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;
Tij=Tji; 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 δℓ;
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 αℓ parameters;
αℓ and the scattering amplitude;
partial and total scattering cross-sections;
absorbtion cross-section;
optical theorem for the EM waves.
Last Modified: April 25, 2024.
Vadim Kaplunovsky
vadim@physics.utexas.edu