Understanding the Electromagnetic Field is an entry level textbook for graduate students with a focus on the electromagnetic field. This book explores the relationship between the field and electric charges. The earlier part of the book deals with the derivation of Maxwell's equations from experimental laws. Next, the electromagnetic field is studied in the light of special relativity, leading logically to the quantum theory of radiation. Quantum mechanics is introduced as a quantum field theory of the electromagnetic field. The rules of quantum mechanics are developed in a systematic way, with relativistic quantum electrodynamics explaining some puzzles that emerge in classical electrodynamics. A chapter is devoted to the study of angular momentum in quantum mechanics, uniquely showing its importance in the understanding of the interaction between the field and charges. The geometry of the space in which the electromagnetic field is embedded is shown to be significant. General relativity provides a relationship between the geometry of space and the presence of matter. The final chapter is devoted to deriving the fundamental equations of general relativity. Mathematical expressions are derived for the effect of gravity on the electromagnetic field, and measurable results are calculated. The prerequisites of this book are Newtonian physics, calculus and linear algebra. Exercises are provided throughout the book.
Author(s): Basil S. Davis
Publisher: World Scientific Publishing
Year: 2023
Language: English
Pages: 404
Contents
Preface
1. The Study of Empty Space
1.1 Introduction
1.2 Real Numbers
1.3 Scalar Fields
1.4 Delta Function
1.5 Curvilinear Coordinates
1.5.1 Spherical Coordinates
1.5.2 Cylindrical Coordinates
1.6 Vector Integral Calculus
1.6.1 Line Integrals
1.6.2 Stokes' Theorem
1.6.3 Divergence Theorem
1.6.4 Gradient Theorem
1.7 General Orthogonal Coordinates
1.7.1 Differential Operations in Orthogonal Curvilinear Coordinates
1.8 Imaginary Numbers
1.8.1 The Argand Plane
1.8.2 Gamma Function
1.8.3 Physical Reality of Imaginary Numbers
2. Fields Produced by Stationary Charges
2.1 Coulomb's Law
2.1.1 An Inverse Square Law
2.1.2 Geometries of Space
2.1.3 Higher Dimensions
2.1.4 Round Objects
2.2 The Electric Field
2.2.1 Electric Flux
2.2.2 Field Near a Uniformly Charged Infinite Plane
2.2.2.1 Method A: Coulomb's Law
2.2.2.2 Alternative method using Coulomb's Law
2.2.2.3 Method B: Gauss's Law
2.3 Conservative Force
2.4 Potential and Field
2.4.1 Equipotential Surfaces
2.4.2 Calculation of Potential at a Point
3. Electric Fields and Potentials
3.1 Solutions to Laplace's Equation
3.1.1 Boundary Value Solutions
3.1.2 Rectangular Coordinates
3.1.3 Spherical Polar Coordinates
3.1.4 Spherical Harmonics
3.1.5 Cylindrical Coordinates
3.2 A Puzzle
3.3 Electric Field Lines
3.3.1 Parallel Plate Capacitor
3.3.2 Properties of Field Lines
3.3.3 Conductors and Potential
3.3.4 Image Charges
3.4 Multipole Expansion of Potentials
3.4.1 Dipoles
3.4.2 Multipoles
3.5 Electrostatic Energy
3.5.1 Potential and Potential Energy
3.5.2 Energy of an Electric Field
3.5.3 Energy of a Charged Capacitor
3.5.4 Potential Energy and Field Energy of a Pair of Charges
4. Magnetostatics
4.1 The Phenomenon of Magnetism
4.1.1 Properties of Magnets
4.1.2 Coulomb's Law for Magnets
