b
y J. W. Keohane and J. P. Foy

## Part I: Electricity

### Chapter 1    Charge

#### 1.1    Charge Conservation

Discussion 1.1: Franklin’s Sign Convention for Charge
Problem 1.1: Franklin’s Letter
Thought Experiment 1.1: Charging an Electroscope
Thought Experiment 1.2: Franklin’s Leyden Jar Experiments
Fundamental Law 1.1: The Conservation of Charge
Problem 1.2: Franklin’s Bells
Problem 1.3: Franklin’s Tower
Problem 1.4: A Pointed Lightning Rod
Problem 1.5: The Dissipation Law of Coulomb
Discussion 1.2: Fundamental Laws and Charge Conservation
Discussion 1.3: Modern Tests of Charge Conservation

#### 1.2    Current

Definition 1.1: Current
Discussion 1.4: Units of Charge
Thought Experiment 1.3: A Charged Arrow Shaft
Thought Experiment 1.4: Current Flow in a Neutral Medium
Problem 1.6: A Wheatstone Bridge I
Problem 1.7: The Discovery of Cosmic Radiation
Definition 1.2: The Area Vector
Definition 1.3: Current Density
Example 1.1: Unidirectional Current Density
Example 1.2: Current Through a Constriction
Problem 1.8: Drift Velocity
Problem 1.9: Electrolysis of Water
Problem 1.10: A Fuel Cell Car
Problem 1.11: The Large Hadron Collider
Thought Experiment 1.5: Charge of a Parallel Plate Capacitor
Problem 1.12: A Leaky Parallel Plate Capacitor
Thought Experiment 1.6: Charge Leakage from a Balloon.

#### 1.3    The Continuity Equation

Example 1.3: A Current Pulse
Thought Experiment 1.7: Linear Charge Continuity
Discussion 1.5: Intensive and Extensive Properties
Definition 1.4: Charge Density
Thought Experiment 1.8: Charge Continuity

#### 1.4    The Divergence in Curvilinear Coordinates

Definition 1.5: The Divergence
Thought Experiment 1.9: The Divergence in Cartesian Coordinates
Thought Experiment 1.10: The Divergence in Cylindrical Coordinates
Thought Experiment 1.11: The Divergence in Spherical Coordinates

#### 1.5    Solving The Continuity Equation

Example 1.4: The Spherical Continuity Equation
Problem 1.13: The Current Around a Star
Problem 1.14: The Leaky Cable
Problem 1.15: An AC Transmission Line
Problem 1.16: A Cylindrical AC Current Wave
Problem 1.17: A Point AC Current Source
Thought Experiment 1.12: A Two-Pole Circuit
Example 1.5: Electroplating a Cylindrical Object
Example 1.6: Current Density Inside a Box
Problem 1.18: A Discharging Long Rod
Problem 1.19: An AC Spherical Shell
Problem 1.20: A Discharging Cloud
Problem 1.21: A Plasma Globe
Problem 1.22: The Heliospheric Current Sheet
Problem 1.23: Electrotyping a Sculpture
Problem 1.24: Electroplating a Globe
Problem 1.25: Electroplating a Cone
Thought Experiment 1.13: Algorithmically Solving the Continuity Equation

### Chapter 2    The Electrostatic Force

#### 2.1    Coulomb’s Law

Discussion 2.1: Gaussian Units
Fundamental Law 2.1: Coulomb’s Electrostatic Force Law
Problem 2.1: The ESU and the Coulomb
Discussion 2.2: Spooky Action at a Distance

#### 2.2    The Electric Field

Definition 2.1: The Electric Field
Problem 2.2: The Electric Body Force
Example 2.1: A Charge on a Spring
Problem 2.3: A Driven Oscillator
Problem 2.4: The Net Force on an Electric Dipole
Problem 2.5: The Torque on an Electric Dipole

#### 2.3    The Electric Field Surrounding Point Charges

Thought Experiment 2.1: Coulomb’s Law for a Collection of Charges
Example 2.2: A Single Hoop of Wire
Example 2.3: The Electric Field Near a Flat Disk

#### Problem 2.6: A Coil of Charge

Example 2.4: An Idealized Parallel Plate Capacitor
Example 2.5: A Spherical Shell
Problem 2.7: The Torque on an Electric Dipole
Problem 2.8: The Net Force on an Electric Dipole

### Chapter 3    Electrical Potential Energy

#### 3.1    Work and Energy in Mechanical Systems

Thought Experiment 3.1: The Work-Energy Theorem
Example 3.1: Lifting and Dropping a Mass
Thought Experiment 3.2: The Potential Energy
Problem 3.1: A Point Charge and a Flat Sheet
Discussion 3.1: Energy Concepts in Electrodynamics
Fundamental Law 3.1: The Conservation of Energy
Thought Experiment 3.3: Force Laws from Potential Energy

#### 3.2    The Gradient in Curvilinear Coordinates

Derivation 3.1: The Gradient in Cartesian Coordinates
Derivation 3.2: The Gradient in Cylindrical Coordinates
Derivation 3.3: The Gradient in Spherical Coordinates
Discussion 3.2: Directional Derivatives
Definition 3.2: The Directional Derivative of a Scalar
Problem 3.2: The Directional Derivative and Unit Vectors
Problem 3.3: Directional Derivative Identities
Problem 3.4: The Directional Derivative in Curvilinear Coordinates

#### 3.3    Energy Continuity

Thought Experiment 3.4: The Energy Continuity Equation
Example 3.2: The Luminosity of the Sun
Problem 3.5: Visual Brightness of a Camera Flash
Example 3.3: A Hydropower Dam

