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6.013 Electromagnetics and Applications, Fall 2005
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CONTENTS
Chapter 1-REVIEW OF VECTOR ANALYSIS
1.1 COORDINATE SYSTEMS
1.1.1 Rectangular(Cartesian)Coordinates
1.1.2 CircularCylindricalCoordinates
1.1.3 Spherical Coordinates
1.2 VECTOR ALGEBRA
1.2.1 Scalarsand Vectors
1.2.2 Multiplicationof a Vector by a Scalar
1.2.3 Addition and Subtraction
1.2.4 The Dot (Scalar)Product
1.2.5 The Cross (Vector) Product
1.3 THE GRADIENT AND THE DEL
OPERA TOR
1.3.1 The Gradient
1.3.2 CurvilinearCoordinates
(a) Cylindrical
(b) Spherical
1.3.3 The Line Integral
1.4 FLUX AND DIVERGENCE
1.4.1 Flux
1.4.2 Divergence
1.4.3 CurvilinearCoordinates
(a) Cylindrical Coordinates
(b) SphericalCoordinates
1.4.4 The Divergence Theorem
1.5 THE CURL AND STOKES' THEOREM
1.5.1 Curl
1.5.2 The Curlfor CurvilinearCoordinates
(a) CylindricalCoordinates
(b) Spherical Coordinates
1.5.3 Stokes' Theorem
1.5.4 Some Useful Vector Relations
(a) The Curl of the Gradient is Zero
IVx (Vf)= O]
(b) The Divergence of the Curl is Zero
[V - (Vx A)= 0]
PROBLEMS
Chapter 2-THE ELECTRIC FIELD
2.1 ELECTRIC CHARGE
2.1.1 Chargingby Contact
2.1.2 ElectrostaticInduction
2.1.3 Faraday's"Ice-Pail"Experiment
2.2 THE COULOMB FORCE LAW
BETWEEN STATIONARY CHARGES
2.2.1 Coulomb's Law
X
Contents
2.2.2 Units
2.2.3 The Electric Field
2.2.4 Superposition
2.3 CHARGE DISTRIBUTIONS
2.3.1 Line, Surface, and Volume Charge Distributions
2.3.2 The Electric Field Due to a Charge Distribution
2.3.3 Field Due to an Infinitely Long Line
Charge
2.3.4 Field Due to Infinite Sheets of Surface
Charge
(a) Single Sheet
(b) ParallelSheets of Opposite Sign
(c) Uniformly Charged Volume
2.3.5 Hoops of Line Charge
(a) Single Hoop
(b) Disk of Surface Charge
(c) Hollow Cylinderof Surface Charge
(d) Cylinder of Volume Charge
2.4 GAUSS'S LAW
2.4.1 Propertiesof the Vector Distance Between
two Points rQp
(a) rQp
(b) Gradientof the Reciprocal Distance,
V(1/rQp)
2.4.2
2.4.3
2.4.4
2.4.5
2.4.6
2.5
THE
2.5.1
2.5.2
2.5.3
2.5.4
2.5.5
(c) Laplacianof the Reciprocal Distance
Gauss's Law In Integral Form
(a) Point Charge Inside or Outside a
Closed Volume
(b) ChargeDistributions
SphericalSymmetry
(a) Surface Charge
(b) Volume ChargeDistribution
CylindricalSymmetry
(a) Hollow Cylinder of Surface Charge
(b) Cylinderof Volume Charge
Gauss'sLaw and the Divergence Theorem
ElectricField DiscontinuityAcross a Sheet
of Surface Charge
ELECTRIC POTENTIAL
Work Required to Move a Point Charge
The ElectricField and Stokes' Theorem
The Potentialand the Electric Field
FiniteLength Line Charge
ChargedSpheres
(a) Surface Charge
(b) Volume Charge
(c) Two Spheres
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Xi
2.5.6 Poisson'sand Laplace's Equations
THE METHOD OF IMAGES WITH
LINE CHARGES AND CYLINDERS
2.6.1 Two ParallelLine Charges
2.6.2 The Method of Images
(a) GeneralProperties
