Electricity & Magnetism

About this course

A full course in classical electromagnetism, in six modules. Builds the four laws from Coulomb and Gauss, works through potentials, conductors, dielectrics and boundary-value problems, turns to steady currents and magnetic materials, then completes Maxwell's equations and the conservation laws. The last three modules develop electromagnetic waves in vacuum and in media, waveguides, retarded potentials, radiation, and the relativistic unification of E and B.

What you will be able to do

• Apply Gauss's law and symmetry arguments to find fields from charge distributions
• Solve boundary-value problems with separation of variables and the method of images
• Compute magnetic fields from currents with Biot-Savart and Ampère's law
• State Maxwell's equations and derive the wave equation from them
• Analyse reflection, refraction and dispersion at material boundaries
• Work with retarded potentials and gauge freedom to compute radiation fields
• Express electrodynamics covariantly and treat E and B as one field tensor

What this course covers

• The Magnetic Force Law: F = qv × B
• Magnetic Force Does No Work
• Circular Motion of a Charged Particle: Radius r = mv/qB
• Cyclotron Frequency: Period Independent of Speed
• The Cyclotron: Application of Circular Motion
• Velocity Selector and the Full Lorentz Force Law
• Worked Example: Proton in a Magnetic Field
• Conductors in Electrostatic Equilibrium
• Faraday Cage Shielding
• Capacitance and Parallel-Plate Geometry
• Energy Stored in a Capacitor
• Energy Density of the Electric Field
• Full Worked Problem: Parallel-Plate Capacitor
• The Electric Field and Coulomb's Law
• The Magnetic Field and the Lorentz Force
• Maxwell's Four Laws — Building the Complete Rule Book
• Electromagnetic Waves and the Speed of Light
• Monochromatic plane wave: anatomy and constraints
• Polarization: linear and circular
• Energy density and intensity of a light wave
• Polarizers project rather than block (Malus's law)
• The electromagnetic spectrum as one family
• Worked example: electric field amplitude of sunlight
• Multi-wavelength astronomy: the spectrum as observational tool
• Magnetic Flux and Faraday's Flux Rule
• Two Faces of Faraday's Law: Motional and Transformer Voltage
• Lenz's Law and Energy Conservation
• Mutual Inductance and the Transformer Principle
• Self-Inductance, Back-EMF, and Stored Energy
• Generator Calculation: Faraday's Law as Mains Voltage
• Scalar and vector potentials generating all fields
• Two Maxwell equations satisfied automatically
• Gauge freedom and gauge transformations
• Gauge choices: Coulomb and Lorenz gauges
• Causality selects retarded over advanced solutions
• Retarded potentials and the retarded time
• Jefimenko's equations as the retarded generalisation of static laws
• Ampère's Law Contradiction with Changing Fields
• Maxwell's Displacement Current
• Maxwell's Equations: The Complete Set
• Symmetry of the Curl Laws
• Charge Conservation from the Completed Equations
• Electromagnetic Waves and the Speed of Light
• Maxwell's 1865 Unification of Light and Electromagnetism
• Moving charge as source of magnetic field
• Biot-Savart law and field of a straight wire and loop
• Ampère's law and the solenoid field
• Curl and divergence of B (local form of magnetostatics)
• Force between two parallel current-carrying wires
• Closing synthesis and preview of vector potential
• What makes a charge radiate: acceleration criterion
• Radio antenna as acceleration-driven radiator
• Electric dipole radiation and the sine-squared donut
• Rayleigh scattering and the blue sky
• Larmor formula for a radiating point charge
• Classical atomic collapse: Larmor applied to the hydrogen atom
• Radiation reaction and limits of classical electrodynamics
• The Electric Field as a Physical Reality
• Coulomb's Law and the Inverse-Square Force
• Superposition of Electric Fields
• From Discrete Sums to Continuous Charge Distributions
• Electric Field on the Axis of a Charged Ring
• Electric Field Line Maps
• Polarization P and bound charge
• Bound charge opposing the applied field
• Electric displacement D and modified Gauss's law
• Linear dielectrics, susceptibility, and dielectric constant
• Dielectric constant multiplying capacitance
• Water as universal solvent via high ε_r
• Maxwell's boundary conditions at an interface
• Three laws at a boundary: frequency, reflection angle, Snell's law
• Amplitude coefficients at normal incidence and energy conservation
• Fresnel equations for oblique incidence
• Brewster's angle and perfect polarization of reflected light
• Total internal reflection and the critical angle
• Unifying the straw, mirage, fiber, and diamond under one law
• The continuity equation as conservation law template
• Electromagnetic field energy density
• Poynting's theorem and the Poynting vector
• Energy flow through empty space — the resistor solved
• Electromagnetic field momentum density
• Radiation pressure
• Physical reality of the field — capstone synthesis
• Electric Flux and Gauss's Law
• Flux vs. Field: a Critical Distinction
• Symmetry as the Key to Using Gauss's Law
• Field of a Uniformly Charged Solid Sphere
• Differential Form of Gauss's Law (Maxwell's First Equation)
• Zero Divergence Does Not Mean Zero Field
• Special Relativity and Maxwell's Hidden Relativism
• The Wire-and-Charge Thought Experiment
• Magnetism as Electrostatics Plus Relativity
• Field Transformation Rules Between Frames
• Quantitative Derivation: Electric Force Reproduces QVB
• The Electromagnetic Field Tensor F^μν
• Einstein's 1905 Paper: Relativity Born from E&M
• Existence of the Magnetic Vector Potential
• Gauge Freedom and the Coulomb Gauge
• Poisson's Equation for the Vector Potential
• Vector Potential of a Wire and Solenoid
• The Aharonov–Bohm Effect and Physical Reality of A
• The Magnetic Dipole Moment and Its Potential
• Work to assemble a charge configuration
• Energy rewritten in terms of the field
• Energy density of the electric field
• Two energy formulas: agreement in total, not point-by-point
• Forces from energy: F = −dW/dx
• Defibrillator capacitor: energy in the field
• Wave speed and index of refraction in a dielectric
• Frequency dependence of permittivity and dispersion
• Electron-oscillator model of dispersion
• Complex wave number and skin depth
• Phase velocity vs. group velocity
• Electromagnetic waves in conductors and skin depth
• Metal as a mirror: reflection and opacity
• Dispersion as the internet's speed limit
• Magnetization M and bound currents
• MRI scanner as magnetization application
• Auxiliary field H and linear magnetic media
• B versus H: which field is fundamental
• Ferromagnetism: domains, saturation, and hysteresis
• Iron-core solenoid: quantitative field amplification
• Magnetic memory and hard drives via hysteresis
• Maxwell's Equations in Empty Space
• Deriving the Wave Equation via the Curl Trick
• The Electromagnetic Spectrum as One Family
• Reading the Wave Speed: v = 1/√(μ₀ε₀) = c
• Refuting the Luminiferous Aether
• Anatomy of a Plane Electromagnetic Wave
• Light as an Electromagnetic Wave: Course Capstone
• Laplace's Equation and Boundary Value Problems
• Uniqueness Theorem
• Method of Images
• Induced Surface Charge Distribution
• Image Force, Energy, and the Factor-of-Two Trap
• Separation of Variables as the General Method
• Why and how waves are guided
• Perfect-conductor boundary conditions and mode quantisation
• TE and TM modes and the cutoff frequency
• Phase velocity, group velocity, and relativity in a guide
• TEM mode and the need for two conductors
• Transmission-line model: L, C, impedance, and signal speed
• Cutoff frequency worked example: X-band waveguide
• Path independence and the conservative electric field
• Electric potential V: definition and units
• Two-way relationship between E and V
• Equipotentials and their geometry
• Potential of point charges and continuous distributions
• Energy stored in a charge configuration
• The field tensor F^μν: construction and structure
• Four-vectors: four-current and four-potential
• Maxwell's four equations as two tensor lines
• Tensor transformation: E and B mixing under a boost
• Tensor form reveals manifest Lorentz covariance
• Two Lorentz invariants of the field tensor
• The field tensor as gateway to gauge theory and general relativity

