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Continuum Electromechanics

To Janet Damman Melcher

Continuum Electromechanics

James R. Melcher

The MIT PressCambridge, Massachusetts, and London, England

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

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.

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

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

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

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

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

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

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

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

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

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

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

Continuum Electromechanics

1

Introduction to ContinuumElectromechanics

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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