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MIT OpenCourseWare http://ocw.mit.edu Continuum Electromechanics
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Continuum Electromechanics
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To Janet Damman Melcher
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Continuum Electromechanics
James R. Melcher
The MIT PressCambridge, Massachusetts, and London, England
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Copyright 1981 by
The Massachusetts Institute of Technology
All rights reserved. No part of this book may be reproduced
inany form or by any means, electronic or mechanical,
includingphotocopying, recording, or by any information storage
andretrieval system, without permission in writing from the
publisher.
Printed and bound in the United States of America
Library of Congress Cataloging in Publication Data
Melcher, James R.Continuum electromechanics.
Includes index.1. Electric engineering. 2. Electrodynamics.
3. Continuum mechanics. 4. Electromagnetic fields.I.
Title.TK145.M616 621.3 81-1578ISBN 0-262-13165-X AACR2
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Preface
The three stages in which this text came into being give some
insight as to how the materialhas matured. As "notes" written in
the early 1960's, it was intended to serve as an introductionto the
subject of electrohydrodynamics. Thus, it reflected the author's
early research interests.During this period, the author had the
privilege of collaborating with Herbert H. Woodson (nowUniversity
of Texas) on the development of an undergraduate subject, "Fields,
Forces and Motion".That effort resulted in the text
Electromechanical Dynamics (Wiley, 1968). There has also beena
strong influence from Hermann A. Haus, with whom the author has
collaborated for a number ofyears in the development and teaching
of an undergraduate electromagnetic field theory subject.Both
Woodson, with his interests in rotating machinery and
magnetohydrodynamics, and Haus, whothen worked in areas ranging
from electron beam engineering and plasmas to the electrodynamicsof
continuous media, stimulated the notion that there was a set of
fundamental ideas that perme-ated many different "specialty areas".
To be taught were widely applicable basic laws, approachesto
modeling and mathematical techniques for disclosing what the models
had to say.
The text took its second form in 1972-1973, when the objective
was to achieve this broaderand more enduring aspect of the
material. Much of the writing was done while the author was on
aGuggenheim Fellowship and a Fellow of Churchill College, Cambridge
University, England. Duringthat year, as a guest of George
Batchelor's Department of Applied Mathematics and
TheoreticalPhysics, and with the privilege of working with Sir
Geoffrey Taylor, there was the opportunityto further broaden the
perspective. Here, the influences were toward the disciplines of
contin-uum mechanics.
Unfortunately, the manuscript resulting from this second writing
was more in the nature oftwo books than one. More integration and
culling of material was required if the self-imposedobjective was
to be achieved of helping to define a discipline rather than simply
covering anumber of interrelated topics.
The third version, this text, would probably not have come into
being had it not been forthe active encouragement of Aina Sils. Her
editorial help and typewriter artistry provided teach-ing material
that was immediately sufficiently attractive to serve as an
incentive to commit nightsand weekends to yet another rewrite.
As a close colleague who has been instrumental in establishing
as an area the continuumelectromechanics of biological systems,
Alan J. Grodzinsky has been both a source of technicalinsight and
an inspiration to complete the publication of material that for so
many years hadbeen referenced in theses as "notes."
Research carried out by still other colleagues at MIT will be
seen to have influenced thescope and content. The Electric Power
Systems Engineering Laboratory, directed by Gerald L.Wilson,is an
example with its activities in superconducting machinery (James L.
Kirtley, Jr.)and its model power system (Steven D. Umans). Others
are the High Voltage Laboratory (John G.Trump and Chathan M.
Cooke), the National Magnet Laboratory (Ronald R. Parker and
Richard D.Thornton), the Research Laboratory of Electronics (Paul
Penfield, Jr. and David H. Staelin),the Materials Processing Center
(Merton C. Flemings), the Energy Laboratory (Janos M. Beer andJean
F. Louis), the Polymer Processing Program (Nam P. Suh), and the
Laboratory for InsulationResearch,(Arthur R. Von Hippel and William
B. Westphal).
A great satisfaction and motivation has come from seeing the
ideas promolgated here servethe needs of industry. The author's
consulting activities, for more than 30 different
companies,provided many useful examples. In the face of an
increasing awareness of the importance of energyto our societal
institutions and our way of life, it has been satisfying to see the
concepts pre-sented here applied not only to the development of new
energy systems, but to the conflictingproblem of environmental
control as well.
Where possible, examples have intentionally been chosen that can
be illustrated with gen-erally available films. Referenced in
Appendix C, these are in two series. The series from theNational
Committee on Fluid Mechanics Films was being developed at the
Education DevelopmentCenter while the author was active in making
three films in the series from the National Commit-tee on
Electrical Engineering Films. Interaction with such individuals as
Ascher H. Shapiro andJ. A. Shercliff fostered an interest in using
films to enliven and undergird classroom education.
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While graduate students involved with the subject or carrying
out their PhD theses, a numberof people have made substantial
contributions. Some of these are James F. Hoburg (Secs. 8.17
and8.18), Jose Ignacio Perez Arriaga (Secs. 4.5 and 4.8), Peter W.
Dietz (Sec. 5.17), Richard S.Withers (Secs. 5.8 and 5.9), Kent R.
Davey (Sec. 8.5), and Richard M. Ehrlich (Sec. 5.9).
Problems at the ends of chapters were typed by Eleanor J.
Nicholson. Figures were drawnby the author.
Solutions to the problems have been prepared in the form of a
manual. Intended as an aid tothose either presenting this material
in the classroom or using it for self-study, this manual
isavailable for the cost of reproduction from the author. Requests
should be over the signature ofeither a member of a university
faculty or the industrial equivalent.
James R. Melcher
Cambridge, MassachusettsJanuary, 1981
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Contents
1. INTRODUCTION TO CONTINUUM ELECTROMECHANICS
Background 1.1Applications 1.2Energy Conversion Processes
1.4Dynamical Processes and Characteristic Times 1.4Models and
Approximations 1.4Transfer Relations and Continuum Dynamics of
Linear Systems 1.6
2. ELECTRODYNAMIC LAWS, APPROXIMATIONS AND RELATIONS
2.1 Definitions 2.12.2 Differential Laws of Electrodynamics
2.12.3 Quasistatic Laws and the Time-Rate Expansion 2.22.4
Continuum Coordinates and the Convective Derivative 2.62.5
Transformations between Inertial Frames 2.72.6 Integral Theorems
2.92.7 Quasistatic Integral Laws 2.102.8 Polarization of Moving
Media 2.112.9 Magnetization of Moving Media 2.132.10 Jump
Conditions 2.14
Electroquasistatic Jump Conditions 2.14Magnetoquasistatic Jump
Conditions 2.18Summary of Electroquasistatic and Magnetoquasistatic
Jump Conditions 2.19
2.11 Lumped Parameter Electroquasistatic Elements 2.192.12
Lumped Parameter Magnetoquasistatic Elements 2.202.13 Conservation
of Electroquasistatic Energy 2.22
Thermodynamics 2.22Power Flow 2.24
2.14 Conservation of Magnetoquasistatic Energy 2.26
Thermodynamics 2.26Power Flow 2.28
2.15 Complex Amplitudes; Fourier Amplitudes and Fourier
Transforms 2.29
Complex Amplitudes 2.29Fourier Amplitudes and Transforms
2.30Averages of Periodic Functions 2.31
2.16 Flux-Potential Transfer Relations for Laplacian Fields
2.32
Electric Fields 2.32Magnetic Fields 2.32Planar Layer
2.32Cylindrical Annulus 2.34Spherical Shell 2.38
2.172.182.19
Energy Conservation and Quasistatic Transfer Relations
2.40Solenoidal Fields, Vector Potential and Stream Function
2.42Vector Potential Transfer Relations for Certain Laplacian
Fields 2.42
Cartesian Coordinates 2.45Polar Coordinates 2.45Axisymmetric
Cylindrical Coordinates 2.45
2.20 Methodology 2.46
PROBLEMS 2.47
3. ELECTROMAGNETIC FORCES, FORCE DENSITIES AND STRESS
TENSORS
Macroscopic versus Microscopic Forces 3.1The Lorentz Force
Density 3.1Conduction 3.2Quasistatic Force Density
3.4Thermodynamics of Discrete Electromechanical Coupling 3.4
Electroquasistatic Coupling 3.4Magnetoquasistatic Coupling
3.6
3.6 Polarization and Magnetization Force Densities on Tenuous
Dipoles 3.6
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3.7 Electric Korteweg-Helmholtz Force Density 3.9
Incompressible Media 3.11Incompressible and Electrically Linear
3.12Electrically Linear with Polarization Dependent on Mass Density
Alone 3.12Relation to the Kelvin Force Density 3.12
3.8 Magnetic Korteweg-Helmholtz Force Density 3.13
Incompressible Media 3.15Incompressible and Electrically Linear