Syllabus

1. Electrostatics

   1. L1 · Fields & the Four Laws
   2. L2 · The Electric Field & Coulomb's Law
   3. L3 · Gauss's Law & Symmetry
   4. L4 · Electric Potential

2. Potentials & Fields in Matter

   1. L5 · Conductors & Capacitance
   2. L6 · Dielectrics & Polarization
   3. L7 · Electrostatic Energy
   4. L8 · Boundary-Value Problems & the Method of Images

3. Magnetostatics & Magnetic Materials

   1. L9 · The Magnetic Field & the Lorentz Force
   2. L10 · Biot-Savart & Ampère's Law
   3. L11 · The Magnetic Vector Potential
   4. L12 · Magnetic Materials

4. Electrodynamics & Conservation Laws

   1. L13 · Faraday's Law & Induction
   2. L14 · Maxwell's Equations Complete
   3. L15 · Conservation Laws
   4. L16 · The Wave Equation

5. Electromagnetic Waves

   1. L17 · Waves in Vacuum
   2. L18 · Reflection & Refraction
   3. L19 · Electromagnetic Waves in Media
   4. L20 · Waveguides & Transmission Lines

6. Radiation & Relativistic Electrodynamics

   1. L21 · Potentials, Gauges & Retardation
   2. L22 · Radiation
   3. L23 · Relativity & Electrodynamics
   4. L24 · E & B as One Field