(b) Line Charge Near a Conducting
Plane
2.6.3 Line Chargeand Cylinder
2.6.4 Two Wire Line
(a) Image Charges
(b) Force of Attraction
(c) CapacitancePer Unit Length
2.7 THE METHOD OF IMAGES WITH
POINT CHARGES AND SPHERES
2.7.1 Point Chargeand a Grounded Sphere
2.7.2 Point ChargeNear a GroundedPlane
2.7.3 Sphere With Constant Charge
2.7.4 Constant Voltage Sphere
PROBLEMS
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Chapter 3-POLARIZATION AND CONDUCTION
3.1 POLARIZATION
3.1.1 The Electric Dipole
3.1.2 PolarizationCharge
3.1.3 The DisplacementField
3.1.4 LinearDielectrics
(a) Polarizability
(b) The Local ElectricField
3.1.5 Spontaneous Polarization
(a) Ferro-electrics
(b) Electrets
3.2 CONDUCTION
3.2.1 Conservationof Charge
3.2.2 ChargedGas ConductionModels
(a) Governing Equations
(b) Drift-DiffusionConduction
(c) Ohm's Law
(d) Superconductors
3.3 FIELD BOUNDARY CONDITIONS
3.3.1 Tangential Component of E
3.3.2 Normal Component of D
3.3.3 Point ChargeAbove a DielectricBoundary
3.3.4 Normal Componentof P and eoE
3.3.5 Normal Component of J
3.4 RESISTANCE
3.4.1 Resistance Between Two Electrodes
3.4.2 ParallelPlateResistor
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3.5
3.6
3.7
3.8
3.9
3.4.3 Coaxial Resistor
3.4.4 SphericalResistor
CAPACITANCE
3.5.1 ParallelPlateElectrodes
3.5.2 Capacitancefor any Geometry
3.5.3 CurrentFlow Through a Capacitor
3.5.4 Capacitanceof Two ContactingSpheres
LOSSY MEDIA
3.6.1 Transient ChargeRelaxation
3.6.2 Uniformly ChargedSphere
3.6.3 Series Lossy Capacitor
(a) ChargingTransient
(b) Open Circuit
(c) Short Circuit
(d) SinusoidalSteady State
3.6.4 DistributedSystems
(a) GoverningEquations
(b) Steady State
(c) TransientSolution
3.6.5 Effects of Convection
3.6.6 The Earth and Its Atmosphere as a Leaky
SphericalCapacitor
FIELD-DEPENDENT SPACE CHARGE
DISTRIBUTIONS
3.7.1 Space Charge Limited Vacuum Tube
Diode
3.7.2 Space Charge Limited Conduction in
Dielectrics
ENERGY STORED IN A DIELECTRIC
MEDIUM
3.8.1 Work Necessary to Assemble a Distribution
of Point Charges
(a) Assembling the Charges
(b) BindingEnergy of a Crystal
3.8.2 Work Necessary to Form a Continuous
ChargeDistribution
3.8.3 Energy Density of the Electric Field
3.8.4 Energy Stored in ChargedSpheres
(a)
Volume Charge
(b) Surface Charge
(c) BindingEnergy of an Atom
3.8.5 Energy Stored In a Capacitor
FIELDS AND THEIR FORCES
3.9.1 Force Per Unit Area On a Sheet of Surface
Charge
3.9.2 Forces On a PolarizedMedium
(a) Force Density
(b) Permanently PolarizedMedium
(c) Linearly PolarizedMedium
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3.10
3.9.3 ForcesOn a Capacitor
ELECTROSTATIC GENERATORS
3.10.1 Van de GraaffGenerator
3.10.2 Self-Excited ElectrostaticInduction
Machines
3.10.3 Self-Excited Three-PhaseAlternating
Voltages
3.10.4 Self-Excited Multi-FrequencyGenerators
PROBLEMS
Chapter 4-ELECTRIC FIELD BOUNDARY
VALUE PROBLEMS
4.1 THE UNIQUENESS THEOREM
4.2 BOUNDARY VALUE PROBLEMS IN