#### 3.4    Energy and Work in DC Circuits

Thought Experiment 3.5: The Potential Energy of an Electrochemical Cell
Problem 3.6: The Oxford Electric Bell
Thought Experiment 3.6: Power in a Circuit
Problem 3.7: The O’Shaughnessy Dam
Problem 3.8: A Tandem Van de Graaff Accelerator
Discussion 3.3: Electromotive Force
Thought Experiment 3.7: Ohm’s Law
Thought Experiment 3.8: The Heaviside Interpretation of Ohm’s Law
Thought Experiment 3.9: The Reynolds Number from Ohm’s Law
Discussion 3.4: Classical Models of Conductivity
Problem 3.9: The Earth’s Electric Field
Example 3.4: A Cylindrical Resistor
Problem 3.10: A Conical Resistor
Thought Experiment 3.10: Voltmeters and Ammeters
Example 3.5: Kirchhoff’s Loop and Junction Rules
Problem 3.11: A Wheatstone Bridge II
Problem 3.12: An Infinite Resistor Ladder
Discussion 3.5: Non-linear Electronic Components
Problem 3.13 An Incandescent Light Bulb

#### 3.5    Electrostatic Potential Energy

Thought Experiment 3.11: Universal Gravitational Potential Energy
Problem 3.14: Universal Electrostatic Potential Energy
Example 3.6: The Impact of Two Astronomical Objects
Problem 3.15: Skew Line Charges
Example 3.7: The Self-Gravity of the Sun
Problem 3.16: Building up a Sphere of Charge
Example 3.8: The Electrostatic Self-Energy of a Disk

#### 3.6    The Coulomb Potential

Definition 3.3: The Coulomb Potential
Problem 3.17: The Lattice Energy of Table Salt
Thought Experiment 3.12: The Monopole Electric Potential
Thought Experiment 3.13: The Electric Dipole
Thought Experiment 3.14: The Electric Quadrupole
Definition 3.4: The Quadrupole Moment Tensor
Thought Experiment 3.15: Multipole Expansion I
Example 3.9: The Water Molecule
Problem 3.20: The Geometry of Hydrogen Cyanide
Problem 3.21: The Octopole
Problem 3.22: Four Charges
Problem 3.23: Six Charges

### Chapter 4    Gauss’s Law

#### 4.1    Capacitors

Example 4.1: Faraday’s Spherical Leyden Jar
Thought Experiment 4.1: Capacitance
Example 4.2: An RC Circuit
Problem 4.1: A Photoflash
Thought Experiment 4.2: The Permittivity of Dielectrics
Problem 4.2: The Electric Field in a Capacitor
Thought Experiment 4.3: The Permittivity of Free Space
Discussion 4.1: Permittivity, the Aether, and Toy Models of Matter
Definition 4.1: The Polarization Vector
Thought Experiment 4.4: The Dumbbell Model of Dielectrics
Problem 4.3: Polarization Vector in a Capacitor
Thought Experiment 4.5: The Electric Field in a Polarized Medium
Thought Experiment 4.6: The Continuity of Molecular Charge

#### 4.2    Gauss’s Law

Example 4.3: Gauss’s Law of Gravity
Problem 4.4: The Disk of the Milky Way
Thought Experiment 4.7: Gauss’s Law
Fundamental Law 4.1: Gauss’s Law
Problem 4.5: An Atmospheric Electric Field
Problem 4.6: Gauss’s Law in Integral Form
Problem 4.7: A Dielectric Sphere in an Electric Field
Thought Experiment 4.8: Electric Flux
Discussion 4.2: The Term Flux in Physics
Thought Experiment 4.9: The Electric Flux Through a Cube
Thought Experiment 4.10: Electric Flux Through an Empty Surface
Discussion 4.3: Gauss’s Law in Free Space
Discussion 4.4: Gauss’s Law vs. Coulomb’s Law
Problem 4.8: A Finite Sheet of Charge
Example 4.4: The Parallel Plate Capacitor and Gauss’s Law
Example 4.5: A Conductor Inside a Capacitor
Example 4.6: A Spherically Symmetrical Charge Distribution
Example 4.7: The Motion of a Dielectric in a Capacitor
Example 4.8: The Capacitance of a Coaxial Cable
Problem 4.9: The Capacitance of Two Parallel Wires

#### 4.3    Classical Models of Dielectrics

Thought Experiment 4.11: A Single Atom in an Electric Field
Thought Experiment 4.12: The Clausius-Mosotti Model
Thought Experiment 4.13: The Langevin Formula for polar dielectrics

### Chapter 5    The Equations of Laplace and Poisson

#### 5.1    Equations for the Electrostatic Potential

Thought Experiment 5.1: Gauss’s Law in Potential Form
Thought Experiment 5.2: Linearity and Superposition
Problem 5.1: Non-Linearity and Poisson’s Equation

#### 5.2    Vector Second Derivatives

Definition 5.1: The Laplacian
Thought Experiment 5.3: The Laplacian in Cartesian Coordinates
Thought Experiment 5.4: The Laplacian in Cylindrical Coordinates
Thought Experiment 5.5: The Laplacian in Spherical Coordinates

#### 5.3    Multipole Solutions to Laplace’s Equation

Problem 5.2: The Monopole Electric Potential
Problem 5.3: The Electric Dipole
Example 5.1: A Conducting Ball in an Electric Field
Thought Experiment 5.6: A Faraday Cage
Thought Experiment 5.7: Multipole Expansion with Legendre Polynomials
Problem 5.5: Legendre Polynomials
Problem 5.6: A Dielectric Sphere in an Electric Field

#### 5.4    Separation of Variables

Example 5.2: The Potential in a Long Coaxial Cable
Problem 5.7: The Capacitance of Two Parallel Wires
Problem 5.8: A Spherical Capacitor
Example 5.3: The Electric Potential Inside a 2D Conducting Box
Example 5.4: A Conducting Ball in an Electric Field Revisited
Example 5.5: A Capped Coaxial Cable