3.15Electrically Linear with Magnetization Dependent on Mass
Density Alone 3.15Relation to Kelvin Force Density 3.15
3.9 Stress Tensors 3.153.10 Electromechanical Stress Tensors
3.173.11 Surface Force Density 3.193.12 Observations-3.21
PROBLEMS 3.23
4. ELECTROMECHANICAL KINEMATICS: ENERGY-CONVERSION MODELS AND
PROCESSES
4.1 Objectives 4.14.2 Stress, Force, and Torque in Periodic
Systems 4.14.3 Classification of Devices and Interactions 4.2
Synchronous Interactions 4.4D-C InteractionsSynchronous
Interactions with Instantaneously Induced Sources 4.5
4.4 Surface-Coupled Systems: A Permanent Polarization
Synchronous Machine 4.8
Boundary Conditions 4.8Bulk Relations 4.10Torque as a Function
of Voltage and Rotor Angle (v,Or) 4.10Electrical Terminal Relations
4.11
4.5 Constrained-Charge Transfer Relations 4.13
Particular Solutions (Cartesian Coordinates) 4.14Cylindrical
Annulus 4.15Orthogonality of Hi's and Evaluation of Source
Distributions 4.16
4.6 Kinematics of Traveling-Wave Charged-Particle Devices
4.17
Single-Region Model 4.19Two-Region Model 4.20
4.7 Smooth Air-Gap Synchronous Machine Model 4.21
Boundary Conditions 4.23Bulk Relations 4.23Torque as a Function
of Terminal Currents and Rotor Angle 4.23Electrical Terminal
Relations 4.25Energy Conservation 4.25
4.8 Constrained-Cufrent Magnetoquasistatic Transfer Relations
4.264.9 Exposed Winding Synchronous Machine Model 4.28
Boundary Conditions 4.30Bulk Relations 4.30Torque as a Function
of Terminal Variables 4.30Electrical Terminal Relations 4.31
4.10 D-C Magnetic Machines 4.33
Mechanical Equations 4.36Electrical Equations 4.37The Energy
Conversion Process 4.39
4.11 Green's Function Representations 4.404.12
Quasi-One-Dimensional Models and the Space-Rate Expansion 4.414.13
Variable-Capacitance Machines 4.44
Synchronous Condition 4.46
viii
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4.14 Van de Graaff Machine 4.49
Quasi-One-Dimensional Fields 4.49Quasistatics 4.51Electrical
Terminal Relations 4.52Mechanical Terminal Relations 4.52Analogy to
the Magnetic Machine 4.52The Energy Conversion Process 4.53
4.15 Overview of Electromechanical Energy Conversion Limitations
4.53
Synchronous'Alternator 4.54Superconducting Rotating Machine
4.54Variable-Capacitance Machine 4.55Electron-Beam Energy
Converters 4.56
PROBLEMS 4.57
5. CHARGE MIGRATION, CONVECTION AND RELAXATION
5.1 Introduction 5.15.2 Charge Conservation with Material
Convection 5.25.3 Migration in Imposed Fields and Flows 5.5
Steady Migration with Convection 5.6Quasistationary Migration
with Convection 5.7
5.4 Ion Drag Anemometer 5.75.5 Impact Charging of Macroscopic
Particles: The Whipple and Chalmers Model 5.9
Regimes (f) and (i) for Positive Ions; (d) and (g) for Negative
Ions 5.14Regimes (d) and (g) for Positive Ions; (f) and (i) for
Negative Ions 5.14Regimes (j) and (k) for Positive Ions; (b) and
(c) for Negative Ions 5.15Regime (k) for Positive Ions; (a) for
Negative Ions 5.15Regime (e), Positive Ions; Regime (h), Negative
Ions 5.15Regime (h) for Positive Ions; (e) for Negative Ions
5.15Positive and Negative Particles Simultaneously 5.16Drop
Charging Transient 5.16
5.6 Unipolar Space Charge Dynamics: Self-Precipitation 5.17
General Properties 5.18A Space-Charge Transient 5.19Steady-State
Space-Charge Precipitator 5.20
5.7 Collinear Unipolar Conduction and Convection: Steady D-C
Interactions 5.22
The Generator Interaction 5.24The Pump Interaction 5.24
5.8 Bipolar Migration with Space Charge 5.26
Positive and Negative Ions in a Gas 5.26Aerosol Particles
5.27Intrinsically Ionized Liquid 5.27Partially Dissociated Salt in
Solvent 5.27Summary of Governing Laws 5.27Characteristic Equations
5.28One-Dimensional Characteristic Equations 5.28Numerical Solution
5.29Numerical Example 5.30
5.9 Conductivity and Net Charge Evolution with Generation and
Recombination:Ohmic Limit 5.33
Maxwell's Capacitor 5.35Numerical Example 5.36
DYNAMICS OF OHMIC CONDUCTORS
5.10 Charge Relaxation in Deforming Ohmic Conductors 5.38
Region of Uniform Properties 5.39Initial Value Problem
5.39Injection from a Boundary 5.40
5.11 Ohmic Conduction and Convection in Steady State: D-C
Interactions 5.42
The Generator Interaction 5.43The Pump Interaction 5.43
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5.12 Transfer Relations and Boundary Conditions for Uniform
Ohmic Layers 5.44
Transport Relations 5.44Conservation of Charge Boundary
Condition 5.44
5.13 Electroquasistatic Induction Motor and Tachometer 5.45
Induction Motor 5.461 i. T U C .6a
5.14 An Electroquasistatic Induction Motor; Von Quincke's Rotor
5.495.15 Temporal Modes of Charge Relaxation 5.54
Temporal Transients Initiated from State of Spatial Periodicity
5.54Transient Charge Relaxation on a Thin Sheet 5.55Heterogeneous
Systems of Uniform Conductors 5.56
5.16 Time Average of Total Forces and Torques in the Sinusoidal
Steady State 5.60
Fourier Series Complex Amplitudes 5.60Fourier Transform Complex
Amplitudes 5.60
5.17 Spatial Modes and Transients in the Sinusoidal Steady State
5.61
Spatial Modes for a Moving Thin Sheet 5.62Spatial Transient on
Moving Thin Sheet 5.66Time-Average Force 5.67
PROBLEMS 5.71
6. MAGNETIC DIFFUSION AND INDUCTION INTERACTIONS
6.1 Introduction 6.16.2 Magnetic Diffusion in Moving Media
6.16.3 Boundary Conditions for Thin Sheets and Shells 6.4
Translating Planar Sheet 6.5
6.4 Magnetic Induction Motors and a Tachometer 6.6
Two-Phase Stator Currents 6.6Fields 6.7Time-Average Force
6.8Balanced Two-Phase Fields and Time-Average Force 6.8Electrical
Terminal Relations 6.8Balanced Two-Phase Equivalent Circuit
6.10Single-Phase Machine 6.10Tachometer 6.11
6.5 Diffusion Transfer Relations for Materials in Uniform
Translationor Rotation 6.11
Planar Layer in Translation 6.13Rotating Cylinder
6.13Axisymmetric Translating Cylinder 6.14
6.6 Induction Motor with Deep Conductor: A Magnetic Diffusion
Study 6.15
Time-Average Force 6.15Thin-Sheet Limit 6.17Conceptualization of
Diffusing Fields 6.17
6.7 Electrical Dissipation 6.196.8 Skin-Effect Fields,
Relations, Stress and Dissipation 6.20
Transfer Relations 6.21Stress 6.21Dissipation 6.22
6.9 Magnetic Boundary Layers 6.22
Similarity Solution 6.24Normal Flux Density 6.25Force 6.26
6.10 Temporal Modes of Magnetic Diffusion 6.26
Thin-Sheet Model 6.26Modes in a Conductor of Finite Thickness
6.27Orthogonality of Modes 6.29
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6.11 Magnetization Hysteresis Coupling: Hysteresis Motors
6.30
PROBLEMS 6.35
7. LAWS, APPROXIMATIONS AND RELATIONS OF FLUID MECHANICS
7.1 Introduction 7.17.2 Conservation of Mass 7.1
Incompressibility 7.1
7.3 Conservation of Momentum 7.27.4 Equations of Motion for an
Inviscid Fluid 7.27.5 Eulerian Description of the Fluid Interface
7.37.6 Surface Tension Surface Force Density 7.4
Energy Constitutive Law for a Clean Interface 7.4Surface Energy
Conservation 7.5Surface Force Density Related to Interfacial
Curvature 7.5Surface Force Density Related to Interfacial
Deformation 7.6
7.7 Boundary and Jump Conditions 7.87.8 Bernoulli's Equation and
Irrotational Flow of Homogeneous Inviscid Fluids 7.9
A Capillary Static Equilibrium 7.10
7.9 Pressure-Velocity Relations for Inviscid, Incompressible
Fluid 7.11
Streaming Planar Layer 7.11Streaming Cylindrical Annulus
7.13Static Spherical Shell 7.13
7.10 Weak Compressibility 7.137.11 Acoustic Waves and Transfer
Relations 7.13
Pressure-Velocity Relations for Planar Layer
7.14Pressure-Velocity Relations for Cylindrical Annulus 7.15
7.12 Acoustic Waves, Guides and Transmission Lines 7.15
Response to Transverse Drive 7.16Spatial Eigenmodes 7.17Acoustic
Transmission Lines 7.17
7.13 Experimental Motivation for Viscous Stress Dependence on
Strain Rate 7.187.14 Strain-Rate Tensor 7.20
Fluid Deformation Example 7.21Strain Rate as a Tensor 7.21
7.15 Stress-Strain-Rate Relations 7.21
Principal Axes 7.22Strain-Rate Principal Axes the Same as for
Stress 7.22Principal Coordinate Relations 7.23Isotropic Relations
7.23
7.16 Viscous Force Density and the Navier-Stokes's Equation
7.247.17 Kinetic Energy Storage, Power Flow and Viscous Dissipation
7.257.18 Viscous Diffusion 7.26
Convection Diffusion of Vorticity 7.26Perturbations from Static
Equilibria 7.27Low Reynolds Number Flows 7.27
7.19 Perturbation Viscous Diffusion Transfer Relations 7.28
Layer of Arbitrary Thickness 7.29Short Skin-Depth Limit
7.31Infinite Half-Space of Fluid 7.31
7.20 Low Reynolds Number Transfer Relations 7.32
Planar Layer 7.32Axisymmetric Spherical Flows 7.33
7.21 Stokes's Drag on a Rigid Sphere 7.367.22 Lumped Parameter
Thermodynamics of Highly Compressible Fluids 7.36
Mechanical Equations of State 7.37Energy Equation of State for a
Perfect Gas 7.37Conservation of Internal Energy in CQS Systems
7.37
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7.23 Internal Energy Conservation in a Highly Compressible Fluid
7.38
Power Conversion from Electromagnetic to Internal Form 7.39Power
Flow Between Mechanical and Internal Subsystems 7.39Integral
Internal Energy Law 7.39Combined Internal and Mechanical Energy
Laws 7.39Entropy Flow 7.40
7.24 Overview 7.41
PROBLEMS 7.43
8. STATICS AND DYNAMICS OF SYSTEMS HAVING A STATIC
EQUILIBRIUM
8.1 Introduction 8.1
STATIC EQUILIBRIA
8.2 Conditions for Static Equilibria 8.18.3 Polarization and
Magnetization Equilibria: Force Density and Stress
Tensor Representations 8.4
Kelvin Polarization Force Density 8.4Korteweg-Helmholtz