4.2 Current and Magnetic Field
4.2.1 Electric Current
4.2.2 Oersted's Law
4.3 Force on a Charge in a Magnetic Field
4.3.1 Forces between Wires carrying Current
4.3.2 The Lorentz Force
4.4 The Biot-Savart Law
4.5 Magnetic Field of a Circular Circuit
4.5.1 Magnetic Field inside a Long Solenoid
4.6 The Vector Potential A
4.7 Magnetic Moment and Rotating Charges
4.7.1 Magnetic Dipole Moment of a Current Loop
4.7.2 Gyromagnetic Ratio
4.8 Magnetic Monopoles
5. Fields Produced by Time Varying Sources
5.1 Equation of Continuity and Applications
5.2 Faraday's Law
5.2.1 Magnetic Flux Increase and EMF in a Circuit
5.2.2 Magnetic Flux Inertia
5.3 Sources and Fields
5.3.1 Maxwell's Equations
5.3.2 Fields and Potentials
5.3.3 Causality
5.3.4 Speed of Electromagnetic Fields
5.3.5 Plane Wave Electric and Magnetic Fields
5.3.6 The Lienard-Wiechert Potentials
5.4 Energy in a Magnetic Field
5.5 Simple Circuits
5.5.1 RC Circuits
5.5.2 LC Circuits
6. Energy and Momentum of Fields
6.1 Tensor Analysis
6.1.1 Transformation Laws
6.1.2 The Metric Tensor
6.2 Energy and Momentum Flow in Fields
6.2.1 Poynting Vector
6.2.2 Maxwell Stress Tensor
6.2.3 Momentum of a Field
6.2.4 Momentum and Energy of a Field
6.3 Energy Flow in Simple Circuits
7. Special Relativity and Electromagnetism
7.1 Detection of the Ether
7.1.1 Medium of Electromagnetic Waves
7.1.2 Motion of Detector Relative to the Medium
7.1.3 Michelson-Morley Experiment
7.2 Einstein's Theory of Special Relativity
7.2.1 Postulates of Special Relativity
7.3 The Geometry of Special Relativity
7.3.1 World Lines
7.3.2 Space-like, Time-like and Light-like Intervals
7.3.3 Limiting Cases of Time-like and Space-like Intervals
7.4 The Algebra of Special Relativity
7.4.1 Transformation of Space and Time Intervals
7.4.1.1 Lorentz contraction
7.4.1.2 Time dilatation
7.4.1.3 Proper time
7.4.2 Energy and Mass
7.5 Electromagnetic Fields
7.5.1 Covariant Forms
7.5.2 Electromagnetic Field Tensor
7.6 Lorentz Transformation of Tensors
7.7 Doppler Effect
7.7.1 Classical Formula
7.7.2 Relativistic Formula
7.7.3 Alternative Derivation of Relativistic Doppler Formula for Light
7.8 Relativistic Dynamics of Charged Particles
7.9 Lagrangian Dynamics of the Electromagnetic Field
7.9.1 The Principle of Least Action
7.9.2 Examples
7.9.2.1 Harmonic oscillator
7.9.2.2 Ring around a sphere
7.9.3 LC Circuit
7.9.4 Energy of an Electromagnetic Field
7.9.5 Electric and Magnetic Fields in a Propagating Wave
7.9.6 Lagrangian for a Charge Interacting with a Field
7.9.6.1 Free particle
7.9.6.2 Electron in an electromagnetic eld
7.9.6.3 Hamiltonian formulation
7.9.6.4 Gauge invariance
7.10 Abstract Covariant Formulation of the Lagrangian
7.11 Maxwell's Equations from the Lagrangian
8. Microscopy of the Electromagnetic Field
8.1 Charges and Fields
8.2 Macroscopic and Microscopic Domains
8.3 Frequency and Energy of an Electromagnetic Wave
8.4 Determination of the Value of h
8.5 Thermodynamics
8.6 Statistical Mechanics
8.6.1 One-dimensional Gas
8.6.2 Equipartition of Energy
8.6.3 Second Law of Thermodynamics
8.7 Entropy and Temperature
8.8 Electromagnetic Radiation Gas
8.8.1 Transition to Quantum Theory
8.8.2 Correspondence with Classical Formulas
8.8.3 Stefan-Boltzmann Law
8.9 The Photoelectric E ect
8.9.1 Einstein's Explanation
