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*; stress tensor and momentum flow; local conservation of momentum.^{ij}=T^{ji}

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=I
_{0}×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∝
*k*^{4}, 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. - Plan for 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; maybe summary of intermediate-zone fields for all the modes.
- Tentative plan for 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 cross-sections.

Last Modified: April 18, 2024. Vadim Kaplunovsky

vadim@physics.utexas.edu