#### 5.5    The Method of Images

Example 5.6: A Point Charge Near a Conducting Plane
Problem 5.9: A Line Charge Near a Conducting Plane
Problem 5.10: A Point Charge and Two Conducting Planes
Problem 5.11: A Point Charge and a Conducting Paraboloid

#### 5.6    The Method of Relaxation

Thought Experiment 5.8: Poisson’s Equation with Relaxation
Example 5.7: A Polarized Layer of Water
Example 5.8: A 2D Dipole in a Grounded Box
Example 5.9: A 3D Octopole in a Grounded Box
Thought Experiment 5.9: Relaxation in 2D Cylindrical Coordinates
Example 5.10: A Coaxial Cable with Charge

## PART II: Magnetism

### Chapter 6    Permanent Magnets

#### 6.1    The Magnetic Field

Thought Experiment 6.1: Magnetic Field Lines
Thought Experiment 6.2: The Torque on a Permanent Magnet
Definition 6.1: The Magnetic Field
Thought Experiment 6.3: The Potential Energy of a Magnet
Problem 6.1: Zeeman Splitting
Thought Experiment 6.4: The Force on a Magnet in a Non-Uniform Field
Problem 6.2: A Magnet on a Torsion Spring
Discussion 6.1: The Stern-Gerlach Experiment
Problem 6.3: A Spherical Magnet in an External Field
Problem 6.4: Random Spherical Magnets
Discussion 6.2: Operational Definitions and Fields

#### 6.2    Peregrinus’s Principle

Thought Experiment 6.5: There are no Magnetic Monopoles
Thought Experiment 6.6: The Continuity of Magnetic Flux
Fundamental Law 6.1: Peregrinus’s Principle
Thought Experiment 6.7: Faraday’s Magnetic Field Lines
Problem 6.5: An Inverse Square Magnetic Field
Problem 6.6: A Thin Magnetic Needle

#### 6.3    The Field Surrounding Permanent Magnets

Problem 6.7: Magnetic Monopoles
Thought Experiment 6.8: Atomic Magnetic Moments
Problem 6.8: The Divergence of a Dipole Field
Thought Experiment 6.9: A Collection of Small Magnets
Problem 6.9: Two Small Magnets
Thought Experiment 6.10: The Attractive Force Between Magnets
Example 6.1: The Force Between Two Repulsive Magnets
Problem 6.10: The Force Between Magnetic Rods
Thought Experiment 6.11: The Interaction Energy of Two Magnets
Problem 6.11: The Interaction Force Between Two Magnets
Problem 6.12: The Force Between Flat Bar Magnets
Problem 6.13: Hyperfine Structure
Problem 6.14: The Torque Between Magnets
Thought Experiment 6.12: Magnetization
Thought Experiment 6.13: The Magnetic Moment of a Spherical Magnet
Problem 6.15: The Force Between Cubic Magnets
Thought Experiment 6.14: The Field Outside a Magnet
Problem 6.16: A Cylindrical Magnet
Problem 6.17: A Magnetized Sphere
Problem 6.18: The Neocube
Problem 6.19: A Toroidal Magnet

### Chapter 7    The Vector Potential and the Curl

#### 7.1    Magnetic Flux and the Vector Potential

Problem 7.1: The Vector Potential of a Very Long Magnet
Thought Experiment 7.1: The Vector Potential of a Tiny Magnet
Problem 7.2: Odd Shaped Surfaces
Thought Experiment 7.2: The Vector Potential Everywhere
Discussion 7.1: Peregrinus’s Principle and the Vector Potential

#### 7.2    The Curl in Curvilinear Coordinates

Problem 7.3: Azimuthal Vectors and the Curl
Thought Experiment 7.3: Conservative Forces and the Curl of the Force
Definition 7.1: The Curl
Thought Experiment 7.4: The Curl on a 3×3 Numerical Grid
Derivation 7.1: The Curl in Cartesian Coordinates
Problem 7.4: The 1854 Smiths Exam Problem 8
Derivation 7.2: The Curl in Cylindrical Coordinates
Problem 7.5: Algebraic Derivation of the Cylindrical Curl
Derivation 7.3: The Curl in Spherical Coordinates
Problem 7.6: Algebraic Derivation of the Spherical Curl
Thought Experiment 7.5: Newton’s Law of Gravity is Path Independent
Problem 7.7: The Electrostatic Force is Path independent
Problem 7.8: Conservative Force Fields

#### 7.3    The Vector Potential and Permanent Magnets

Thought Experiment 7.6: The Vector Potential of a Permanent Magnet
Thought Experiment 7.7: A Long Thin Magnetic Needle
Problem 7.9: The Long Thin Needle and the Dipole Field
Thought Experiment 7.8: A Flat Magnetic Disk
Thought Experiment 7.9: The Magnetic Field of a Very Thick Magnet
Thought Experiment 7.10: A Square 2D Magnet
Problem 7.10: A 2D Square Magnet
Thought Experiment 7.11: The Magnetic Field Inside a Cylindrical Magnet
Problem 7.11: A Magnetized Sphere

### Chapter 8    Electromagnetism

#### 8.1    Hans Christian Ørsted’s Discovery

Thought Experiment 8.1: The Magnetic Field Surrounding a Vertical Wire
Thought Experiment 8.2: Parallel Wires
Definition 8.1: The Ampere

#### 8.2    The Law of Laplace

Thought Experiment 8.3: The Magnetic Force on a Wire
Definition 8.2: The Tesla
Problem 8.1: Raising the Bar
Thought Experiment 8.4: A Loudspeaker
Thought Experiment 8.5: A Galvanometer
Problem 8.2: A Pivoting Wire Loop
Problem 8.3: A Leaning Wire Loop
Problem 8.4: A Rectangular Current Loop
Problem 8.5: A Spinning Charged Sphere
Thought Experiment 8.6: The Magnetic Field Surrounding a Current Loop