Polarization Force Density 8.6
Korteweg-Helmholtz Magnetization Force Density 8.6
8.4 Charge Conserving and Uniform Current Static Equilibria
8.8
Uniform Charged Layers 8.8Uniform Current Density 8.10
8.5 Potential and Flux Conserving Equilibria 8.11
Antiduals 8.12Bulk Relations 8.13Stress Equilibrium
8.13Evaluation of Surface Deflection 8.13Evaluation of Stress
Distribution 8.14
HOMOGENEOUS BULK INTERACTIONS
8.6 Flux Conserving Continua and Propagation of Magnetic Stress
8.16
Temporal Modes 8.19Spatial Structure of Sinusoidal Steady-State
Response 8.19
8.7 Potential Conserving Continua and Electric Shear Stress
Instability 8.20
Temporal Modes 8.23
8.8 Magneto-Acoustic and Electro-Acoustic Waves 8.25
Magnetization Dilatational Waves 8.27
PIECEWISE HOMOGENEOUS SYSTEMS
8.9 Gravity-Capillary Dynamics 8.28
Driven Response 8.29Gravity-Capillary Waves 8.30Temporal
Eigenmodes and Rayleigh-Taylor Instability 8.30Spatial Eigenmodes
8.31
8.10 Self-Field Interfacial Instabilities 8.338.11 Surface Waves
with Imposed Gradients 8.38
Bulk Relations 8.38Jump Conditions 8.39Dispersion Equation
8.39Temporal Modes 8.39
8.12 Flux Conserving Dynamics of the Surface Coupled z-0 Pinch
8.40
Equilibrium 8.42Bulk Relations 8.42Boundary and Jump Conditions
8.42Dispersion Equation 8.43
8.13 Potential Conserving Stability of a Charged Drop:
Rayleigh's Limit 8.44
Bulk Relations 8.45Boundary Conditions 8.45Dispersion Relation
and Rayleigh's Limit 8.45
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8.14 Charge Conserving Dynamics of Stratified Aerosols 8.46
Planar Layer 8.46Boundary Conditions 8.47Stability of Two Charge
Layers 8.47
8.15 The z Pinch with Instantaneous Magnetic Diffusion 8.50
Liquid Metal z Pinch 8.51Bulk Relations 8.51Boundary Conditions
8.52Rayleigh-Plateau Instability 8.53z Pinch Instability 8.54
8.16 Dynamic Shear Stress Surface Coupling 8.54
Static Equilibrium 8.54Bulk Perturbations 8.55Jump Conditions
8.55Dispersion Equation 8.56
SMOOTHLY INHOMOGENEOUS SYSTEMS AND THEIR INTERNAL MODES
8.17 Frozen Mass and Charge Density Transfer Relations 8.57
Weak-Gradient Imposed Field Model 8.59
Reciprocity and Energy Conservation 8.60
8.18 Internal Waves and Instabilities 8.62
Configuration 8.62Normalization 8.62Driven Response 8.63Spatial
Modes 8.66Temporal Modes 8.66
PROBLEMS 8.69
9. ELECTROMECHANICAL FLOWS
9.1 Introduction 9.19.2 Homogeneous Flows with Irrotational
Force Densities 9.2
Inviscid Flow 9.2Uniform Inviscid Flow 9.2Inviscid Pump or
Generator with Arbitrary Geometry 9.4Viscous Flow 9.5
FLOWS WITH IMPOSED SURFACE AND VOLUME FORCE DENSITIES
9.3 Fully Developed Flows Driven by Imposed Surface and Volume
Force Densities 9.59.4 Surface-Coupled Fully Developed Flows
9.7
Charge-Monolayer Driven Convection 9.7EQS Surface Coupled
Systems 9.10MQS Systems Coupled by Magnetic Shearing Surface Force
Densities 9.10
9.5. Fully Developed Magnetic Induction Pumping 9.119.6 Temporal
Flow Development with Imposed Surface and Volume Force Densities
9.13
Turn-On Transient of Reentrant Flows 9.13
9.7 Viscous Diffusio4 Boundary Layers 9.16
Linear Boundary Layer 9.17Stream-Function Form of Boundary Layer
Equations 9.17Irrotational Force Density; Blasius Boundary Layer
9.18Stress-Constrained Boundary Layer 9.20
9.8 Cellular Creep Flow Induced by Nonuniform Fields 9.22
Magnetic Skin-Effect Induced Convection 9.22
Charge-Monolayer Induced Convection 9.24
SELF-CONSISTENT IMPOSED FIELD
9.9 Magnetic Hartmann Type Approximation and Fully Developed
Flows 9.25
Approximation 9.25Fully Developed Flow 9.26
xiii
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9.10 Flow Development in the Magnetic Hartmann Approximation
9.289.11 Electrohydrodynamic Imposed Field Approximation 9.329.12
Electrohydrodynamic "Hartmann" Flow 9.339.13 Quasi-One-Dimensional
Free Surface Models 9.35
Longitudinal Force Equation 9.36Mass Conservation 9.36Gravity
Flow with Electric Surface Stress 9.37
9.14 Conservative Transitions in Piecewise Homogeneous Flows
9.37
GAS DYNAMIC FLOWS AND ENERGY CONVERTERS
9.15 Quasi-One-Dimensional Compressible Flow Model 9.419.16
Isentropic Flow Through Nozzles and Diffusers 9.429.17 A
Magnetohydrodynamic Energy Converter 9.45
MHD Model 9.46Constant Velocity Conversion 9.47
9.18 An Electrogasdynamic Energy Converter 9.48
The EGD Model 9.49Electrically Insulated Walls 9.51Zero Mobility
Limit with Insulating Wall 9.51Constant Velocity Conversion
9.52
9.19 Thermal-Electromechanical Energy Conversion Systems
9.53
PROBLEMS 9.57
10. ELECTROMECHANICS WITH THERMAL AND MOLECULAR DIFFUSION
10.1 Introduction 10.110.2 Laws, Relations and Parameters of
Convective Diffusion 10.1
Thermal Diffusion 10.1Molecular Diffusion of Neutral Particles
10.2Convection of Properties in the Face of Diffusion 10.3
THERMAL DIFFUSION
10.3 Thermal Transfer Relations and an Imposed Dissipation
Response 10.5
Electrical Dissipation Density 10.5Steady Response
10.6Traveling-Wave Response 10.6
10.4 Thermally Induced Pumping and Electrical Augmentation of
Heat Transfer 10.8
Electric Relations 10.8Mechanical Relations 10.8Thermal
Relations 10.9
10.5 Rotor Model for Natural Convection in a Magnetic Field
10.10
Heat Balance for a Thin Rotating Shell 10.11Magnetic Torque
10.12Buoyancy Torque 10.12Viscous Torque 10.12Torque Equation
10.12Dimensionless Numbers and Characteristic Times 10.13Onset and
Steady Convection 10.14
10.6 Hydromagnetic B~nard Type Instability 10.15
MOLECULAR DIFFUSION
10.7 Unipolar-Ion Diffusion Charging of Macroscopic Particles
10.1910.8 Charge Double Layer 10.2110.9 Electrokinetic Shear Flow
Model 10.23
Zeta Potential Boundary Slip Condition 10.24Electro-Osmosis
10.24Electrical Relations; Streaming Potential 10.25
10.10 Particle Electrophoresis and Sedimentation Potential
10.25
Electric Field Distribution 10.26Fluid Flow and Stress Balance
10.27
xiv
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10.11 Electrocapillarity 10.2710.12 Motion of a Liquid Drop
Driven by Internal Currents 10.32
Charge Conservation 10.33Stress Balance 10.34
PROBLEMS 10.37
11. STREAMING INTERACTIONS
11.1 Introduction 11.1
BALLISTIC CONTINUA
11.2 Charged Particles in Vacuum; Electron Beams 11.1
Equations of Motion 11.1Energy Equation 11.2Theorems of Kelvin
and Busch 11.2
11.3 Magnetron Electron Flow 11.311.4 Paraxial Ray Equation:
Magnetic and Electric Lenses 11.6
Paraxial Ray Equation 11.6Magnetic Lens 11.8Electric Lens
11.8
11.5 Plasma Electrons and Electron Beams 11.10
Transfer Relations 11.10Space-Charge Dynamics 11.10Temporal
Modes 11.11Spatial Modes 11.12
DYNAMICS IN SPACE AND TIME
11.6 Method of Characteristics 11.13
First Characteristic Equations 11.13Second Characteristic Lines
11.14Systems of First Order Equations 11.14
11.7 Nonlinear Acoustic Dynamics: Shock Formation 11.16
Initial Value Problem 11.16The Response to Initial Conditions
11.17Simple Waves 11.18Limitation of the Linearized Model 11.20
11.8 Nonlinear Magneto-Acoustic Dynamics 11.21
Equations of Motion 11.21Characteristic Equations 11.21Initial
Value Response 11.22
11.9 Nonlinear Electron Beam Dynamics 11.2311.10 Causality and
Boundary Conditions: Streaming Hyperbolic Systems 11.27
Quasi-One-Dimensional Single Stream Models 11.28Single Stream
Characteristics 11.29Single Stream Initial Value Problem
11.30Quasi-One-Dimensional Two-Stream Models 11.32Two-Stream
Characteristics 11.33Two-Stream Initial Value Problem
11.34Causality and Boundary Conditions 11.35
LINEAR DYNAMICS IN TERMS OF COMPLEX WAVES
11.11 Second-Order Complex Waves 11.37
Second Order Long-Wave Models 11.37Spatial Modes 11.39Driven
Response of Bounded System 11.40Instability of Bounded System
11.41Driven Response of Unbounded System 11.45
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11.12 Distinguishing Amplifying from Evanescent Modes 11.46
Laplace and Fourier Transform Representation in Time and Space
11.47Laplace Transform on Time as the Sum of Spatial Modes:
Causality 11.48Asymptotic Response in the Sinusoidal Steady State
11.50Criterion Based on Mapping Complex k as Function of Complex W
11.53
11.13 Distinguishing Absolute from Convective Instabilities
11.54
Criterion Based on Mapping Complex k as Function of Complex W
11.54Second Order Complex Waves 11.55
11.14 Kelvin-Helmholtz Types of Instability 11.5611.15
Two-Stream Field-Coupled Interactions 11.1511.16 Longitudinal
Boundary Conditions and Absolute Instability 11.1611.17
Resistive-Wall Electron Beam Amplification 11.68
PROBLEMS 11.71
APPENDIX A. Differential Operators in Cartesian, Cylindrical,
andSpherical Coordinates
APPENDIX B. Vector and Operator IdentitiesAPPENDIX C. Films
INDEX
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Continuum Electromechanics
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1
Introduction to ContinuumElectromechanics
w
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1.1 Background
There are two branches to the area of electromagnetics. One is
primarily concerned with electro-magnetic waves. Typically of
interest are guided and propagating waves ranging from radio to
opticalfrequencies. These may propagate through free space, in
plasmas or through optical fibers. Althoughthe interaction of
electromagnetic waves with media of great variety is of essential
interest, and in-deed the media modify these waves, it is the
electromagnetic wave that is at center stage in thisbranch.