8.10 Uncertainty Principle
8.10.1 Wave Particle Duality
8.10.2 Quantum Theory of Light
8.11 Electron Waves
8.11.1 Photon Angular Momentum
9. The Quantum Mechanics of the Field
9.1 Feynman Graphs
9.2 Quantum Interactions
9.3 Complex Numbers in Quantum Mechanics
9.4 States and Operators
9.5 Physical Meaning of Symbols
9.5.1 Creation and Annihilation of Photons
9.5.2 Propagation of a Photon
9.5.3 Probability Amplitudes
9.5.4 Addition of Paths
9.5.5 Classical and Quantum Probabilities
9.5.6 Constructive Interference
9.5.7 Destructive Interference
9.6 Matrix Representation
9.6.1 States, Operators and Matrices
9.6.2 A Highly Simplified Special Case
9.6.3 Operators and Square Matrices
9.7 Orthonormal Vectors
9.8 Operators and State Vectors
9.8.1 Changing a Vector by an Operator
9.8.1.1 The Number Operator
9.8.2 Eigenfunctions and Eigenvalues
9.9 Energy of a Photon Field
9.10 Hermitian Operators
9.10.1 Eigenvalues of Hermitian Operators
9.11 Other Operators in Quantum Mechanics
9.12 The Spatial Wave Function
9.12.1 Probability Density
9.12.2 Amplitude and Probability
9.12.3 Field Operators and Probability Amplitude
9.12.4 Wave Function of a Plane Wave Photon
9.13 Operators and Eigenvalues
9.13.1 Physical Observables
9.13.2 Application to Electrons
9.13.3 Eigenvalues and Eigenfunctions
9.13.4 Electron in a General State
9.14 Operator Algebra
9.14.1 Momentum Operator is Hermitian
9.14.2 Commutation of Operators
10. Quantum Angular Momentum and the Field
10.1 Quantum Mechanics of a Stable Hydrogen Atom
10.2 Quantization of Angular Momentum
10.2.1 Potential and Kinetic Energy
10.2.2 Angle between Orbital Angular Momentum and Measuring Field
10.3 Spin
10.3.1 Stern-Gerlach Experiment
10.3.2 Spin and Polarization of the Photon
10.4 The Pauli Spin Matrices
10.4.1 Wave Function and Electron Spin
10.4.2 Trace of a Hermitian Matrix
10.4.3 Unitary Matrices
10.4.4 Spin Angular Momentum
10.5 Spin of an Electron
10.6 Pauli Equation
10.6.1 Pauli Equation for a Free Electron
10.6.2 Electron in an Electromagnetic Field
10.7 Interaction of an Electron with a Photon
11. Relativistic Quantum Electrodynamics
11.1 Applying Special Relativity to Pauli's Equation
11.2 Interpretation of the Negative Energy States
11.3 Anomalous Velocity of the Electron
11.3.1 Dirac's Explanation
11.3.2 Position of an Electron
11.4 Electron Velocity and Electron Charge
11.4.1 The Electron as a Spherical Shell
11.4.2 Quantization of Maxwell's Equations
11.4.3 Interaction with Potentials
11.4.3.1 Experimental Evidence for Zitterbewegung
12. Gravity and Electromagnetism
12.1 Dimensions and Their Relationships
12.2 Curved Space
12.3 Curvature of Spaces
12.3.1 Covariant Derivatives
12.3.2 Christo el Symbols for Covariant Tensors
12.3.3 Torsion Tensor
12.3.4 Metric Compatibility
12.3.5 Expression for the Christoffel Symbol
12.3.6 Divergence Theorem
12.3.7 Parallel Transport
12.3.7.1 Parallel transport in at space
12.3.7.2 Parallel transport in curved space
12.3.8 Curvature of Space
12.4 Equation for Curved Space
12.4.1 Space Containing Matter
12.4.2 The Constant k
12.4.3 A Solution to Einstein's Equation
12.5 Some Consequences of General Relativity
12.5.1 Equivalence Principle
12.5.2 Doppler Effect due to Gravity
12.5.3 Deflection of Light by the Sun
12.6 Quantization of Gravity
Index
