- January 19 (Wednesday):
- Syllabus and admin: course content, textbook, prerequisites, homework, exams, grades, etc.
(see the main web page for the class).

Laplace or Poisson equations for the Φ(**x**): boundary conditions; methods of solving (outline); image charges. - January 21 (Friday):
- Separation of variables: outline of the method; spherical shell example; using spherical harmonics.

Green's functions: inverse operators; Dirichlet and Neumann boundary conditions; using Green's functions. - Regular lecture on January 24 (Monday):
- Green's functions and their uses: symmetry; Green theorem (for non-trivial bounday potentials or fields); half-space example.
- Likbez lecture on January 24 (Monday):
- Separation of variables in 2D:
Solving Laplace equation for
*Φ(x,y)=A(x)B(y)*; general solution as an infinite series; finding the coefficients; examples.

Separation of variables in 3D (Cartesian coordinates).

Separation of variables in spherical coordinates. - January 26 (Wednesday):
- Electric multipole expansion:
potentials of compact charged bodies; expanding 1/|
**x−y**|; Legendre polynomials; spherical harmonics, spherical multipole moments, and their potentials; dipole moment in detail. - January 28 (Friday):
- Finished electric multipole expansion:
quadrupole term; quadrupole moment tensor in detail; examples;
~~octupole terms and octupole tensor~~; higher multipoles. - January 31 (Monday):
- 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. - February 2 (Wednesday):
- Highlights of magnetostatics:
vector potential
**A**(**x**) and equations it obeys; examples of calculating**A**(**x**) and**B**(**x**); multipole expansion; magnetic dipole moment in detail. - February 4 (Friday):
- Canceled due to bad weather.
- Regular lecture on February 7 (Monday):
- Magnetic dipoles:
multipole expansion for the volume current; magnetic dipole moment in detail;
gyromagnetic ratio; fields of point dipoles (electic and magnetic);
forces and torques on dipoles;
~~magnetic effects on atoms~~. - Make-up lecture on February 7 (Monday):
- Finished magnetic dipoles.

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 conditions in dielectrics and magnets. - February 9 (Wednesday):
- Polarization and Magnetization:
dielectric sphere example;
scalar magnetic potential Ψ; permanent magnet examples;
multivalued Ψ(
**x**) in presence of wires. - February 11 (Friday):
- Electrostatic energy and forces on dielectrics:
electrostatic energy; self-energy and interaction energy; energy in linear
~~and non-linear~~dielectrics; energy and forces in capacitors; started forces on dielectrics in electric fields. - Regular lecture on February 14 (Monday):
- Finished electrostatic energy:
forces on dielectrics in electric fields; energy in non-linear dielectrics; hysteresis and energy loss (briefly).

Started Faraday Induction Law: Faraday's flux rule; motional EMF, and its relation to the flux rule. - Extra lecture on February 14 (Monday):
- 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 16 (Wednesday):
- Finished Faraday Induction Law:
induced non-potential electric field; ∇×
**E**=−∂**B**/∂*t*; scalar and vector potentials for time-dependent fields; gauge transforms and gauge-fixing.

Magnetic energy: energy of inductor coil; energy of magnetic field; energy loss to hysteresis; forces on magnetic materials. - February 18 (Friday):
- Complex amplitudes and impedance.

Mutual inductance and transformers (briefly). - February 21 (Monday):
- Eddy currents and skin effect: demo#1; demo#2; diffusion equation for the current and the magnetic field; solving the diffusion equation: how the field penetrates a conductor; skin effect for AC currents.
- February 23 (Wednesday):
- Maxwell equations:
the displacement current; Maxwell equations and electromagnetic waves;
equations for the potentials
**A**and Φ; transverse gauge gauge; Landau gauge. - February 25 (Friday):
- 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. - Regular lecture on February 28 (Monday):
- 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.

Began plane EM waves: wave vectors; electric and magnetic amplitudes. - Extra lecture on February 28 (Monday):
- 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.
- March 2 (Wednesday):
- Plane electromagnetic waves: electric and magnetic amplitudes; wave impedance; wave energy; linear, circular and elliptic polarizations.
- March 4 (Friday):
- Geometric laws for general waves:
law of reflection and Snell's law of refraction; total internal reflection and evanescent waves.

Reflection and refraction of electromagnetic waves: 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~~. - Regular lecture on March 7 (Monday):
- Finish Reflection and refraction of EM waves:
phase shift in total internal reflection.

Dispersion and attenuation of EM waves: time lag and complex ε(ω) and μ(ω); power dissipation due to Im(ε) and Im(μ); complex conductivity; attenuation of plane EM waves. - Extra lecture on March 7 (Monday):
- Superfluids: Bose–Einstein condensation and the condensate field; density and velocity of the superfluid.

Superconductivity: Cooper pairs and their condensation; charged superfluid; Meissner effect. - March 9 (Wednesday):
- Gaussian wave packets.