#### 8.3    Ampère’s Law

Thought Experiment 8.7: The Magnetic Field Surrounding a Wire
Problem 8.6: Ørsted’s Current Source
Thought Experiment 8.8: The Magnetic Field Around a Capacitor
Thought Experiment 8.9: The Maxwell-Ampère Law in Free Space
Thought Experiment 8.10: The Magnetic Field Inside an Empty Solenoid
Problem 8.7: The Magnetic Field in a Long Solenoid Magnet
Problem 8.8: An Large Current Sheet
Thought Experiment 8.11: A Hollow Magnetic Ring
Thought Experiment 8.12: A Magnetic Ring
Thought Experiment 8.13: The Maxwell-Ampère Law in General
Fundamental Law 8.1: The Maxwell-Ampère Law

#### 8.4    Electromagnets

Thought Experiment 8.14: The Magnetizing Field
Problem 8.9: Magnetic Potential Energy
Problem 8.10: The Vector Potential of a Solenoid
Discussion 8.1: The Confusing Magnetizing Field
Thought Experiment 8.15: Magnetic Hysteresis
Discussion 8.2: Models of Magnetic Matter
Problem 8.11: Classically Modeling Ferromagnetism
Thought Experiment 8.16: The Magnetic Moment of an Electromagnet
Problem 8.12: Work Done by an Electromagnet
Thought Experiment 8.18: A Chain of Paperclips and Magnetic Flux
Thought Experiment 8.19: The Magnetizing Field in a Permanent Magnet

### Chapter 9    Faraday’s Law of Induction

Thought Experiment 9.1: The Magnetic Flux through a Magnetic Rod
Problem 9.1: The Flux Gradient through a Long Cylindrical Magnet
Thought Experiment 9.2: Faraday’s Wire Spool Experiment
Thought Experiment 9.3: Faraday’s Law in Various Forms
Problem 9.2: A Time Dependent Magnet
Problem 9.3: Heating a Ring
Thought Experiment 9.4: The Induced Electric Field in Free Space
Thought Experiment 9.5: A Loop Inside a Changing Magnetic Field
Problem 9.4: The Vector Potential in a Current Loop
Problem 9.5: Induced Voltage in a Coil
Problem 9.6: A Coil in a Capacitor
Problem 9.7: The Current Density in a Conducting Wire
Thought Experiment 9.6: Magnetic Heating of a Coin
Problem 9.8: A Spinning Hoop
Thought Experiment 9.7: The Brush Dynamo-Electric Machine
Problem 9.9: The Wallace Machine
Problem 9.11: Faraday’s Generator and Capacitor
Problem 9.12: A Spinning Coil of Wire Surrounding a Magnet

#### 9.2    Inductors

Thought Experiment 9.8: A Toroidal Inductor
Thought Experiment 9.9: The Energy Density from Faraday’s Law
Problem 9.13: The Energy Density of a Ferromagnet
Problem 9.14: Energy Cycle of a Hot Magnet
Problem 9.15: The Energy Density Inside of a Toroidal Inductor
Problem 9.16: Current Density Induced by a Toroidal Inductor
Problem 9.17: A Toroidal Inductor around an AC Current
Problem 9.18: A Toroidal Inductor with an Iron Core
Problem 9.19: Electrical Steel
Thought Experiment 9.13: An LC Circuit
Problem 9.20: An RLC Circuit
Problem 9.21: A Driven Series RLC Circuit
Problem 9.22: A Mutual Inductor
Problem 9.23: Power Converters

#### 9.3    Inductance in Free Space

Thought Experiment 9.14: Measuring Permeability
Thought Experiment 9.15: Mutual Inductance in Free Space
Thought Experiment 9.16: The Inductance Matrix
Problem 9.24: Inductance of Two Loops
Thought Experiment 9.17: A Coupled LC Circuit
Thought Experiment 9.18: The Inductance of a Free Space Coaxial Cable
Problem 9.25: The Inductance of Two Parallel Wires

#### 9.4    Maxwell’s Equations in Potential Form

Derivation 9.1: Peregrinus’s Principle in Potential Form
Derivation 9.2: Faraday’s Law in Potential Form
Discussion 9.1: Choice of Gauge
Thought Experiment 9.19: A Gauge Transformation
Definition 9.1: The Lorenz Gauge
Thought Experiment 9.20: Gauss’s Law in Potential Form
Thought Experiment 9.21: The Maxwell-Ampère Law in Potential Form
Thought Experiment 9.22: The Vector Potential of a Long Wire
Problem 9.26: The Long Wire
Problem 9.27: Three Long Wires
Problem 9.28: Four Long Wires

### Chapter 10    The Electron

#### 10.1    The Hall Effect

Thought Experiment 10.1: The Magnetic Force on a Charged Fluid
Thought Experiment 10.2: The Electromagnetic Force on a Charged Fluid
Thought Experiment 10.3: The Hall Effect
Problem 10.1: A Hall Effect Thruster

#### 10.2    The Lorentz Force

Thought Experiment 10.4: The Lorentz Force
Fundamental Law 10.1: The Electromagnetic Force
Problem 10.2: Interplanetary Dust Grains

#### 10.3    The Discovery of the Electron

Problem 10.4: The Ratio of Forces
Problem 10.5: Jupiter’s Magnetic Field
Thought Experiment 10.5: Deflection of Electrons in an Oscilloscope
Problem 10.6: An Oscilloscope
Problem 10.7: A Mass Spectrometer
Problem 10.8: A Betatron Accelerator
Discussion 10.1: Practically Solving Equations of Motion
Thought Experiment 10.6: A Free Electron in an Electromagnetic Field
Thought Experiment 10.7: A Free Particle in an Azimuthal Electric Field
Problem 10.9: A Magnetic Bottle