CARTESIAN GEOMETRIES
4.2.1 Separationof Variables
4.2.2 Zero Separation Constant Solutions
(a) Hyperbolic Electrodes
(b) ResistorIn an Open Box
4.2.3 Nonzero Separation Constant Solutions
4.2.4 Spatially PeriodicExcitation
4.2.5 RectangularHarmonics
4.2.6 Three-DimensionalSolutions
4.3 SEPARATION OF VARIABLES IN
CYLINDRICAL GEOMETRY
4.3.1 PolarSolutions
4.3.2 Cylinder in a Uniform Electric Field
(a) Field Solutions
(b) Field Line Plotting
4.3.3 Three-DimensionalSolutions
4.3.4 High Voltage InsulatorBushing
4.4 PRODUCT SOLUTIONS IN SPHERICAL
GEOMETRY
4.4.1 One-DimensionalSolutions
4.4.2 Axisymmetric Solutions
4.4.3 Conducting Spheres in a Uniform Field
(a) Field Solutions
(b) FieldLine Plotting
4.4.4 Charged Particle Precipitation Onto a
Sphere
4.5 A NUMERICAL METHODSUCCESSIVE RELAXATION
4.5.1 FiniteDifference Expansions
4.5.2 Potential Insidea Square Box
PROBLEMS
Chapter
5-THE
5.1
FORCES
MAGNETIC
ON
FIELD
MOVING
CHARGES
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Contents
5.1.1
5.1.2
5.2
5.3
5.4
5.5
5.6
5.7
5.8
The Lorentz Force Law
Charge Motions in a Uniform Magnetic
Field
5.1.3 The Mass Spectrograph
5.1.4 The Cyclotron
5.1.5 Hall Effect
MAGNETIC FIELD DUE TO CURRENTS
5.2.1 The Biot-Savart Law
5.2.2 Line Currents
5.2.3 CurrentSheets
(a) Single Sheet of Surface Current
(b) Slab of Volume Current
(c) Two ParallelCurrentSheets
5.2.4 Hoops of Line Current
(a) Single Hoop
(b) Two Hoops (Helmholtz Coil)
(c) Hollow Cylinder of Surface Current
DIVERGENCE AND CURL OF THE
MAGNETIC FIELD
5.3.1 Gauss's Law for the Magnetic Field
5.3.2 Ampere's CircuitalLaw
5.3.3 Currents With CylindricalSymmetry
(a) Surface Current
(b) Volume Current
THE VECTOR POTENTIAL
5.4.1 Uniqueness
5.4.2 The Vector Potential of a Current Distribution
5.4.3 The Vector Potentialand Magnetic Flux
(a) FiniteLength Line Current
(b) Finite Width Surface Current
(c) Flux Through a Square Loop
MAGNETIZATION
5.5.1 The MagneticDipole
5.5.2 Magnetization Currents
5.5.3 MagneticMaterials
(a) Diamagnetism
(b) Paramagnetism
(c) Ferromagnetism
BOUNDARY CONDITIONS
5.6.1 TangentialComponent of H
5.6.2 TangentialComponent of M
5.6.3 Normal Component of B
MAGNETIC FIELD BOUNDARY
VALUE PROBLEMS
5.7.1 The Method of Images
5.7.2 Sphere in a Uniform Magnetic Field
MAGNETIC FIELDS AND FORCES
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5.8.1 Magnetizable Media
5.8.2 Force on a CurrentLoop
(a) Lorentz Force Only
(b) Magnetization Force Only
(c) Lorentz and Magnetization Forces
PROBLEMS
Chapter 6-ELECTROMAGNETIC INDUCTION
6.1 FARADAY'S LAW OF INDUCTION
6.1.1 The ElectromotiveForce (EMF)
6.1.2 Lenz's Law
(a) Short CircuitedLoop
(b) Open CircuitedLoop
(c) Reaction Force
6.1.3 Laminations
6.1.4 Betatron
6.1.5 Faraday'sLaw and Stokes' Theorem
6.2 MAGNETIC CIRCUITS
6.2.1 Self-Inductance
6.2.2 Reluctance
(a) Reluctances in Series
(b) Reluctances in Parallel
6.2.3 TransformerAction
(a) Voltages are Not Unique