Dynamical phenomena of interest to this branch are typified by
times, T, shorter than thetransit time of an electromagnetic wave
propagating over a characteristic length of the "system"
beingconsidered. For a characteristic length Z and wave velocity c
(in free space, the velocity of light),this transit time is
k/c.
In the chapters that follow, it is the second branch of
electromagnetics that plays the majorrole. In the sense that
electromagnetic wave transit times are short compared to times of
interest,the electric and magnetic fields are quasistatic: T
>> R/c. The important dynamical processes relateto conduction
phenomena, to the mechanics of ponderable media, and to the two-way
interaction createdby electromagnetic forces as they elicit a
mechanical response that in turn alters the fields.
Because the mechanics can easily upstage the electromagnetics in
this second branch, it is likelyto be perceived in terms of a few
of its many parts. For example, from the electromagnetic point
ofview there is much in common between issues that arise in the
design of a synchronous alternator andof a fusion experiment. But,
on the mechanical side, the rotating machine, with its problems of
vibra-tion and fatigue, seems to have little in common with the
fluid-like plasma continuum. So, the twoareas are not generally
regarded as being related.
In this text, the same fundamentals bear on a spectrum of
applications. Some of these are re-viewed in Sec. 1.2. The unity of
these widely ranging topics hinges on concepts, principles
andtechniques that can be traced through the chapters that follow.
By way of a preview, Secs. 1.3-1.7are outlines of these chapters,
based on themes designated by the section headings.
Chapters 2 and 3 are concerned with fundamentals. First the laws
and approximations are intro-duced that account for the effect of
moving media on electromagnetic fields. Then, the-force den-sities
and associated stress tensors needed to account for the return
influence of the fields on themotion are formulated.
Chapter 4 takes up the class of devices and phenomena that can
be described by models in whichthe distributions (or the relative
distributions) of both the material motion and of the field
sourcesare constrained. This subject of electromechanical
kinematics embraces lumped parameter electro-mechanics. The
emphasis here is on using the field point of view to determine the
relationship betweenthe lumped parameters and the physical
attributes of devices, and to determine the distribution of
stress and force density.
Chapters 5 and 6 retain the mechanical kinematics, but delve
into the self-consistent evolutionof fields and sources. Motions of
charged microscopic and macroscopic particles entrained in
movingmedia are of interest .in their own right, but also underlie
the limitations of commonly used conduc-tion constitutive laws.
These chapters both introduce basic concepts, such as the Method of
Charac-teristics and temporal and spatial modes, and model
practical devices ranging from the electrostaticprecipitator to the
linear induction machine.
Chapters 7-11 treat interactions of fields and media where not
only the field sources are freeto evolve in a way that is
consistent with the effect of deforming media, but the mechanical
systemsrespond on a continuum basis to the electric and magnetic
forces.
Chapter 7 introduces the basic laws and approximations of fluid
mechanics. The formulation oflaws, deduction of boundary conditions
and use of transfer relations is a natural extension of
theviewpoint introduced in the context of electromagnetics in Chap.
2.
Chapter 8 is concerned with electromechanical static equilibria
and the dynamics resulting fromperturbing these equilibria.
Illustrated are a range of electromechanical models motivated by
Chaps.5 and 6. It is here that temporal instability first comes to
the fore.
Chapter 9 is largely devoted to electromechanical flows.
Included is a discussion of flowdevelopment, understood in terms of
the same physical processes represented by characteristic times
-
in the previous four chapters. Flows that display super- and
sub-critical behavior presage causaleffects of wave propagation
taken up in Chap. 11. The last half of this chapter is an
introductionto "direct" thermal-to-electric energy conversion.
Chapter 10 is divided into parts that are each concerned with
diffusion processes. Thermal diffu-
sion, together with convective heat transfer, is considered
first. Electrical dissipation accompanies
almost all electromechanical processes, so that heat transfer
often poses an essential limitation on in-
vention and design. Because fields are often used for dielectric
or induction heating, this is a subjectin its own right. This part
begins with examples where the coupling is "one-way" and ends by
consideringsome of the mechanisms for two-way coupling between the
thermal and electromechanical subsystems. The
second part of this chapter serves as an introduction to
electromechanical processes that occur on a spa-tial scale small
enough that molecular diffusion processes come into play. Here
introduced is the inter-
play between electric and mechanical stresses that makes it
possible for particles to undergo electro-
phoresis rather than migrate in an electric field. The concepts
introduced in this second part are ap-
plicable to physicochemical systems and point to the
electromechanics of biological systems.
Chapter 11 brings together models and concepts from Chaps. 5-10,
emphasizing streaming interac-
tions, in which ordered kinetic energy is available for
participation in the energy conversion process.Included are
fluid-like continua such as electron beams and plasmas.
1.2 Applications
Transducers and rotating machines that are described by the
lumped parameter models of Chap. 4are so pervasive a part of modern
day technology that their development might be regarded as
complete.
But, with new technologies outside the domain of
electromechanics, there come new needs for electro-
mechanical devices. The transducers used to drive high-speed
computer print-outs are an example. New
devices in other areas also result in electromechanical
innovations. For example, high power solid-
state electronics is revolutionizing the design and utilization
of rotating machines.
As energy needs press the capabilities of electric power
systems, rotating machines continue to be
the mainstay of energy conversion to electrical form.
Synchronous generators are subject to in-
creasingly stringent demands. To improve capabilities,
superconducting windings are being incorpo-rated into a new class
of generators. In these synchronous alternators, magnetic materials
no longer
play the essential role that they do in conventional machines,
and new design solutions are required.
The Van de Graaff machine also considered in Chap. 4 should not
be regarded as a serious approach
to bulk power generation, but nevertheless represents an
important approach to the generation of ex-
tremely high potentials. It is also the grandfather of proposed
energy conversion approaches. An
example is the electrogasdynamic "thermal-to-electric" energy
converter of Chap. 9, Sec. 9.
Chapters 5 and 6 begin to hint at the diversity of applications
outside the domain of lumped
parameter electromechanics. The behavior of charged particles in
moving fluids is important for under-
standing liquid insulation in transformers and cables. Again, in
the area of power generation and dis-
tribution, ions and charged macroscopic particles contribute to
the contamination of high-voltage in-
sulators. Also related to the overhead line transmission of
electric power is the generation of audible
noise. In this case, the charged particles considered in Chap. 5
contribute to the transduction of
electrical energy into acoustic form, the result being a
sufficient nuisance that it figures in the de-
termination of rights of way.
Some examples in Chap. 5 are intended to give basic background
relevant to the control of particu-
late air pollution. The electrostatic precipitator is widely
used for air pollution control. Gasescleaned range from the
recirculating air within a single room to the exhaust of a utility.
With
industries of all sorts committed to the use of increasingly
dirtier fuels, new devices that also ex-
ploit electrical forces are under development. These include not
only air pollution control equipment,
but devices for painting, agricultural spraying, powder
deposition and the like.
Image processing is an application of charged particle dynamics,
as are other matters taken up in
later chapters. Charged droplet printing is under development as
a means of marrying the computer
to the printed page. Xerographic and aerosol printing of
considerable variety exploit electrical forces
on particles.
A visit to a printing plant, to a paper mill or to a textile
factory makes the importance of
charges and associated electrical forces on moving materials
obvious. The charge relaxation processes
considered in Chap. 5 are fundamental to understanding such
phenomena.
The induction machines considered in Chap. 6 are the most common
type of rotating motor. But
related interactions between moving conductors and magnetic
fields also figure in a host of other
applications. The development of high-speed ground
transportation has brought into play the linear
induction machine as a means of propulsion, and induced magnetic
forces as a means of producing mag-
Secs. 1.1 & 1.2
-
netic lift. Even if these developments do not reach maturity,
the induction type of interaction would
remain important because of its application to material
transport in manufacturing processes, and to
melting, levitation and pumping in metallurgical operations. The
application of induced magnetic
forces to the sorting of refuse is an example of how such
processes can figure in seemingly unrelated
areas.
Chapter 7 plays a role relative to fluid mechanics that Chap. 2
does with respect to electromag-
netics. Without a discourse on the applications of this material
in its own right, consider the rele-
vance of topics that are taken up in the subsequent
chapters.
Fields can be used to position, levitate and shape fluids. In
many cases, a static equilibriumis desired. Examples treated in
Chap. 8 include the levitation of liquid metals for
metallurgicalpurposes, shaping of interfaces in the processing of
plastics and glass, and orientation of ferrofluid
--- C i4----i idi...4 isea s an o cryogen c qu s n zero grav ty
env ronments.
The electromechanics of systems having a static equilibrium is
often dominated by instabilities.The insights gained in Chap. 8 are
a starting point in understanding atomization processes induced
bymeans of electric fields. Here, droplets formed by means of
electric fields figure in electrostaticpaint spraying and corona
generation from conductors under foul weather conditions. Internal
in-stabilities also taken up in Chap. 8 are basic to mixing of
liquids by electrical means and for elec-trical control of liquid
crystal displays. Both two-phase (boiling and condensation) and
convectiveheat transfer can be augmented by electromechanical
coupling, usually through the mechanism of in-stability. Perhaps
not strictly in the engineering domain is thunderstorm
electrification. Thestability of charged drops and the
electrohydrodynamics of air entrained collections of charged
drops
are topics touched upon in Chap. 8 that have this meteorological
application.
The statics and dynamics of hydromagnetic equilibria is now a
subject in its own right. Largelybecause of its relevance to fusion
machines, the discussion of hydromagnetic waves and surface
insta-bilities serves as an introduction to an area of active
research that, like other applications, hasimportant implications
for the energy posture. Internal modes taken up in Chap. 8 also
have counter-parts in hydromagnetics.
Magnetic pumping of liquid metals, taken up in Chap. 9, has
found application in nuclear reac-tors and in metallurgical
operations. Electrically induced pumping of semi-insulating and
insulating
liquids, also discussed in Chap. 9, has seen application, but in
a range of modes. A far wider range
of fluids have properties consistent with electric approaches to
pumping and hgnce there is the promise
of innovation in manufacturing and processing.
Magnetohydrodynamic power generation is being actively developed
as an approach to convertingthermal energy (from burning coal) to
electrical form. The discussion of this approach in Chap. 9 is
not only intended as an introduction to MHD energy conversion,
but to the general issues confronted in
any approach to thermal-to-electrical energy conversion,
including turbine-generator systems. The elec-
trohydrodynamic converter also discussed there is an alternative
to the MHD approach that sees periodic
interest. For that reason, its applicability is a matter that
needs to be understood.