#### 10.4    The Elementary Charge

Discussion 10.2: Conserved Quantities and the Navier-Stokes Equation
Thought Experiment 10.8: Newton’s Second Law in a Fluid
Definition 10.1: Viscosity
Problem 10.10: Water Through a Cylindrical Pipe
Thought Experiment 10.9: Irrotational Flow
Problem 10.11: Stirring a Glass of Water
Problem 10.12: The Double Curl and the Vector Laplacian
Thought Experiment 10.10: Stoke’s Law
Problem 10.13: Stokes Law in the Millikan Experiment
Problem 10.14: Conserving Mass in Stokes Law
Problem 10.15: The Ratio of the Forces Between Electrons

### Chapter 11    Galilean Relativity in Electrodynamics

#### 11.1    Galilean Relativity

Problem 11.1: Newtonian Galilean Transforms
Thought Experiment 11.1: The Current Density in a Neutral Medium
Thought Experiment 11.2: Galilean Transformation of the Current Density
Discussion 11.1: The Principle of Relativity
Fundamental Law 11.1: The Principle of Relativity
Thought Experiment 11.3: Cycloid Motion of a Wheel
Thought Experiment 11.4: The Electromagnetic Force in a Moving Frame
Thought Experiment 11.5: A Galilean Transformation of the Electric Field
Problem 11.2: Galilean Transformation of the Scalar Potential
Thought Experiment 11.6: The Magnetic Field is a Galilean Invariant

#### 11.2    Breakdown of Galilean Relativity

Thought Experiment 11.7: A Paradoxical Line Charge
Problem 11.3: A Moving Capacitor
Thought Experiment 11.8: The Vector Potential Galilean Transformation
Problem 11.4: Transformation of the Charge Density
Thought Experiment 11.9: The Galilean Transform of the Magnetic Field
Problem 11.5: The Magnetic Field Transformation
Problem 11.6: The Galilean Transformation of the Electromagnetic Force
Table 11.1: The Galilean Transformations
Problem 11.7: Galileo’s Cannon
Problem 11.8: A Moving Dielectric Rod
Problem 11.9: A Rotating Dielectric Rod
Discussion 11.2: Galilean Relativity and Modern Physics
Problem 11.10: A Velocity Selector
Thought Experiment 11.10: The Free Particle Simulator
Problem 11.11: Shock Acceleration
Problem 11.12: A Magnetic Bullet

#### 11.3    Motional EMF

Thought Experiment 11.11: The Induced Electric Field
Problem 11.13: An Azimuthal Magnetic Field
Problem 11.14: An Airplane
Problem 11.15: A Spinning Magnet
Problem 11.16: A Falling Bar
Thought Experiment 11.12: Lenz’s Law
Thought Experiment 11.13: Magnetic Backpressure
Problem 11.17: A Magic Pipe
Problem 11.19: A Conducting Sphere in a Magnetic Field
Problem 11.20: A Rotating Rod
Problem 11.21: A Falling Hoop of Metal
Problem 11.22: Faraday’s Waterloo Bridge Experiment
Thought Experiment 11.14: Motional EMF and Faraday’s Law
Problem 11.23: Motional EMF and the Vector Potential
Problem 11.24: A Moving Loop in a Changing Magnetic Field

#### 11.4    The Law of Biot and Savart

Thought Experiment 11.15: A Moving Point Charge
Thought Experiment 11.16: The Law of Biot-Savart
Thought Experiment 11.17: A Circular Loop of Wire
Problem 11.25: On Axis Field of a Wire Loop
Problem 11.26: A Spiral Current
Problem 11.27: The Magnetic Field of a Short Solenoid
Discussion 11.3: Ampère and the Law of Biot and Savart
Thought Experiment 11.18: The Potentials of a Moving Point Charge
Thought Experiment 11.19: Vector Potential of a Very Short Wire
Problem 11.28: The Vector Potential of a Current Element
Thought Experiment 11.20: Vector Potential of a Finite Length Wire
Thought Experiment 11.21: The Vector Potential of a Current Loop
Problem 11.29: The Magnetic Field of a Current Loop
Thought Experiment 11.22: The Magnetic Field of a Single Current Loop
Thought Experiment 11.23: A Spinning Charged Spherical Shell
Problem 11.30: The Vector Potential of a Charged Shell
Thought Experiment 11.24: The Classical Gyromagnetic Ratio
Problem 11.31: The Electron g Factor
Problem 11.32: Gyromagnetic Ratio of a Spinning Charged Sphere
Problem 11.33: A Spinning Long Cylinder
Problem 11.34: A Spinning Charged Disk
Problem 11.35: A Spinning Non-Uniform Charged Sphere
Problem 11.36: A Precessing Top, MRI and NMR
Thought Experiment 11.25: A Spinning Steel Rod
Discussion 11.4: The Einstein-de Haas and Barnett Effects
Problem 11.37: The Barnett Effect and Interstellar Dust Grains
Problem 11.38: Measuring the g-factor of NiFe Film

### Chapter 12    Superconductors

#### 12.1    Turbulence and the Reynolds Number

Problem 12.1: The Reynolds Number in Everyday Life
Example 12.1: A Pump in a Pool and Numerical Viscosity
Problem 12.2: The Cost of a Hydrodynamic Simulation

#### 12.2    Magnetic Fields and Plasmas

Thought Experiment 12.1: Magnetic Plasma Tubes
Problem 12.3: Solar Physics
Problem 12.4: Deflection of a Salt Water Jet
Problem 12.5: An Azimuthal Current Density
Problem 12.6: The Magnetic Field in a Spinning Plasma

#### 12.3    The Meissner Effect

Thought Experiment 12.2: The Meissner Effect
Problem 12.7: A Superconducting Sphere
Problem 12.8: Modeling the Meissner Effect
Thought Experiment 12.3: A Trapped Magnetic Field
Problem 12.9: A Spherical Hole

