Electromagnetic Theory: Lecture Log

This is the lecture log for the graduate ElectroMagnetic Theory class PHY 387 K, taught in Spring 2024 by professor Vadim Kaplunovsky, unique=56180.

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.

January 16 (Tuesday):

Canceled.

January 18 (Thursday):

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.

Make-up lecture on January 19 (Friday):

Separation of variables method: finish 2D slot example, 3D rectangular pipe example; separation of variables in polar and spherical coordinates.

January 23 (Tuesday):

Green's functions and their uses: Dirichlet and Neumann boundary conditions; inverse Laplacian operators in different spaces; Green's functions and their symmetries; examples; Green theorem (for non-trivial bounday potentials or fields).
Started multipole expansion: potentials of compact charged bodies; expanding 1/|x−y|; Legendre polynomials.

January 25 (Thursday):

Electric multipole expansion: Legendre polynomials and spherical harmonics; spherical multipole moments and their potentials; axially symmetric charges and their multipole moments.
Multipole moments as tensors: spherical harmonics and symmetric traceless tensors; dipole moment vector in detail; quadrupole moment tensor in detail.

January 30 (Tuesday):

Finished the multipole expansion: octupole moment as a 3 index tensor; higher multipole moments as tensors.
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.

February 1 (Thursday):

Highlights of magnetostatics: examples of calculating A(x) and B(x); 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.
Started Polarization and Magnetization: macroscopic fields; polarization and magnetization; bound charges in a dielectric; electric dicplacement field D; dielectric constant.

Extra lecture on February 2 (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.

February 6 (Tuesday):

Polarization and Magnetization: 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.

February 8 (Thursday):

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.
Forces on dielectrics: energy of inserting a dielectric piece; force on that piece; example; forces in capacitors.

February 13 (Tuesday):

FinishEnergy and forces in dielectrics: forces on dielectrics in capacitors; energy in non-linear dielectrics; hysteresis and energy dissipation (friefly).
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.
Demos of eddy currents: demo#1, demo#2.

February 15 (Thursday):

Magnetic energy: energy of inductor coil; energy of magnetic field; energy loss to hysteresis; forces on magnetic materials.
Complex amplitudes and impedance.
Skin effect.

Extra lecture on February 16 (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.

February 20 (Tuesday):

Mutual inductance and transformers (briefly).
Diffusion of magnetic fields into a conductor: diffusion equation for the current and the magnetic field; solving the diffusion equation: how the field penetrates a conductor.
Maxwell equations: the displacement current; Maxwell equations and electromagnetic waves; equations for the potentials A and Φ; transverse gauge; Landau gauge.

February 22 (Thursday):

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.
Electromagnetic momentum: 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 radiation.

February 27 (Tuesday):

Plane EM waves: wave vectors; electric and magnetic amplitudes; wave impedance; wave energy; linear, circular and elliptic polarizations; Stokes parameters; partially polarized waves.

February 29 (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 likbez: Gaussian wave packets.

Extra lecture on March 1 (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.

March 5 (Tuesday):

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.
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.

March 7 (Thursday):

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

March 12 and 14:

Spring Break, no classes.

March 19 (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.

March 21 (Thursday):

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

March 26 (Tuesday):

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

for March 28 (Thursday):

Microwave cavities: 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 March 29 (Friday):

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

April 2 (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; non-linear dipoles; Rutherford atom example.

April 4 (Thursday):

Radiation by higher multipoles: first subleading order; magnetic dipole radiation; electric quadrupole radiation; higher subleading orders (briefly).
Quantum radiation of photons: quantum transitions; Fermi's Golden rule; intro to quantum EM fields and photons (very brief); transition rate in the dipole approximation; classical-quantum correspondence; classical amplitudes as limits of quantum matrix elements.

Extra lecture on April 5 (Friday):

Superconductivity (III): Josephson junctions: tunneling of Cooper pairs; I=I₀×sin(Δφ); voltage and oscillations.

April 9 (Tuesday):

Quantum radiation of photons: allowed and forbidden transitions in atoms; selection rules for the allowed transitions; gamma decays and selection rules in nuclear physics.
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.

April 11 (Thursday):

Radiation by a long antenna: 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.
Receiving antennas: reciprocity theorem; directionality and gain; effective aperture; short dipole example; impedance matching; general antennas.

April 16 (Tuesday):

Introduction to scattering: induced multipoles and re-radiation; partial and total cross-sections; polarized cross-sections; small dielectric sphere example: σ∝k⁴, angular dependence; polarization; Thomson scattering by a free electron.

April 18 (Thursday):

Multiple scatterers of EM waves: interference and the form factor; Rayleight scattering by gases; attenuation by scattering; Bragg scattering by crystals.

Extra lecture on April 19 (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.

April 23 (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.

April 25 (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.