Electromagnetic Theory: Lecture Log

This is the lecture log for the graduate ElectroMagnetic Theory class PHY 387 K, taught in Fall 2024 by professor Vadim Kaplunovsky, unique=56940.

Most lectures should be video recorded and the records available on Canvas. For the few lectures that did not get recorded because of technical glitches, I shall scan the notes I have used in class and links the scans to this page.

August 27 (Tuesday):

Syllabus and admin: course content, textbook, prerequisites, homework, exams, grades, etc.
Laplace and Poisson equations for the Φ(x): boundary conditions; methods of solving (outline); image charge method (briefly).
Separation of variables method: 2D rectangular example.

August 29 (Thursday):

Separation of variables method: 3D rectangular pipe example; separation of spherical coordinates; spherical harmonics; spherical cavity, space outside a sphere, and spherical shell.
Begin Green's functions: definition and use; examples of G_D and G_N in a half-space.

September 3 (Tuesday):

Green's functions: inverse Laplace operator in different Hilbert spaces; G(y,x)=G(x,y); Green's theorem (for non-trivial boundary potentials or fields).
Electric multipole expansion: potentials of compact charged bodies; expanding 1/|x−y|; Legendre polynomials; multipole moments as tensors; leading terms: the net charge and the dipole moment.

September 5 (Thursday):

Electric multipole expansion: ℓ=1 term and the electric dipole; quadrupole moment in detail; octupole tensor in detail; higher multipole moments as tensors; spherical harmonic expansion; multipole moments of axially symmetric systems.

September 10 (Tuesday):

Steady currents: continuity equation and local charge conservation; divergenceless steady currents; Kirchhoff Law.
Introduction to magnetostatics: Biot–Savart–Laplace Law and Ampere's Force Law; Newton's Third Law for magnetic forces; field equations for the magnetic field; Ampere's circuital law; vector potential A(x) and gauge transforms; equations for the vector potential; examples of calculating A(x) and B(x).

September 12 (Thursday):

Introduction to magnetostatics: finish examples of calculating A(x) and B(x).
Multipole expansion of magnetic fields: multipole expansion for A(x) of a current loop; magnetic dipole moment in detail; multipole expansion for the volume current: monopole moment=0 and dipole moment in detail; gyromagnetic ratio; fields of point dipoles (electic and magnetic); forces and torques on dipoles; magnetic effects on atoms.

Extra lecture on September 13 (Friday):

Classical and quantum mechanics of a charged particle: Classical Lagrangian and equations of motion; classical Hamiltonian; quantum Hamiltonian; gauge transforms and their effects of the wave function; generalization to the quantum field theory.

September 17 (Tuesday):

Polarization and Magnetization: macroscopic fields; polarization and magnetization; bound charges in a dielectric; electric dicplacement field D; dielectric constant; bound currents in magentic materials; B and H magnetic fields; magnetic equation of state.
Boundary problems with dielectric and magnetic materials: boundary conditions; dielectric ball example; scalar magnetic potential Ψ; permanent magnet examples; multivalued Ψ around wires.

September 19 (Thursday):

Illustrate bound charges and bound currents; electromagnets.
Electrostatic energy: energy of continuous charges; energy of the electric field; self-energy and interaction energy of discrete charges; electrostatic energy in linear dielectrics; capacitor energy; energy in non-linear dielectrics; hysteresis and energy dissipation (friefly). Begin Forces on dielectrics: energy of inserting a dielectric piece; force on that piece; example.

September 24 (Tuesday):

Forces in dielectrics: energy of inserting a dielectric piece; force on that piece; example.
Faraday Induction Law: Faraday's flux rule; motional EMF, and its relation to the flux rule; induced non-potential electric field; ∇×E=−∂B/∂t; scalar and vector potentials for time-dependent fields; gauge transforms and gauge-fixing.
Eddy currents; demo#1, demo#2.

September 26 (Thursday):

Magnetic energy: energy of inductor coil; energy of magnetic field; energy loss to hysteresis; forces on magnetic materials.
Complex amplitudes and impedance.
Mutual inductance and transformers (briefly).