Inductive and dielectric heating, even of materials at rest and
with no electromechanical con-siderations, are the basis for
important technologies. These topics, as well as the generation
andtransport of heat in electromechanical systems where thermal
effects often pose primary design limi-tations, are part of the
point of the first half of Chap. 10. But, thermal effects can also
becentral to the electromechanical coupling itself. Examples where
thermally induced property inhomo-geneities result in such coupling
include electrothermally induced convection of liquid
insulation.
Electromechanical coupling seated in double layers, also taken
up in Chap. 10, relates to proc-
esses (such as electrophoretic particle motions) that see
applications ranging from the painting ofautomobiles to the
chemical analysis of large molecules. One of the reasons for
including electro-
kinetic and electrocapillary interactions is the suggestion it
gives of mechanisms that can come into
play in biological systems, a subject that draws heavily on
physicochemical considerations. The
purely electromechanical models considered here serve to
identify this developing area.
The electromechanics of streaming fluids and fluid-like systems,
taken up in Chap. 11, has per-haps its best known applications in
the domain of electron beam engineering. Klystrons,
traveling-wave
tubes, resistive-wall amplifiers and the like are examples of
interactions between streams of chargedparticles (electrons) and
various types of structures. The space-time issues of Chap. 11 have
generalapplication to problems ranging from the stimulation of
liquid jets used to form drops, to electro-mechanical processes for
making synthetic fibers, to understanding liquid flow through
"wall-less"pipes (in which electric or magnetic fields play the
role of a duct wall), to beam-plasma interactionsthat result in
instabilities that are used as a mechanism for heating plasmas.
Sec. 1.2
-
1.3 Energy Conversion Processes
A theme of the chapters to follow is conversion of energy
between electrical and mechanical forms.
The relation between electromechanical power flow and the
product of electric or magnetic stress and
material velocity is first emphasized in Chap. 4. Rotating
machines deserve to be highlighted in this
basic sense, because for bulk power generation they are a
standard for comparison. But, even where kine-matic systems are
superseded by those involving self-consistent interactions, there
is value in con-sidering the kinematic examples. They make clear
the basic objectives governing the engineering ofmaterials and
fields even when the objectives are achieved by more devious
methods. For example, thesynchronous interactions with constrained
charged particles are not directly applicable to practicaldevices,
but highlight the basically electroquasistatic electric shear
stress interaction that under-lies electron beam interactions in
Chap. 11.
The classification of energy conversion processes made in Chap.
4 provides a frame of referencefor many of the self-consistent
interactions described in later chapters. Thus, d-c rotating
machinesfrom Chap. 4 have counterparts with fluid conductors in
Chap. 9, and the Van de Graaff generator is a
prototype for the gasdynamic models developed in Chaps. 5 and 9.
Electric and magnetic induction ma-chines, respectively taken up in
Chaps. 5 and 6, are a prototype for induction interactions with
fluids
in Chap. 9, And, the synchronous interactions of Chap. 4
motivate the self-consistent electron beaminteractions of Chap.
11.
1.4 Dynamical Processes and Characteristic Times
Rate processes familiar from electrical circuits are the
discharge of a capacitor (C) or an in-ductor (L) through a resistor
(R), or the oscillation of energy between a ca citor and an
inductor.One way to characterize the dynamics is in terms of the
times RC, L/R and C, respectively.
Characteristic times describing rate processes on a continuum
basis are a recurring theme. The
electromagnetic times summarized in Table 1.4.1 are the field
analogues of those familiar from circuit
theory. Rather than defining the variables, reference is made to
the section where the characteristic
times are introduced. Some of the mechanical and thermal ones
also have lumped parameter counter-
parts. For example, the viscous diffusion time, which represents
the mechanical damping of ponder-
able material, is the continuum version of the damping rate for
a dash-pot connected to a mass.
The electromechanical characteristic times represent the
competition between electric or magnetic
forces and viscous or inertial forces. In specialized areas,
they may appear in a different guise.
For example, with the electric field intensity that due to the
bunching of electrons in a plasma,
the electro-inertial time is the reciprocal plasma frequency. In
a highly conducting fluid stressed
by a magnetic field intensity H, the magneto-inertial time is
the transit time for an Alfvyn wave.
Especially in fluid mechanics, these characteristic times are
often brought into play as dimension-less ratios of times. Table
1.4.2 gives some of these ratios, again with references to the
sectionswhere they are introduced.
1.5 Models and Approximations
There are three classes of approximation, used repeatedly in the
following chapters, that shouldbe recognized as a recurring theme.
Formally, these are based on time-rate, space-rate and
amplitude-
parameter expansions of the relevant laws.
The time-rate approximation gives rise to a quasistatic model,
and exploits the fact that
temporal rates of change of interest are slow compared to one or
more times characterizing certain
dynamical processes. Some possible times are given in Table
1.4.1. Both for electroquasistaticsand magnetoquasistatics, the
critical time is the electromagnetic wave transit time, Tem (Sec.
2.3).
Space-rate approximations lead to quasi-one-dimensional (or
two-dimensional) models. These are
also known as long-wave models. Here, fields or deformations in
a "transverse" direction can be approxi-
mated as being slowly varying with respect to a "longitidunal"
direction. The magnetic field in a
narrow but spatially varying air gap and the flow of a gas
through a duct of slowly varying cross
section are examples.
Amplitude parameter expansions carried to first order result in
linearized models. Often theyare used to describe dynamics
departing from a static or steady equilibrium. Long-wave and
linearized
models are discussed and exemplified in Sec. 4.12, and are
otherwise used repeatedly without formality.
Secs. 1.3, 1.4 & 1.5
-
Table 1.4.1. Characteristic times for systems having a typical
length k.
Time Nomenclature Section reference
Electromagnetic
T em = l/c Electromagnetic wave transit time 2.3emT = 1/a Charge
relaxation time 2.3, 5.10
T = ax2 Magnetic diffusion time 2.3, 6.2m
T .mig= /bE Particle migration time 5.9
Mechanical and thermal
T = R/a Acoustic wave transit time 7.11
T = pt2/n Viscous diffusion time 7.18, 7.24
S= f/pa2 Viscous relaxation time 7.24
S= 2/K Molecular diffusion time 10.2D
T= 2pCv/kT Thermal diffusion time 10.2
Electromechanical
TV = n/cE2 Electro-viscous time 8.7
TMV = r/pHH2 Magneto-viscous time 8.6
T = p/E 2 Electro-inertial time 8.7
TMI= V/H 2 Magneto-inertial time 8.6
Table. 1.4.2. Dimensionless numbers as ratios of characteristic
times. The material transitor residence time is T = R/U, where U is
a typical material velocity.
Secs. 1.4 & 1.5
Number Symbol Nomenclature Sec. ref.
Electromagnetic
Te/T = EU/ka Re Electric Reynolds number 5.11
Tm/T = UckU Rm Magnetic Reynolds number 6.2
Mechanical and thermal
Ta/T = U/a M Mach number 9.19
Tv/T = pkU/n Ry Reynolds number 7.18
TD/T = ZU/K R Molecular Peclet number 10.2
TT/T = pcp U/kT RT Thermal Peclet number 10.2
TD/Tv = -/kD PD Molecular-viscous Prandtl number 10.2
TT/Tv = cp /kT PT Thermal-viscous Prandtl number 10.2
Electromechanical
` =-HEfy Hm Magnetic Hartmann number 8.6TEEV e EV He Electric
Hartmann number 9.12
Tm /T = no/p Pm Magnetic-viscous Prandtl number 8.6
-
1.6 Transfer Relations and Continuum Dynamics of Linear
Systems
Fields, flows and deformations in systems that are uniform in
one or more "longitudinal" direc-
tions can have the dependence on the associated coordinate
represented by complex amplitudes, Fourier
series, Fourier transforms, or the apDronriate extension of
these in various coordinate systems.Typically, configurations are
nonuniform in the remaining "transverse" coordinate. The dependence
ofvariables on this direction is represented by "transfer
relations." They are first introduced inChap. 2 as flux-potential
relations that encapsulate Laplacian fields in coordinate systems
for whichLaplace's equation is variable separable.
At the risk of having a forbidding appearance, most chapters
include summaries of transfer rela-tions in the three common
coordinate systems. This is done so that they can be a resource,
helping toobviate tedious manipulations that tend to obscure what
is essential in the derivation of a model. Thetransfer relations
help in organizing a development. Once the way in which they
represent the space-time dynamics of a given medium is appreciated,
they are also a way of quickly communicating thephysical nature of
a continuum.
Applications in Chap. 4 begin to exemplify how the transfer
relations can help to organize therepresentation of configurations
involving piece-wise uniform media. The systems considered there
arespatially periodic in the "longitudinal" direction.
With each of the subsequent chapters, the application of the
transfer relations is broadened. InChap. 5, the temporal transient
response is described in terms of the temporal modes. Then,
spatialtransients for systems in the temporal sinusoidal steady
state are considered. In Chap. 6, magneticdiffusion processes are
represented in terms of transfer relations, which take a form
equally applicableto thermal and particle diffusion.
Much of the summary of fluid mechanics given in Chap. 7 is
couched in terms of transfer relations.There, the variables are
velocities and stresses. In a wealth of electromechanical examples,
couplingbetween fields and media can be represented as occurring at
boundaries and interfaces, where there arediscontinuities in
properties. Thus, in Chap. 8, the purely mechanical relations of
Chap. 7 are com-
bined with the electrical relations from Chap. 2 to represent
electromechanical systems. More spe-cialized are electromechanical
transfer relations representing charged fluids, electron beams,
hydro-magnetic systems and the like, derived in Chaps. 8-11.
A feature of many of the examples in Chap. 8 is instability, so
that again the temporal modes
come to the fore. But with effects of streaming brought into
play in Chap. 11, there is a question
of whether the instability is absolute in the sense that the
response becomes unbounded with time at
a given point in space, or convective (amplifying) in that a
sinusoidal steady state can be
established but with a response that becomes unbounded in space.
These issues are taken up in Chap. 11.