#### 12.4    Acceleration Theory

Thought Experiment 12.4: The First London Equation
Problem 12.10: The Kinetic Inductivity
Problem 12.11: Electrons in a Superconductor
Thought Experiment 12.5: The Second London Equation
Discussion 12.1: The Development of Superconductivity

#### 12.5    Magnetic Levitation

Thought Experiment 12.6: The Magnetic Force on a Superconductor
Problem 12.12: Magnetic Levitation I
Problem 12.13: Magnetic Levitation II

#### 12.6    Type II Superconductors

Discussion 12.2: Ampére’s Current Loop Model of Magnetic Matter
Problem 12.14: Type II Superconductors and Vortex Lattices

## Part III: Light

### Chapter 13    Transmission Lines

#### 13.1    RLC Circuits

Thought Experiment 13.1: A Telegraph Signal Generator
Problem 13.1: A Series RLC Circuit
Problem 13.2: A Stack of Capacitors and Resistors
Thought Experiment 13.2: A Chain of LC Circuits
Problem 13.3:   An RLC Circuit Chain

#### 13.2    Continuous Transmission Lines

Thought Experiment 13.3: An Ideal Coaxial Cable Transmission Line
Thought Experiment 13.4: The Telegrapher’s Wave Equation
Thought Experiment 13.5: Heaviside’s Lossy Transmission Line
Problem 13.4: 100 Ohm Twisted Pair Telephone Cable
Problem 13.5: The Characteristic Speed of a Signal in a Coaxial Cable
Problem 13.6: A Parallel Wire Transmission Line

#### 13.3    Sinusoidally Driven Circuits

Definition 13.1: The Complex Impedance
Example 13.1: The Impedance of Resistors
Example 13.2: The Impedance of an Inductor
Problem 13.7: Resistors and Inductors in Series and Parallel
Problem 13.8: The Impedance of a Capacitor
Problem 13.9: The Impedance of RLC Circuits
Problem 13.10: The Impedance of an RC Stack
Problem 13.11: The Impedance of an LC Chain
Problem 13.12: The Twisted Pair Telephone Cable
Discussion 13.1: Complex Amplitudes in Phase Space

### Chapter 14    Light in an Optical Medium

#### 14.1    Maxwell’s Field Equations

Thought Experiment 14.1:   The Electric Displacement Vector
Thought Experiment 14.2: Conservation of Charge and the Magnetizing Field
Definition 14.1: The Auxiliary Fields
Problem 14.1 : The Maxwell-Ampère Law in Auxiliary Form
Discussion 14.1: Maxwell’s Four-field Approach
Problem 14.2: A Dielectric Capacitor
Problem 14.3: A Steel Inductor
Thought Experiment 14.3: A Cylindrical Electromagnetic Wave Algorithm
Problem 14.4: The Cylindrical Wave Algorithm
Example 14.1: The Cylindrical Wave Solved Differentially
Thought Experiment 14.4: The Energy of a Cylindrical Wave
Problem 14.5: A Cylindrical Wave in the Intermediate Limit

#### 14.2    Light in a Linear Medium

Derivation 14.1: The Wave Equation in a Linear Medium
Problem 14.6: The Wave Speed in an Insulative Linear Medium
Thought Experiment 14.5: The Plane Wave Solution in a Linear Medium
Problem 14.7: The Wave Vector

#### 14.3    The Refraction of Light

Definition 14.2: The Index of Refraction
Problem 14.8: Dispersion of Light in Glass
Thought Experiment 14.6: Snell’s Law
Problem 14.9: Making a Lens
Problem 14.10: A Rectangular Prism
Thought Experiment 14.7: Fermat’s Principle
Example 14.2: A Spherical Lens
Problem 14.11: The Spherical Lens
Problem 14.12: A Rectangular Prism Revisited
Problem 14.13: An Exponential Atmosphere

#### 14.4    Fresnel’s Theory of Light

Thought Experiment 14.8: Linearly Polarized Light
Thought Experiment 14.9: Boundary Conditions at an Optical Surface
Thought Experiment 14.10: The Fresnel Equations
Thought Experiment 14.11: Brewster’s Angle
Problem 14.14: Brewster’s Experiment
Problem 14.15: Dashboard Glare

#### 14.5    Reflection off of Good Conductors

Thought Experiment 14.12: Radiation from a Large Conducting Sheet
Thought Experiment 14.13: How Light Reflects off a Conducting Surface
Thought Experiment 14.14: Huygens’s Principle
Example 14.3: The Law of Reflection
Example 14.4: The Poisson Spot
Discussion 14.2: Huygens’s Principle and Refraction
Problem 14.16: The Principles of Fermat and Huygens

#### 14.6    The Propagation of Light in a Conductor

Thought Experiment 14.15: A Plane Wave in a Conductive Medium
Problem 14.17: The Reduced Wavelength
Problem 14.18: The Wave Speed in a Conductor
Problem 14.19: Good Conductors Driven at Ultra High Frequencies
Example 14.5: The Magnetic Field Inside a Conductor
Problem 14.20: The Magnetic Field Inside a Conductor
Problem 14.21: Skin Depth and Reflectivity

### Chapter 15    Light in Free Space

#### 15.1    The Wave Equation in Free Space

Discussion 15.1: Maxwell’s Field Equations in Free Space
Thought Experiment 15.1: The Field Equations in a Vacuum
Problem 15.1: The Magnetic Field Wave Equation
Discussion 15.2: The Michelson-Morley Experiment
Fundamental Law 15.1: The Constancy of the Speed of Light
Discussion 15.3: The Road to Relativity
Thought Experiment 15.2: The Speed of Light in a Moving Frame
Problem 15.2: The Michelson-Morley Experiment

#### 15.2    The Inverse Square Law

Thought Experiment 15.3: The Inverse Square Law of Light
Problem 15.3: The Solar Constant, Albedo, the Seasons, and Sea Ice
Problem 15.4: Dwarf planets
Problem 15.5: The Cosmic Distance Ladder