Extra lecture on September 27 (Friday):

Electric–Magnetic duality and Dirac Monopoles: duality of EM fields; duality of charges and currents; magnetic monopoles and troubles with their vector potentials; Dirac monopoles and charge quantization; electric-magnetic duality in QFT; angular momentum of a dyon.

October 1 (Tuesday):

Finished Mutual inductance and transformers.
Magnetic duffusion and skin effect: diffusion equation for the current and the magnetic field; solving the diffusion equation: how the field penetrates a conductor; skin effect for AC currents.
Maxwell equations: the displacement current; Maxwell equations and electromagnetic waves; started equations for the potentials

October 3 (Thursday):

Maxwell equations: equations for A and Φ in transverse gauge and in Landau gauge.
Green's functions of the d'Alembert operator: Fourier transformed Green's functions; causality; retarded and advanced Green's functions; retarded potentials and retarded fields; Efimenko equations.
Electromagnetic energy: local conservation of energy; local work-energy theorem; EM energy density, flow density, and power density; Poynting vector and Poynting theorem.
Intro to stress tensor: pressure and stress forces in continuous media; stress tensor; T^ij=T^ji; stress tensor and momentum flow; local conservation of momentum.

October 8 (Tuesday):

EM momentum and stress tensor: EM force density, momemntum density, and Maxwell's stress tensor; proof of local momentum conservation; tension and compression of magnetic fields; pressure of thermal EM waves.
Plane EM waves: wave vectors; electric and magnetic amplitudes; wave impedance; wave energy.
Polarizations of plane EM waves: linear polarizations; circular polarizations; elliptic polarizations; polarization bases; Stokes parameters; partially polarized light.

October 10 (Thursday):

Reflection and refraction of electromagnetic waves: geometric law of reflection and Snell's law of refraction; total internal reflection and evanescent waves; amplitudes and boundary conditions for the EM waves; coefficients of reflection and transmission; calculations for waves polarized normally to the plane of incidence; calculations for waves polarized within the plane of incidence; Brewster's angle; phase shift in total internal reflection.
Math reminder: Gaussian wave packets.

Extra lecture on October 11 (Friday):

Superconductivity (I): Cooper pairs and their condensation; Landau–Ginzburg theory of a superfluid Bose–Einstein condensate; density and velocity of the superfluid; LG theory of a charged superfluid; supercurrent and Messner effect.

October 15 (Tuesday):

Dispersion of waves: fequency-dependent n(ω) and its effects; phase velocity of a wave; wave packets and the group velocity; phase and group velocities in terms of the refraction index. dispersion and spreading out of wave packets; signal rate.
Attenuation of EM waves: complex n(ω) and attenuation; origins of power loss in attenuation: time lag of polarization and complex ε(ω); complex conductivity; time lag of magnetization and complex μ(ω); attenuation in water.

October 17 (Thursday):

Microscopic origin of dispersion: single-resonance toy model; multi-resonance model; normal and anomalous dispersion; attenuation in water. low frequency behavior α(ω); Drude conductivity in metals; high-frequency α(ω): plasmas and plasma frequency; plasma frequency in metals.

Extra lecture on October 18 (Friday):

Superconductivity (II): flux quantization; magnetic vortices; type I and type II superconductors.

October 22 (Tuesday):

Symmetries of mechanics and electromagnetism: Rotations: scalar, vectors, and tensors; Reflections: polar and axial vectors, cross product rule, mechanical and EM examples, true scalars and pseudoscalars, parity; Time reversal symmetry: examples of T-even and T-off quantities. Optical activity: chirality and birefringence; polarization rotation; Faraday affect; Faraday effect in plasma. Introduction to waveguides: Maxwell equations and boundary conditions; dispersion relations and cutoff frequencies; transverse and longitudinal components of EM fields.

October 24 (Thursday):

Waveguides: 2d Maxwell equation; TEM waves; TE waves; TM waves; wave power; wave energy and wave speed; waves in rectangular waveguides; waves in circular waveguides.

October 29 (Tuesday):

Attenuation in waveguides: effects of wall resistivity on the boundary conditions; wave attenuation due to wall resistivity; frequency dependence of the attenuation rate.
Quality factor of a resonator: mechanical example; resonance width; LRC circuit example; superheterodyne; microwave cavities as high-Q resonators.