Sec. 1.6
-
2
Electrodynamic Laws,Approximations and Relations
:14
-
2.1 Definitions
Continuum electromechanics brings together several disciplines,
and so it is useful to summarizethe definitions of electrodynamic
variables and their units. Rationalized MKS units are used not
onlyin connection with electrodynamics, but also in dealing with
subjects such as fluid mechanics and heattransfer, which are often
treated in English units. Unless otherwise given, basic units of
meters (m),kilograms (kg), seconds (sec), and Coulombs (C) can be
assumed.
Table 2.1.1. Summary of electrodynamic nomenclature.
Name Symbol Units
Discrete Variables
Voltage or potential difference v [V] = volts = m2 kg/C
sec2Charge q [C] = Coulombs = CCurrent i [A] = Amperes =
C/secMagnetic flux X [Wb] = Weber = m2 kg/C secCapacitance C [F] =
Farad C2 sec2 /m2 kgInductance L [H] = Henry = m2 kg/C2
Force f [N] = Newtons = kg m/sec2
Field Sources
Free charge density Pf C/m3
Free surface charge density f C/m2
Free current density 4f A/m2
Free surface current density Kf A/m
Fields (name in quotes is often used for convenience)
"Electric field" intensity V/m"Magnetic field" intensity
A/mElectric displacement C/m2
Magnetic flux density Wb/m 2 (tesla)Polarization density
C/m2Magnetization density M A/mForce density F N/m3
Physical Constants
Permittivity of free space 6o = 8.854 x 1012 F/mPermeability of
free space 1o = 4r x 10- 7 H/m
Although terms involving moving magnetized and polarized media
may not be familiar, Maxwell'sequations are summarized without
prelude in the next section. The physical significance of the
un-familiar terms can best be discussed in Secs. 2.8 and 2.9 after
the general laws are reduced to theirquasistatic forms, and this is
the objective of Sec. 2.3. Except for introducing concepts
concernedwith the description of continua, including integral
theorems, in Secs. 2.4 and 2.6, and the dis-cussion of Fourier
amplitudes in Sec. 2.15, the remainder of the chapter is a parallel
development ofthe consequences of these quasistatic laws. That the
field transformations (Sec. 2.5), integral laws(Sec. 2.7), splicing
conditions (Sec. 2.10), and energy storages are derived from the
fundamental quasi-static laws, illustrates the important dictum
that internal consistency be maintained within the frame-work of
the quasistatic approximation.
The results of the sections on energy storage are used in Chap.
3 for deducing the electric andmagnetic force densities on
macroscopic media. The transfer relations of the last sections are
animportant resource throughout all of the following chapters, and
give the opportunity to explore thephysical significance of the
quasistatic limits.
2.2 Differential Laws of Electrodynamics
In the Chu formulation,l with material effects on the fields
accounted for by the magnetizationdensity M and the polarization
density P and with the material velocity denoted by v, the laws
ofelectrodynamics are:
Faraday's law
4+ 3H P-at o M +(o Sto Bt
1. P. Penfield, Jr., and H. A. Haus, Electrodynamics of Moving
Media, The M.I.T. Press, Cambridge,Massachusetts, 1967, pp.
35-40.
-
Ampere's law
V x H = E + + V x (P x v) + J (2)ot t f
Gauss' law
V*E = -V*P + Pf (3)
divergence law for magnetic fields
oV.H = -ioV *M (4)
and conservation of free charge
V'Jf + t = 0 (5)
This last expression is imbedded in Ampere's and Gauss' laws, as
can be seen by taking the diver-gence of-Eq. 2 and exploiting Eq.
3. In this formulation the electric displacement and magnetic
fluxdensity B are defined fields:
D = E + P (6)o
4- -B = o(H + M) (7)
2.3 Quasistatic Laws and the Time-Rate Expansion
With a quasistatic model, it is recognized that relevant time
rates of change are sufficientlylow that contributions due to a
particular dynamical process are ignorable. The objective in
thissection is to give some formal structure to the reasoning used
to deduce the quasistatic field equa-tions from the more general
Maxwell's equations. Here, quasistatics specifically means that
timesof interest are long compared to the time, Tem, for an
electromagnetic wave to propagate through thesystem.
Generally, given a dynamical process characterized by some time
determined by the parameters ofthe system, a quasistatic model can
be used to exploit the comparatively long time scale for proc-esses
of interest. In this broad sense, quasistatic models abound and
will be encountered in manyother contexts in the chapters that
follow. Specific examples are:
(a) processes slow compared to wave transit times in general;
acoustic waves and the model isone of incompressible flow, Alfvyn
and other electromechanical waves and the model is less
standard;
(b) processes slow compared to diffusion (instantaneous
diffusion models). What diffuses canbe magnetic field, viscous
stresses, heat, molecules or hybrid electromechanical effects;
(c) processes slow compared to relaxation of continua
(instantaneous relaxation or constant-potential models). Charge
relaxation is an important example.
The point of making a quasistatic approximation is often to
focus attention on significantdynamical processes. A quasistatic
model is by no means static. Because more than one rate processis
often imbedded in a given physical system, it is important to agree
upon the one with respect towhich the dynamics are quasistatic.
Rate processes other than those due to the transit time of
electromagnetic waves enter throughthe dependence of the field
sources on the fields and material motion. To have in view the
additionalcharacteristic times typically brought in by the field
sources, in this section the free currentdensity is postulated to
have the dependence
Jf = G(r)E + Jv(v,pf,H) (i)
In the absence of motion, Jv is zero. Thus, for media at rest
the conduction model is ohmic, with the
el-ctrical conductivity a in general a funqtion Qf position.
Examples of Jv are a convection currentpfv, or an ohmic
motion-induced current a(v x 0oH). With an underbar used to denote
a normalizedquantity, the conductivity is normalized to a typical
(constant) conductivity a :
a = (r,t) (2)o-
To identify the hierarchy of critical time-rate parameters, the
general laws are normalized.Coordinates are normalized to one
typical length X, while T represents a characteristic dynamical
time:
(x,y,z) = (Zx,kY,kz); t = Tt (3)
Secs. 2.2 & 2.3
-
In a system sinusoidally excited at the angular frequency , T=
W-1lIn a system sinusoidally excited at the angular frequency w,
T=w
The most convenient normalization of the fields depends on the
specific system. Where electro-mechanical coupling is significant,
these can usually be categorized as "electric-field dominated"
and"magnetic-field dominated." Anticipating this fact, two
normalizations are now developed "in parallel,"the first taking e
as a characteristic electric field and the second taking _ as.a
characteristic mag-netic field:
o v T -v
p E f 80H 0 +pf =-9 , H=- - H,MM T-
H = H, M v = (/), = J
= Lf -v
E= P Pf p-,. P=f, - P
It might be appropriate with this step to recognize that the
material motion introduces a characteristic(transport) time other
than T. For simplicity, Eq. 4 takes the material velocity as being
of the orderof R/T.
The normalization used is arbitrary. The same quasistatic laws
will be deduced regardless of thestarting point, but the
normalization will determine whether these laws are "zero-order" or
higher orderin a sense to now be defined.
The normalizations of Eq. 4 introduced into Eqs. 2.2.1-5 result
in
V.1 = -V. + pf
V.H = -V-M
+T +. + E 9P ( x)VxH = - aE + J + + -- +Vx (P xv)
T v +t ~te
H+ 3 xV
~-V]VxE = -s t t + Vx (x
Se F ~fV. E + - V*J t+ ]e v 't J
V.E = -V.p +
V.H = -V.M
Tm
VxH = -- ET
S HVxE = at
+ J + O +
V x (Mx v)
- T DpV. E + V + 0T T t =
m m
where underbars on equation numbers are used to indicate that
the equations are normalized and
Tm 0a 2 , Te 0 0o/
= -em Vo o = Z/c (10)
In Chap. 6, T will be identified as the magnetic diffusion time,
while in Chap. 5 the role of thecharge-relaxation time Te is
developed. The time required for an electromagnetic plane wave to
propa-
gate the distance k at the velocity c is Tem. Given that there
is just one characteristic length,there are actually only two
characteristic times, because as can be seen from Eq. 10
(11)Tme em
Unless Te and Tm, and hence Tem, are all of the same order,
there are only two possibilities for the
relative magnitudes of these times, as summarized in Fig.
2.3.1.
18W(II Ir
Tm
electroquasistati cs
TCe mem
magnetoquasistatics
Fig. 2.3.1. Possible relations between physical time constants
on a time
scale T which typifies the dynamics of interest.
Sec. 2.3
+ Vx(P x v (7)BtA
(8)
( 4((1 ~I _
-
By electroquasistatic (EQS) approximation it is meant that the
ordering of times is as to the left andthat the parameter 08
(Tem/T)Z is much less than unity. Note that T is still arbitrary
relative to Te.In the magnetoquasistatic (MQS) approximation, 0 is
still small, but the ordering of characteristic timesis as to the
right. In this case, T is arbitrary relative to Tm.
To make a formal statement of the procedure used to find the
quasistatic approximation, the normal-ized fields and charge
density are expanded in powers of the time-rate parameter 0.
E = E + E1 + E2+
0 0o+ H01+ 8 +2 (12)
iv - ( v)o + 0v) +0 ( )2 +
Pf = (Pf)o + (Pf) 1 + ()2 +
In the following, it is assumed that constitutive laws relate P
and M to E and H, so that thesedensities are similarly expanded.
The velocity 4 is taken as given. Then, the series are sub-stituted
into Eqs. 5-9 and the resulting expressions arranged by factors
multiplying ascendingpowers of 0. The "zero order" equations are
obtained by requiring that the coefficients of 8vanish. These are
simply Eqs. 5-9 with B = 0:
V.- o = -V.-P + (pf) o
VxHo = - o0
e
apo+ --- +
at(Jv)o + --
Vx (o x V)
4.VxE 0
o
+ e (v = 0V.oE +T- o + at
V.E = -V.o + (P)o
v-H -V-MT
VxH -- a E + (J )
aH aMVxE o Vx(M x V)
o at at 0
V.o E +_V.) =0Eo T V)om
The zero-order solutions are found by solving these equations,
augmented by appropriate
boundary conditions. If the boundary conditions are themselves
time dependent, normalization
will turn up additional characteristic times that must be fitted
into the hierarchy of Fig. 2.3.1.