#### 15.3    Conserved Quantities in Plane Waves

Thought Experiment 15.4: The Energy Density
Problem 15.6: Solar Sails
Problem 15.7: Energy Density and Radiation Pressure
Thought Experiment 15.6: The Momentum Density
Thought Experiment 15.7: Circular Polarization
Thought Experiment 15.8: The Angular Momentum of Light
Problem 15.8: Monochromatic Plane Waves
Problem 15.9: The Photon Model of Light

#### 15.4    Measuring Light with a Photometer

Thought Experiment 15.9: The Photometer
Problem 15.10: S.I. Units of Perceived Brightness
Thought Experiment 15.10: Measuring the Angle of Linear Polarization
Thought Experiment 15.11: Measuring Circular Polarization
Thought Experiment 15.12: Stokes Parameters
Example 15.1: Stokes Parameters of Linear Polarized Light
Problem 15.11: Half Wave Plates
Example 15.2: An Example of Elliptical Polarization
Problem 15.12: Elliptical Polarization and Stokes Parameters
Problem 15.13: Coherence
Problem 15.14: The Poincaré Sphere
Problem 15.15: Mueller Matrices

### Chapter 16    Sources of Electromagnetic Radiation

#### 16.1    Spherical Waves

Thought Experiment 16.1: An Idealized Spherical Wave
Problem 16.1: A Spherical Magnetic Wave
Discussion 16.1: Retarded Time
Thought Experiment 16.2: Electric and Magnetic Field Interdependence
Thought Experiment 16.3: A Beamed Spherical Wave
Example 16.1: A Monochromatic Light Source
Example 16.2: Two Source Interference
Problem 16.2: A Diffraction Grating
Problem 16.3: Young’s Experiment
Thought Experiment 16.4: A Moving Light Source
Example 16.3: Measuring the Velocity of a Monochromatic Light Source

Definition 16.1: The Retarded Potentials
Problem 16.4: The Spherical Wave Solution in Potential Form
Thought Experiment 16.5: The Hertzian Dipole
Example 16.4: The Radiation Resistance of a Dipole Antenna
Example 16.5: The Beam Pattern of a Hertzian Dipole Antenna
Example 16.6: The Power per Solid Angle
Thought Experiment 16.6: A Receiving Hertzian Dipole Antenna
Discussion 16.2: Strong Reciprocity Theorem and CPT Symmetry
Problem 16.5: A Baseball Scout with a Radar Gun
Example 16.7: The Cross Section of a Dielectric Sphere
Discussion 16.3: Why the Sky is Blue and Sunsets are Red
Problem 16.6: Mie Scattering

#### 16.3    Antenna Theory

Thought Experiment 16.7: Simple Plane Waves in Potential Form
Thought Experiment 16.8: Spherical Waves in Potential Form
Thought Experiment 16.9: The Poynting Vector in the Radiation Zone
Thought Experiment 16.10: An Omnidirectional Antenna
Problem 16.8: A Terminated Omnidirectional Antenna
Thought Experiment 16.11: A Parabolic Dish
Problem 16.9: Sea Interferometry
Problem 16.10: Self-calibration
Discussion 16.4: Complex Numbers in Electrodynamics
Problem 16.11: Mapping 3C 279 with the Very Long Baseline Array
Problem 16.12: Superluminal Motion in the Jets of 3C 279
Problem 16.13: The Core of the Quasar 3C 279

#### 16.4    Radiation from Point Charges

Thought Experiment 16.13: The Larmor Formula
Problem 16.14: Radiation from an Oscillating Particle
Thought Experiment 16.14: The Radiation Reaction Force
Discussion 16.5: The Abraham-Lorentz Force
Problem 16.15: The Radiation Reaction Force and Momentum
Example 16.8: The Thomson Cross Section
Problem 16.17: The Eddington Luminosity
Example 16.9: Higher Order Thomson Scattering
Problem 16.18: Free Electrons in the Sun

### Chapter 17    Special Relativity

#### 17.1    Relativistic Kinematics

Problem 17.1: Willem de Sitter’s Argument
Thought Experiment 17.1: The Invariance of Perpendicular Length
Thought Experiment 17.2: A Light Clock
Problem 17.2: Mork from Ork
Problem 17.3: Muon Decay
Thought Experiment 17.3: Length Contraction
Thought Experiment 17.4: The Order of Events
Problem 17.4: Car and the Garage
Problem 17.5: Superman and the Farmer
Thought Experiment 17.5: The Lorentz-FitzGerald Transformations
Example 17.1: A Relativistically Moving Light Source
Problem 17.6: The Relativistic Doppler Effect
Thought Experiment 17.6: Addition of Velocities
Problem 17.7: Fizeau’s Experiment
Problem 17.8: Stellar Aberration
Thought Experiment 17.7: The Magnitude of the Spacetime 4-Vector