(b) Ideal Transformers
(c) Real Transformers
6.3 FARADAY'S LAW FOR MOVING
MEDIA
6.3.1 The Electric Field Transformation
6.3.2 Ohm's Law for Moving Conductors
6.3.3 Faraday'sDisk (Homopolar Generator)
(a) Imposed Magnetic Field
(b) Self-Excited Generator
(c) Self-Excited ac Operation
(d) PeriodicMotor Speed Reversals
6.3.4 Basic Motors and Generators
(a) ac Machines
(b) dc Machines
6.3.5 MHD Machines
6.3.6 Paradoxes
(a) A Commutatorlessdc Machine
(b) Changes In Magnetic Flux Due to
Switching
(c) Time Varying Number of Turns on a
Coil
6.4 MAGNETIC DIFFUSION INTO AN
OHMIC CONDUCTOR
6.4.1 Resistor-InductorModel
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Contents
6.4.2
6.4.3
The Magnetic Diffusion Equation
Transient Solution With No Motion
(U = 0)
6.4.4 The SinusoidalSteady State (Skin Depth)
6.4.5 Effects of Convection
6.4.6 A Linear Induction Machine
6.4.7 Superconductors
6.5 ENERGY STORED IN THE MAGNETIC
FIELD
6.5.1 A Single Current Loop
(a) Electrical Work
(b) Mechanical Work
6.5.2 Energy and Inductance
6.5.3 Current Distributions
6.5.4 Magnetic Energy Density
6.5.5 The Coaxial Cable
(a) External Inductance
(b) InternalInductance
6.5.6 Self-Inductance, Capacitance,and Resistance
6.6 THE ENERGY METHOD FOR FORCES
6.6.1 The Principleof Virtual Work
6.6.2 Circuit Viewpoint
6.6.3 MagnetizationForce
PROBLEMS
Chapter 7-ELECTRODYNAMICS-FIELDS AND
WAVES
7.1 MAXWELL'S EQUATIONS
7.1.1 Displacement Current Correction to
Ampere's Law
7.1.2 Circuit Theory as a Quasi-staticApproximation
7.2 CONSERVATION OF ENERGY
7.2.1 Poynting's Theorem
7.2.2 A Lossy Capacitor
7.2.3 Power in Electric Circuits
7.2.4 The Complex Poynting's Theorem
7.3 TRANSVERSE ELECTROMAGNETIC
WA VES
7.3.1 Plane Waves
7.3.2 The Wave Equation
(a) Solutions
(b) Properties
7.3.3 Sources of Plane Waves
7.3.4 A Brief Introduction to the Theory of
Relativity
7.4 SINUSOIDAL TIME VARIATIONS
7.4.1 Frequency and Wavenumber
. I
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Contents
7.4.2 Doppler FrequencyShifts
7.4.3 Ohmic Losses
(a) Low Loss Limit
(b) Large Loss Limit
7.4.4 High-Frequency Wave Propagationin
Media
7.4.5 Dispersive Media
7.4.6 Polarization
(a) LinearPolarization
(b) CircularPolarization
7.4.7 Wave Propagationin AnisotropicMedia
(a) Polarizers
(b) Double Refraction (Birefringence)
7.5 NORMAL INCIDENCE ONTO A PERFECT CONDUCTOR
7.6 NORMAL INCIDENCE ONTO A
DIELECTRIC
7.6.1 Lossless Dielectric
7.6.2 Time-Average Power Flow
7.6.3 Lossy Dielectric
(a) Low Losses
(b) Large Losses
7.7 UNIFORM AND NONUNIFORM PLANE
WA VES
7.7.1 Propagationat an ArbitraryAngle
7.7.2 The Complex PropagationConstant
7.7.3 Nonuniform Plane Waves
7.8 OBLIQUE INCIDENCE ONTO A PERFECT CONDUCTOR
7.8.1 E Field Parallelto the Interface
7.8.2 H Field Parallelto the Interface
7.9 OBLIQUE INCIDENCE ONTO A
DIELECTRIC
7.9.1 E Parallelto the Interface
7.9.2 Brewster'sAngle of No Reflection
7.9.3 CriticalAngle of Transmission