Higher order contributions to the series of Eq.
equations found by making coefficients of like powers
from setting the coefficients of an to zero are:
12 follow from a sequential solution of theof vanish. The
expressions resulting
V En + V., - )n = 0
V*Fn+VM --0n - n Vn
e
at Vx (n x) = 0)
aM 1atVx n ai Vx(Mi 1 x v)
V* Tn+ , n +I )a =o0n + )n at 0nl
V.* + ~*- (nf)nf = 0
v.* + V 0-n n
Vx. - mmE (J
V. E #n- E Tn tvnat nVAE ++x (mNO = 0
V T-C I a(Pf+(n T n atm m
(13)
(14)
(15)
(16)
(17)
-A
(18)
(19)
(20)
(21)
(22)
Sec. 2.3
-
To find the first order contributions, these equations with n=l
are solved with the zero order
solutions making up the right-hand sides of the equations
playing the role of known driving functions.
Boundary conditions are satisfied by the lowest order fields.
Thus higher order fields satisfy homo-
geneous boundary conditions.Once the first order solutions are
known, the process can be repeated with these forming the
"drives" for the n=2 equations.
In the absence of loss effects, there are no characteristic
times to distinguish MQS and EQSsystems. In that limit, which set
of normalizations is used is a matter of convenience. If a
situa-tion represented by the left-hand set actually has an EQS
limit, the zero order laws become the quasi-static laws. But, if
these expressions are applied to a situation that is actually MQS,
then first-order terms must be calculated to find the quasistatic
fields. If more than the one characteristictime Tern is involved,
as is the case with finite Te and Tm, then the ordering of rate
parameters cancontribute to the convergence of the expansion.
In practice, a formal derivation of the quasistatic laws is
seldom used. Rather, intuition andexperience along with comparison
of critical time constants to relevant dynamical times is used
toidentify one of the two sets of zero order expressions as
appropriate. But, the use of normalizationsto identify critical
parameters, and the notion that characteristic times can be used to
unscrambledynamical processes, will be used extensively in the
chapters to follow.
Within the framework of quasistatic electrodynamics, the
unnormalized forms of Eqs. 13-17conmrise the "exact" field laws
These enuations are reordered to reflect their relative
imnortance:
Electroquasistatic (EQS)
V.-E E= -V'P + Pf
Vx = 0
S apfV.Jf + --= 0
VxH = + +2-- + Vx (P x v)f t at
ViiH = -V PoM
Magnetoquasistatic (MQS)
Vx = f (23)
V.1oH = -V.o M (24)
a4,. allH 1VxE at at -oV x (M x v) (25)
VJ = o (26)
VeoE = -VP + Pf (27)0
The conduction current Jf has been reintroduced to reflect the
wider range of validity of these
equations than might be inferred from Eq. 1. With different
conduction models will come different
characteristic times,exemplified in the discussions of this
section by Te and Tm. Matters are more
complicated if fields and media interact electromechanically.
Then, v is determined to some extent
at least by the fields themselves and must be treated on a par
with the field variables. The result
can be still more characteristic times.
The ordering of the quasistatic equations emphasizes the
instantaneous relation between the
respective dominant sources and fields. Given the charge and
polarization densities in the EQS system,
or given the current and magnetization densities in the MQS
system, the dominant fields are known and
are functions only of the sources at the given instant in
time.
The dynamics enter in the EQS system with conservation of
charge, and in the MQS system with
Faraday'l law of induction. Equations 26a and 27a are only
needed 4f an after-the-fact determina-tion of H is to be made. An
example where such a rare interest in H exists is in the small
mag-netic field induced by electric fields and currents within the
human body. The distribution of in-
ternal fields and hence currents is determined by the first
three EQS equations. Given 1, , andJf, the remaining two
expressions determine H. In the MQS system, Eq. 27b can be regarded
as an
expression for the after-the-fact evaluation of pf, which is not
usually of interest in such systems.
What makes the subject of quasistatics difficult to treat in a
general way,even for a system
of fixed ohmic conductivity, is the dependence of the
appropriate model on considerations not con-
veniently represented in the differential laws. For example, a
pair of perfectly conducting plates,
shorted on one pair of edges and driven by a sinusoidal source
at the opposite pair, will be MQS
at low frequencies. The same pair of plates, open-circuited
rather than shorted, will be electroquasi-
static at low frequencies. The difference is in the boundary
conditions.
Geometry and the inhomogeneity of the medium (insulators,
perfect conductors and semiconductors)
are also essential to determining the appropriate approximation.
Most systems require more than one
Sec. 2.3
-
characteristic dimension and perhaps conductivity for their
description, with the result that more thantwo time constants are
often involved. Thus, the two possibilities identified in Fig.
2.3.1 can inprinciple become many possibilities. Even so, for a
wide range of practical problems, the appropriatefield laws are
either clearly electroquasistatic or magnetoquasistatic.
Problems accompanying this section help to make the significance
of the quasistatic limits moresubstantive by considering cases that
can also be solved exactly.
2.4 Continuum Coordinates and the Convective Derivative
There are two commonly used representations of continuum
variables. One of these is familiarfrom classical mechanics, while
the other is universally used in electrodynamics. Because
electro-mechanics involves both of these subjects, attention is now
drawn to the salient features of the tworepresentations.
Consider first the "Lagrangian representation." The position of
a material particle is a naturalexample and is depicted by Fig.
2.4.1a. When the time t is zero, a particle is found at the
positionro . The position of the particle at some subsequent time
is t. To let t represent the displacement ofa continuum of
particles, the position variable ro is used to distinguish
particles. In this sense, thedisplacement then also becomes a
continuum variable capable of representing the relative
displace-ments of an infinitude of particles.
u) kU)Fig. 2.4.1. Particle motions represented in terms of (a)
Lagrangian coordinates,
where the initial particle coordinate ro designates the particle
ofinterest, and (b) Eulerian coordinates, where (x,y,z) designates
thespatial position of interest.
In a Lagrangian representation, the velocity of the particle is
simply
at
If concern is with only one particle, there is no point in
writing the derivative as a partial deriv-ative. However, it is
understood that, when the derivgtive is taken, it is a particular
particlewhich is being considered. So, it is understood that ro is
fixed. Using the same line of reasoning,the acceleration of a
particle is given by
a at
The idea of representing continuum variables in terms of the
coordinates (x,y,z) connected withthe space itself is familiar from
electromagnetic theory. But what does it mean if the variable
ismechanical rather than electrical? We could represent the
velocit- of the continuum of particlesfilling the space of interest
by a vector function v(x,y,z,t) = v(r,t). The velocity of
particles
having the position (x,y,z,) at a given time t is determined by
evaluating the function v(r,t). Thevelocity appearing in Sec. 2.2
is an example. As suggested by Fig. 2.4.1b, if the function is
the
velocity evaluated at a given position in space, it describes
whichever particle is at that point at
the time of interest. Generally, there is a continuous stream of
particles through the point (x,y,z).
Secs. 2.3 & 2.4
~
)
-
Computation of the particle acceleration makes evident the
contrast between Eulerian and Lagrangianrepresentations. By
definition, the acceleration is the rate of change of the velocity
computed for agiven particle of matter. A particle having the
position (x,y,z) at time t will be found an instantAt later at the
position (x + vxAt,y + vyAt,z + vzAt). Hence the acceleration
is
v(x + v At,y + v At,z + v At,t + At) - v(x,y,z,t)a=lim x y z
(3)
AtOAt
Expansion of tje first term in Eq. 3 about the initial
coordinates of the particle gives the convectivederivative of
v:
+ v av av av _ v + +a + v + v + v + v*Vv (4)
t x ax y y z (4)at
The difference between Eq. 2 and Eq. 4 is resolved by
recognizing the difference in the signi-ficance of the partial
derivatives. In Eq. 2, it is understood that the coordinates being
held fixedare the initial coordinates of the particle of interest.
In Eq. 4, the partial derivative is taken,holding fixed the
particular point of interest in space.
The same steps .show that the rate of change of any vector
variable A, as viewed from a particlehaving the velocity v, is
DAaA 31 + (S- + (V); A = A(x,y,z,t) (5)
The time rate of change of any scalar variable for an observer
moving with the velocity v is obtainedfrom Eq. 5 by considering the
particular case in which t has only one component, say 1 =
f(x,y,z,t)Ax.Then Eq. 5 becomes
Df f +ff- E - -+ v.Vf (6)
Reference 3 of Appendix C is a film useful in understanding this
section.
2.5 Transformations between Inertial Frames
In extending empirically determined conduction, polarization and
magnetization laws to includematerial motion, it is often necessary
to relate field variables evaluated in different referenceframes. A
given point in space can be designated either in terms of the
coordinate 1 or of the co-ordinate V' of Fig. 2.5.1. By "inertial
reference frames," it is meant that the relative velocitybetween
these two frames is constant, designated by '. The positions in the
two coordinate systemsare related by the Galilean
transformation:
r' = r - ut; t' = t (1)
Fig. 2.5.1
Reference frames have constantrelative velocity t. The
co-ordinates t = (x,y,z) and 1' =(x',y',z') designate the
sameposition.
It is a familiar fact that variables describing a given physical
situation in one reference framewill not be the same as those in
the other. An example is material velocity, which, if measured in
oneframe, will differ from that in the other frame by the relative
velocity ~.
There are two objectives in this section: one is to show that
the quasistatic laws are invariantwhen subject to a Galilean
transformation between inertial reference frames. But, of more use
is the
relationship between electromagnetic variables in the two frames
of reference that follows from this
Secs. 2.4 & 2.5
-
proof. The approach is as follows. First, the postulate is made
that
the quasistatic equations take the
same form in the primed and unprimed inertial reference frames.
But, in writing the laws in the primed
frame, the spatial and temporal derivatives must be taken with
respect to the coordinates of that ref-
erence frame, and the dependent field variables are then fields
defined in that reference frame. In
general, these must be designated by primes, since their
relation
to the variables in the unprimed frame
is not known.
For the purpose of writing the primed equations of
electrodynamics in terms of the un rimed co-Forthe nurnose o
writina the nrimd enuations of elctrod-ax cs in tems of the u-
rime
ordinates, recognize that
V' + V
A a )+ - = al"a-)+ (- + u*V)A - + uV*A - Vx (uxA)
+(+ uV9 +Vu( t + E atThe left relations follow by using the
chain rule of differentiation and the transformation of Eq. 1.That
the spatial derivatives taken with respect to one frame must be the
same as those with respectto the other frame physically means that
a single "snapshot" of the physical process would be allrequired to
evaluate the spatial derivatives in either frame. There would be no
way of telling whichframe was the one from which the snapshot was
taken. By contrast, the time rate of change for anobserver in the
primed frame is, by definition, taken with the primed spatial
coordinates held fixed.In terms of the fixed frame coordinates,
this is the convective derivative defined with Eqs. 2.4.5and 2.4.6.