#### 17.2    Relativistic Electrodynamics

Thought Experiment 17.8: The Lorentz Invariance of Charge
Thought Experiment 17.9: The Charge Density
Problem 17.9: The Charge Density Lorentz-FitzGerald Transformation
Thought Experiment 17.10: Transforming the Current Density
Problem 17.10: The Lorentz-FitzGerald Current Density Transformation
Thought Experiment 17.11: The 4-Current
Thought Experiment 17.12: The Continuity Equation in 4-Vectors
Thought Experiment 17.13: The 4-Potential
Thought Experiment 17.14: The Parallel Electric Field Transformation
Thought Experiment 17.15: The Parallel Magnetic Field Transformation
Thought Experiment 17.16: A Moving Line Charge
Thought Experiment 17.17: The Perpendicular Field Transformations
Problem 17.11: Motional EMF and Relativity
Problem 17.12: A Point Charge Moving Parallel to a Line Charge
Problem 17.13: A Moving Capacitor
Problem 17.14: Two Lorentz Invariants
Problem 17.15: Making Fields Parallel and Perpendicular
Problem 17.16: Magnetic Moment Transformations
Problem 17.17: Light From an Arbitrarily Moving Source
Thought Experiment 17.18: The Potentials of a Moving Rod
Thought Experiment 17.19: The Transformation of the Potentials
Problem 17.18: The 4-magnitude of the 4-Potential Vector
Problem 17.19: Maxwell’s Equations
Problem 17.20: The Fields of a Point Charge
Problem 17.21: Parallel Wires
Thought Experiment 17.20: Relativistic Energy
Discussion 17.1: Mass and Potential Energy
Problem 17.22: The Magnetic Field Energy
Thought Experiment 17.21: Relativistic Momentum
Example 17.2: A Charged Particle in a Constant Electric Field
Thought Experiment 17.22: Transformation of Energy and Momentum
Problem 17.23: The Kinetic Energy and Momentum
Problem 17.24: A Dogfight in Space
Problem 17.25: The Velocity 4-Vector
Problem 17.26: The Relativistic Child-Langmuir Law
Problem 17.27: Proton Collisions and the Higgs Boson
Problem 17.28: Pion Decay
Example 17.3: Relativistic Doppler Beaming

#### 17.3    The Electromagnetic Field Tensor

Discussion 17.2: Einstein Notation
Definition 17.1: The Kronecker and Levi-Cevita Functions
Problem 17.29: The Triple Product of Three Vectors
Problem 17.30 A Useful Identity
Discussion 17.3: Polar & Axial Vectors and The Faraday Tensor
Thought Experiment 17.23: Transforming the Field Tensor
Problem 17.31: Properties of the Electromagnetic Tensor

#### 17.4    Radiation from Relativistic Point Charges

Thought Experiment 17.24: The Potentials of a Moving Particle
Thought Experiment 17.25: The Constant Velocity Liénard-Wiechert Potentials
Example 17.4: The Potentials of a Uniformly Moving Point Charge
Example 17.5: The Electric Field of a Uniformly Moving Charge
Discussion 17.4: More on the Electric Field of a Uniformly Moving Charge
Example 17.6 The Fields of a Uniformly Moving Point Charge Revisited
Example 17.7 The electric and magnetic fields of an accelerated particle
Discussion 17.5: Radiation and Fields of an Accelerated Charge
Example 17.8 Radiation from a slowly moving accelerated point charge
Example 17.11 Radiation from an arbitrarily accelerated charge

### Chapter 18    The Photon

#### 18.1    Einstein and the Photoelectric Effect

Thought Experiment 18.1: The Ejection of Electrons from a Metal Surface
Thought Experiment 18.2: The Photoelectric Effect
Discussion 18.1: Einstein’s Heuristic Photoelectric Effect
Discussion 18.2: Millikan’s Photoelectric Experiment
Problem 18.1: Millikan’s Value of Planck’s Constant

#### 18.2    Thermal Radiation and the Hydrogen Atom

Thought Experiment 18.3: Rutherford’s Planetary Model of an Atom
Problem 18.2: The Planetary Atom
Thought Experiment 18.4: The Bohr Hydrogen Atom
Problem 18.3: The Balmer Formula
Problem 18.5: The Orbital Angular Momentum
Problem 18.6: The Bohr Atom and Fine Structure
Thought Experiment 18.5: Spontaneous Emission in the Classical Limit
Problem 18.7: The Planetary Atom
Thought Experiment 18.6: The Driven Classical Atom
Thought Experiment 18.7: Classical Stimulated Emission and Absorption
Problem 18.8: The Ratio of Einstein’s A and B Coefficients
Thought Experiment 18.8: Einstein’s Theory of Radiation
Problem 18.9: Thermal Hydrogen Emission
Problem 18.10: Brightness Temperature
Problem 18.11: Wien’s Displacement Law
Problem 18.12: Astronomical Colors
Problem 18.13: Spectral Energy Distributions

#### 18.3    Does the Photon Exist?

Thought Experiment 18.9: The Momentum and Speed of a Photon
Problem 18.14: The Energy and Speed of a Photon
Problem 18.15: The Non-Relativistic Doppler Effect
Problem 18.16: Ram and Radiation Pressure
Problem 18.17: The Recoil of an Atom
Thought Experiment 18.10: The Spin of a Photon
Problem 18.18: Electromagnetic Waves and Photons
Discussion 18.3: Successes of the Particle Model of Light
Discussion 18.4: The Failure of the Particle Model of Light
Problem 18.19: The Double Slit Power
Discussion 18.5: Compton’s Experiment
Thought Experiment 18.11: Compton Scattering
Problem 18.20: The Compton Wavelength
Problem 18.21: The Inverse Compton Effect
Problem 18.22: The Relativistic Doppler Effect

#### 18.4    Matter Waves

Thought Experiment 18.12: de Broglie Waves
Thought Experiment 18.13: The Davisson–Germer Experiment
Fundamental Law 18.1: Wave-Particle Duality
Discussion 18.6: The Wave Nature of Electrons: What is Waving?
Problem 18.23: Complex Identities
Thought Experiment 18.10: The Probability Density Function of Light
Problem 18.24: Photons and Probability Waves
Discussion 18.7: Photons, Interference, and Young’s Experiment

#### 18.5    The New Electrodynamics

Thought Experiment 18.14: A Square Box of Light
Problem 18.25: Blackbody Radiation and Wave-Particle Duality
Problem 18.26: Polarization Vectors
Thought Experiment 18.15: Quantizing the Electromagnetic Field
Derivation 18.1: The Quantum Mechanical Equations of Motion
Problem 18.27 : Mechanism of Light Propagation
Discussion 18.8: Is a Photon a Particle After All?