October 31 (Thursday):

Microwave cavity resonators: standing TE and TM waves; modes and resonant frequencies of a rectangular cavity; modes and frequencies of a cylindrical cavity; quality of a microwave cavity: general estimate, example of a geometric factor.
Optic fibers as waveguides: overview; fiber types; multiple rays for step-index fibers; signal spread; geometric optics for smooth-index fibers.

Extra lecture on November 1 (Friday):

Josephson junctions: tunneling of Cooper pairs; DC Josephson effect; I=I₀×sin(Δφ); AC Josephson effect: voltage and oscillations; origin of voltage gap.

November 5 (Tuesday):

Optic fibers: wave optics for smooth-index fibers; mode counting.
Radiation by compact antennas: radiation by harmonic currents; near, intermediate, and far zones; spherical waves (briefly); multipole expansion.
Electric dipole radiation: far-zone fields; net radiated power and its angular distribution for a linear dipole; linear antenna example; radiative resistance.

November 7 (Thursday):

Finish Electric dipole radiation: non-linear dipoles: power and its angular distribution; Rutherford atom example.
Radiation by higher multipoles: first subleading order; magnetic dipole radiation; electric quadrupole radiation; higher subleading orders (briefly).
Quantum radiation of photons: quantum transitions; transition rate in the dipole approximation; classical-quantum correspondence; classical amplitudes as limits of quantum matrix elements.

Extra lecture on November 8 (Friday):

Aharonov–Bohm effect: role of the vector potential; gauge transforms of wave functions and of propagation amplitudes; interference and the Aharonov–Bohm effect; cohomology of magentic fluxes.
SQUID magnetometers: intro to the Superconducting Quantum Interferometry Devices; currents through two Josephson junctions; phase analysis in a magnetic field; maximal current as a function of the magnetic flux.

November 12 (Tuesday):

Quantum radiation of photons: classical-quantum correspondence for higher multipoles; transition rates and pre-emption; allowed and forbidden transitions in atoms; selection rules for the allowed transitions; gamma decays of nuclei; selection rules in nuclear physics; metastable nuclear states.

November 14 (Thursday):

Canceled due to Weinberg conference
Will be made up on Novermber 15 (Friday).

Makeup lecture on November 15 (Friday):

Radiation by a long antenna: center-fed linear antenna; standing current wave I(z); integral for the EM radiation and its direction dependence; examples of direction dependence for L/λ=½,1,2,3,4,6,10; general patterns; net radiation power and the input impedance; antenna as a boundary problem.

November 19 (Tuesday):

Receiving antennas: reciprocity theorem; directionality and gain; effective aperture; short dipole example; impedance matching; general antennas.
Introduction to scattering: induced multipoles and re-radiation; partial and total cross-sections; small dielectric sphere example; polarized cross-sections.

November 21 (Thursday):

Finished Introduction to scattering: polarized and unpolarized cross-sections; angular dependence.
Multiple scatterers of EM waves: interference and the form factor; Rayleight scattering by gases; attenuation by scattering; Bragg scattering by crystals.

November 25–29 (whole week):

Fall break, no classes.

December 3 (Tuesday):

Scalar spherical waves: asymptotic behavior of spherical waves; partial waves (ℓ,m) and their radial profiles; spherical bessel functions.
Spherical EM waves: transverse vector waves; TM and TE wave modes; no ℓ=0 modes; EM fields of a TM wave; near-zone limit and the electric multipole sources; far-zone fields, wave power, and its angular distribution; EM fields of TE waves; near-zone limit and the magnetic multipole sources; far-zone fields, wave power, and its angular distribution; summary of intermediate-zone fields for all the modes.

December 5 (Thursday):

Partial wave analysis of scattering: incident and scattered waves, and no interference between them; partial scalar waves and phase shifts δ_ℓ; phase shifts and the scattering amplitude; the total cross-section and the optical theorem; small hard sphere example.
Partial EM waves: radial equations and the α_ℓ parameters; α_ℓ and the scattering amplitude; partial and total scattering cross-sections; absorbtion cross-section; optical theorem for the EM waves.
Final exam reminder.