7.9.4 H Field Parallelto the Boundary
7.10 APPLICATIONS TO OPTICS
7.10.1 Reflectionsfrom a Mirror
7.10.2 LateralDisplacementof a Light Ray
7.10.3 Polarizationby Reflection
7.10.4 Light Propagationin Water
(a) Submerged Source
(b) Fish Below a Boat
7.10.5 Totally.Reflecting Prisms
7.10.6 FiberOptics
(a) StraightLight Pipe
(b) Bent Fibers
PROBLEMS
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Contents
Chapter 8--GUIDED ELECTROMAGNETIC
WAVES
8.1 THE TRANSMISSION LINE EQUATIONS
8.1.1 The ParallelPlate TransmissionLine
8.1.2 General TransmissionLine Structures
8.1.3 DistributedCircuitRepresentation
8.1.4 PowerFlow
8.1.5 The Wave Equation
8.2 TRANSMISSION LINE TRANSIENT
WA VES
8.2.1 Transients on Infinitely Long Transmission Lines
8.2.2 Reflections from Resistive Terminations
(a) Reflection Coeffcient
(b) Step Voltage
8.2.3 Approach to the dc Steady State
8.2.4 Inductors and Capacitorsas Quasi-static
Approximations to Transmission Lines
8.2.5 Reflections from Arbitrary Terminations
8.3 SINUSOIDAL TIME VARIATIONS
8.3.1 Solutions to the TransmissionLine Equations
8.3.2 Lossless Terminations
(a) Short CircuitedLine
(b) Open CircuitedLine
8.3.3 Reactive Circuit Elements as Approximations to Short TransmissionLines
8.3.4 Effects of Line Losses
(a) DistributedCircuitApproach
(b) DistortionlessLines
(c) Fields Approach
8.4 ARBITRARY IMPEDANCE TERMINATIONS
8.4.1 The GeneralizedReflection Coefficient
8.4.2 Simple Examples
(a) Load Impedance Reflected Back to the
Source
(b) Quarter Wavelength Matching
8.4.3 The Smith Chart
8.4.4 Standing Wave Parameters
8.5 STUB TUNING
8.5.1 Use of the Smith Chart for Admittance
Calculations
8.5.2 Single-Stub Matching
8.5.3 Double-Stub Matching
8.6 THE RECTANGULAR WAVEGUIDE
8.6.1 GoverningEquations
8.6.2 TransverseMagnetic (TM) Modes
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8.6.3
8.6.4
8.6.5
TransverseElectric (TE) Modes
Cut-Off
Waveguide PowerFlow
(a) PowerFlow for the TM Modes
(b) Power Flow for the TE Modes
8.6.6 Wall Losses
8.7 DIELECTRIC WA VEGUIDE
8.7.1 TM Solutions
(a) Odd Solutions
(b) Even Solutions
8.7.2 TE Solutions
(a) Odd Solutions
(b) Even Solutions
PROBLEMS
Chapter 9-RADIATION
9.1 THE RETARDED POTENTIALS
9.1.1 Nonhomogeneous Wave Equations
9.1.2 Solutions to the Wave Equation
9.2 RADIATION FROM POINT DIPOLES
9.2.1 The Electric Dipole
9.2.2 Alternate Derivation Using the Scalar
Potential
9.2.3 The Electric and Magnetic Fields
9.2.4 Electric FieldLines
9.2.5 RadiationResistance
9.2.6 RayleighScattering(orwhy is the sky blue?)
9.2.7 Radiationfrom a Point Magnetic Dipole
9.3 POINT DIPOLE ARRAYS
9.3.1 A Simple Two Element Array
(a) BroadsideArray
(b) End-fireArray
(c) ArbitraryCurrentPhase
9.3.2 An N DipoleArray
9.4 LONG DIPOLEANTENNAS
9.4.1 FarFieldSolution
9.4.2 Uniform Current
9.4.3 RadiationResistance
PROBLEMS
SOLUTIONS TO SELECTED PROBLEMS
INDEX
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