SYSTEMS 0F CONDUCTORS
`
`§2.09
`
`so that the capacitance per square meter is
`
`1261 + “262 + "263
`d
`
`C1:
`
`(2)
`
`§2.11
`
`where
`
`FORCE ON CONDUCTOR IN DIELECTRIC
`
`33
`
`(2
`
`I
`
`2
`
`E’2=E;2+E:L2 = (%) +(%)
`
`2.09. Stresses When the Capacitivity Is a Function of Density.—In
`considering stresses in a dielectric heretofore in 1.16 and 1.17, we have
`ignored the possibility that the capacitivity may actually change With
`density T so that there may be a hydrostatic stress tending to expand or
`contract the dielectric. By working with a volume element of the shape
`and orientation used in 2.03, we can simplify the investigation to that of
`a. mall parallel—plate capacitor of area AS and spacmg 5 in which the
`charge on the plates is considered as embedded in the dielectric at the
`boundary. Combining 2.07 (1) with. 2.05 (1) and assuming anisotropic
`dielectric of capacitivity e, we have for the energy of our capacrtor
`
`6
`m
`_ mD2
`,
`AW =
`mm = 267’ ASQ2 — 261’ Ab
`where we have let m be the mass of the dielectric per unit area between
`the plates so that m = 76.
`If m is assumed constant and e is taken as a
`function of 1', the force on an area AS of the plate is
`MAW) =
`6(AW) a? = #D'Zaor) AS
`AF:
`65
`67'
`65
`262
`61'
`Thus the stress or force per unit area pulling on the surface of the con—
`ductor is
`
`fég : E 3(67)
`AS
`2&2
`61'
`Carrying out the differentiation and comparing with 1.16 (2), we see
`that the additional hydrostatic stress is
`
`euE2 6K
`
`(1)
`
`2
`
`At the surface between two dielectrics, we shall now have to add to
`those stresses already considered the difference in this hydrostatic pres—
`sure giving, in place of 1.17 (7), for the total stress directed from K to
`K" the value
`KH
`1 K! _ Kl]
`12 D?
`D/2TI 3K, Dung a
`]
`3
`)
`(
`Fn = 26[
`K1
`(1;! + KI! _ K12
`67',
`l
`KHZ
`67'”
`At the surface between a dielectric and a vacuum set K” = 1 and
`aK”/ar” = 0, we ha