However, v in these equations is in general a function of space and
time. In the contextof this section it is saecialized to the
constant u. Thus, in rewriting the convective derivatives ofEq. 2
the constancy of u and a vector identity (Eq. 16, Appendix B) have
been used.
So far, what has been said in this section is a matter of
coordinates. Now, a physically motivatedpostulate is made
concerning the electromagnetic laws. Imagine one electromagnetic
experiment that isto be described from the two different reference
frames. The postulate is that provided each of theseframes is
inertial, the governing laws must take the same form. Thus, Eqs.
23-27 apply with [V - V',c()/at - a()/at'] and all dependent
variables primed. By way of comparing these laws to those ex-
pressed in the fixed-frame, Eqs. 2 are used to rewrite these
expressions in terms of the unprimed in-dependent variables. Also,
the moving-frame material velocity is rewritten in terms of the
unprimedframe velocity using the relation
v' v- u
Thus, the laws originally expressed in the primed frame of
reference become
V.e E'E -V.P' + p0 f
V x E' = 0
V.(ij + up!) + - 0
V x (' + ux C ') - ( + up )
aE$' + ' x,+ + + V x (P xat at
V~o oM0
V x i' =
V*oH' -V.o0 M'
Vx(' - u x ~i')al H0'at
ay M'- (6)at
- V x (' x V)
- f
V~eo'= V.P'+ !
In writing Eq. 7a, Eq. 4a is used. Similarly, Eq. 5b is used to
write Eq. 6b. For the one experi-ment under consideration, these
equations will.predict the same behavior as the fixed frame
laws,Eqs. 2.3.23-27, if the identification is made:
Sec. 2.5
-
E- ,
PE'= P
4. 4 +
S= J -Upf
H' .'A-i x eEo
and hence, from Eq. 2.2.6
D' =D
MQS
'- A (9)4. +M' = M (10)
J = Jf (11)
E'= E + ux poH (12)
(13)
and hence, from Eq. 2.2.7
B' = B (14)
The primary fields are the same whether viewed from one frame or
the other. Thus, the EQS elec-tric field polarization density and
charge density are the same in both frames, as are the MQS
mag-netic field, magnetization density and current density. The
respective dynamic laws can be associatedwith those field
transformations that involve the relative velocity. That the free
current densityis altered by the relative motion of the net free
charge in the EQS system is not surprising. But, itis the
contribution of this same convection current to Ampere's law that
generates the velocity depend-ent contribution to the EQS magnetic
field measured in the moving frame of reference. Similarly,
thevelocity dependent contribution to the MQS electric field
transformation is a direct consequence ofFaraday's law.
The transformations, like the quasistatic laws from which they
originate, are approximate. Itwould require Lorentz transformations
to carry out a similar procedure for the exact electrodynamiclaws
of Sec. 2.2. The general laws are not invariant in form to a
Galilean transformation, and there-in is the origin of special
relativity. Built in from the start in the quasistatic field laws
is aself-consistency with other Galilean invariant laws describing
mechanical continua that will be broughtin in later chapters.
2.6 Integral Theorems
Several integral theorems prove useful, not only in the
description of electromagnetic fields butalso in dealing with
continuum mechanics and electromechanics. These theorems will be
stated here with-out proof.
If it is recognized that the gradient operator is defined such
that its line integral between twoendpoints (a) and (b) is simply
the scalar function evaluated at the endpoints, thenl
I w4= -M) (1)a
Two more familiar theorems1 are useful in dealing with vector
functions. For a closed surface S, en-closing the volume V, Gauss'
theorem states that
V*AdV = '-nda (2)V S
while Stokes's theorem pertains to an open surface S with the
contour C as its periphery:
SV x A1da = A' (3)S C
In stating these theorems, the normal vector is defined as being
outward from the enclosed voluge forGauss' theorem, and the contour
is taken as positive in a direction such that It is related to n by
theright-hand rule. Contours, surfaces, and volumes are sketched in
Fig. 2.6.1.
A possibly less familiar theorem is the generalized Leibnitz
rule.2 In those cases where thesurface is itself a function of
time, it tells how to take the derivative with respect to time of
theintegral over an open surface of a vector function:
1. Markus Zahn, Electromagnetic Field Theory, a problem solving
approach, John Wiley & Sons, New York,1979, pp. 18-36.
2. H. H. Woodson and J. R. Melcher, Electromechanical Dynamics,
Vol. 1. John Wiley & Sons, New York,1968, pp. B32-B36.(See
Prob. 2.6.2 for the derivation of this theorem.)
Secs. 2.5 & 2.6
-
(a) (b) (c)Fig. 2.6.1. Arbitrary contours, volumes and surfaces:
(a) open contour C;
(b) closed surface S, enclosing volume V; (c) open surface Swith
boundary contour C.
- A~nda = [ + (V.A)v ]-nda + (IAx ).dx (4)dt at sS S C
Again, C is the contour which is the periphery of the open
surface S. The velocity vs is the velocityof the surface and the
contour. Unless given a physical significance, its meaning is
purely geometrical.
A limiting form of the generalized Leibnitz rule will be handy
in dealing with closed surfaces.Let the contour C of Eq. 4 shrink
to zero, so that the surface S becomes a closed one. This process
canbe readily visualized in terms of the surface and contour sketch
in Fig. 2.6.1c if the contour C ispictured as the draw-string on a
bag. Then, if C V-1, and use is made of Gauss' theorem (Eq. 2),Eq.
4 becomes a statement of how to take the time derivative of a
volume integral when the volume is afunction of time:
dV = f dV + s.nda (5)tV Vt S
Again, vs is the velocity of the surface enclosing the volume
V.
2.7 quasistatic Integral Laws
There are at least three reasons for desiring Maxwell's
equations in integral form. First, theintegral equations are
convenient for establishing jump conditions implied by the
differentialequations. Second, they are the basis for defining
lumped parameter variables such as the voltage,charge, current, and
flux. Third, they are useful in understanding (as opposed to
predicting) physicalprocesses. Since Maxwell's equations have
already been divided into the two quasistatic systems, itis now
possible to proceed in a straightforward way to write the integral
laws for contours, surfaces,and volumes which are distorting, i.e.,
that are functions of time. The velocity of a surface S is v .
To obtain the integral laws implied by the laws of Eqs.
2.3.23-27, each equation is either(i) integrated over an open
surface S with Stokes's theorem used where the integrand is a curl
operatorto convert to a line integration on C and Eq. 2.6.4 used to
bring the time derivative outside theintegral, or (ii) integrated
over a closed volume V with Gauss' theorem used to convert
integrationsof a divergence operator to integrals over closed
surfaces S and Eq. 2.6.5 used to bring the timederivative outside
the integration:
(E(E + P).-da = fdVS V
Jnda I fdV = 0
S V
H. = IJf da (1)C S
110 (H + M).nda - 0 (2)S
. -o(H + M)*nda (3)C S
- OPM x (:v- ).h%C
Secs. 2.6 & 2.7 2.10
-
H1. = J i.nda + (eE + P).n'daC S S
+ F P x (v - ')*C
SJo(H + M)-nda = 0S
where
J J - VsPf+ 4 4..H' =H - v x sE
s o
Sf.-nda = 0S
+ P).nda = PfdVV
where4' -E+ s
x.
E' -E+v xIIH0
The primed variables are simply summaries of the variables found
in deducing these equations. However,these definitions are
consistent with the transform relationships found in Sec. 2.5, and
the velocityof these surfaces and contours, vs, can be identified
with the velocity of an inertial frame instan-taneously attached to
the surface or contour at the point in question. Approximations
implicit to theoriginal differential quasistatic laws are now
implicit to these integral laws.
2.8 Polarization of Moving Media
Effects of polarization and magnetization are included in the
formulation of electrodynamicspostulated in Sec. 2.2. In this and
the next section a review is made of the underlying models.
Consider the electroquasistatic systems, where the dominant
field source is the charge density.Not all of this charge is
externally accessible, in the sense that it cannot all be brought
to someposition through a conduction process. If an initially
neutral dielectric medium is stressed by anelectric field, the
constituent molecules and domains become polarized. Even though the
materialretains its charge neutrality, there can be a local accrual
or loss of charge because of the polariza-tion. The first order of
business is to deduce the relation of such polarization charge to
the polari-zation density.
For conceptual purposes, the polarization of a material is
pictured as shown in Fig. 2.8.1.
Fig. 2.8.1. Model for dipoles fixed to deformable material. The
model picturesthe negative charges as fixed to the material, and
then the positivehalves of the dipoles fixed to the negative
charges through internalconstraints.
Secs. 2.7 & 2.82.11
-
Fig. 2.8.2
Polarization results in netcharges passing through asurface.
The molecules or domains are represented by dipoles composed of
positive and negative charges +q,separated by the vector distance
1. The dipole moment is then $ = qp, and if the particles have
anumber density n, the polarization density is defined as
P = nqa (1)
In the most common dielectrics, the polarization results because
of the application of an externalelectric field. In that case, the
internal constraints (represented by the springs in Fig. 2.8.1)make
the charges essentially coincident in the absence of an electric
field, so that, on the average,the material is (macroscopically)
neutral. Then,.with the application of the electric field, thereis
a separation of the charges in some direction which might be
coincident with the applied electricfield intensity. The effect of
the dipoles on the average electric field distribution is
equivalentto that of the medium they model.
To see how the polarization charge density is related to the
polarization density, consider themotion of charges through the
arbitrary surface S shown in Fig. 2.8.2. For the moment, consider
thesurface as being closed, so that the contour enclosing the
surface shown is shrunk to zero. Becausepolarization results in
motion of the positive charge, leaving behind the negative image
charge, the netpolarization charge within the volume V enclosed by
the surface S is equal to the negative of the netcharge having left
the volume across the surface S. Thus,
f pdV = - nq.i~da = - *"-da (2)P J J
S S
Gauss' theorem, Eq. 2.6.2, converts the surface integral to one
over the arbitrary volume V. Itfollows that the integrand must
vanish so that
4.
p = - V.P (3)
This polarization charge density is now added to the free charge
density as a source of the electricfield intensity in Gauss'
law:
V.E = Pf +Pp (4)
and Eqs. 3 and 4 comprise the postulated form of Gauss' law, Eq.
2.3.23a.
By definition, polarization charge is conserved, independent of
the free ch