Microscopic origin of dispersion: single-resonance toy model; multi-resonance model; normal and anomalous dispersion; low frequency behavior α(ω); Drude conductivity in metals. - March 11 (Friday):
- Microscopic origin of dispersion, high-frequencyα(ω): plasmas and plasma frequency;
plasma frequency in metals.

Dispersion in 1D waves: 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 14, 16, and 18:
- Spring break.
- March 21 (Monday):
- 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~~. - March 23 (Wednesday):
- Canceled.
- March 25 (Friday):
- Faraday effect in plasma.

Introduction to waveguides: Maxwell equations and boundary conditions; dispersion relations and cutoff frequencies;~~TEM waves~~. - Regular lecture on March 28 (Monday):
- Waveguides: TEM waves; TE waves; TM waves; wave power; speed of wave energy in a waveguide; waves in a rectangular waveguides.
- Make-up lecture on March 28 (Monday):
- More waveguides: waves in a circular waveguide; effects of wall resistivity on the boundary conditions; wave attenuation due to wall resistivity; frequency dependence of the attenuation rate.
- March 30 (Wednesday):
- Microwave cavities:
TM and TE standing waves; modes and resonant frequencies of a rectangular cavity;
modes and frequencies of a cylindrical cavity.

Quality factor of a resonator: mechanical example; resonance width; LRC circuit example.

Quality of a microwave cavity: general estimate;~~example of a geometric factor~~. - April 1 (Friday):
- Finished Microwave cavities:
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. - Regular lecture on April 4 (Monday):
- Finished optic fibers: wave optics for smooth-index fibers; mode counting; wave equations for single-mode fibers.
- Extra lecture on April 4 (Monday):
- Superconductivity: flux quantization; magnetic vortices; type II and type II superconductors.
- April 6 (Wednesday):
- Radiation by compact antennas: radiation by harmonic currents; near, intermediate, and far zones; spherical waves; multipole expansion; the leading term and the electric dipole moment; the basics of electric dipole radiation.
- April 8 (Friday):
- Electric dipole radiation:
linear antenna example;non-linear dipoles; Rutherford atom example.

Higher multipoles: first subleading order; magnetic dipole radiation; begin electric quadrupole radiation. - April 11 (Monday):
- Finish Radiation by compact antennas:
electric quadrupole radiation; higher miltipoles.

Quantum radiation of photons: quantum transitions; Fermi's Golden rule; intro to quantum EM fields and photons; photon emission by excited atoms in the dipole approximation; quantum-classical correspondence. - April 13 (Wednesday):
- Quantum radiation of photons: classical amplitudes as limits of quantum matrix elements; allowed and forbiddent transitions in atoms; selection rules for the allowed transitions.
- April 15 (Friday):
- Gamma decays and selection rules in nuclear physics.

Scalar spherical waves in detail: separation of variables; spherical Bessel functions*j*and_{ℓ}*n*; small-raius and large-radius asymptotics; divergent spherical waves and the Hankel functions_{ℓ}*h*._{ℓ}

Maybe begin spherical EM waves: coordinating the components*E*(_{i}**x**) and*H*(_{i}**x**); TM waves and TE waves. - Regular lecture on April 18 (Monday):
- Spherical EM waves:
coordinating the component waves
*E*(_{i}**x**) and*H*(_{i}**x**); TM waves and TE waves; no ℓ=0 modes of EM waves. - Extra lecture on April 18 (Monday):
- Josephson junctions:
tunneling of Cooper pairs;
*I*=*I*_{0}×sin(Δφ); voltage and oscillations. - April 20 (Wednesday):
- Spherical EM waves:
long-distance and short-distance limits;
electric multipoles source TM waves, magnetic multipoles source TE waves;
power and its angular distribution for the waves with specific ℓ and
*m*.

Maybe begin: radiation by a long antenna. - April 22 (Friday):
- Radiation by a long antenna:
center-fed long 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. - April 25 (Monday):
- 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; polarized cross-sections; begin example of a small dielectric sphere. - April 27 (Wednesday):
- Scattering examples:
small dielectric sphere: σ∝
*k*^{4}, angular dependence, and polarization; Thomson scattering by a free electron.

Started multiple scatterers. - April 29 (Friday):
- Multiple scatterers of EM waves: interference and the form factor; Rayleight scattering by gases; attenuation by scattering; Bragg scattering by crystals.
- Regular lecture on May 2 (Monday):
- Partial wave analysis (for the scalar waves): partial waves; radial waves and sphase shifts; scattering amplitude and total cross-section in terms of phase shifts; scattering off a hard sphere; small sphere limit; large sphere limit.
- Extra lecture on May 2 (Monday):
- 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. - May 4 (Wednesday):
- Diffraction: Introduction; Green's theorem; Kirchhoff approximation; integrals over the aperture(s); Fresnel and Fraunhofer diffraction; Fraunhofer limit in detail; rectangular aperture as an example.
- May 6 (Friday):
- Diffraction: multiple apertures, interference, and diffraction gratings; diffraction in a circular aperture; Airy disk; laser Moon ranging example; telescopes and resolution.

Last Modified: May 6, 2022. Vadim Kaplunovsky

vadim@physics.utexas.edu