Electromagnetic Theory II
Time-dependent electromagnetism developed from Maxwell's equations: why the static field equations are incomplete, the full Maxwell equations and the quasistatic approximation; electromotive force, Ohm's law, motional EMF, Faraday's law and Lenz's law with generators and eddy currents; mutual and self-inductance from the Neumann formula, RL and LC circuits, magnetic energy and the displacement current required by charge conservation; Maxwell's equations in vacuum and in matter with the polarization current and the boundary conditions at surfaces; the conservation of charge, energy and momentum, with the Poynting vector, energy flow in cables and resistors, the Maxwell stress tensor and radiation pressure; plane electromagnetic waves, their energy, intensity, momentum and polarization, with sunlight and solar sails; reflection and refraction at surfaces, the Fresnel equations, Brewster's angle and total internal reflection; waves in conductors and the skin effect, dispersion and absorption from the Lorentz oscillator model, plasmas, the ionosphere and group velocity; rectangular waveguides, their modes and cutoff frequencies, and transmission lines with characteristic impedance, reflection, standing waves and impedance matching; scalar and vector potentials, gauge transformations, the Coulomb and Lorenz gauges and the retarded and Liénard–Wiechert potentials; radiation from oscillating dipoles and accelerated charges, the Larmor formula, Rayleigh scattering, synchrotron light and antennas; and a capstone showing that electrodynamics is consistent with special relativity, with the transformation of fields between frames, the field invariants and the field tensor, all supported by runnable Python examples.
PHYS-382 Year 3 3 credits 12 lessons · 4 h 43 min ▶ Video lessons coming soon
Offered by HSM University · Physics
This is the Canadian version of this course. See the American version.
What you'll learn
- Explain why the static field equations are incomplete, state the full Maxwell equations, and decide when the quasistatic approximation applies using the ratio \(\omega\varepsilon/\sigma\) and the size of a system compared with a wavelength.
- Apply Ohm's law, compute motional and induced EMFs, and use Faraday's law and Lenz's law to find induced electric fields and currents.
- Calculate mutual and self-inductance, including with the Neumann formula, analyse RL and LC circuits, find magnetic energy as \(\int B^2/2\mu_0\,d\tau\), and derive the displacement current from charge conservation.
- Write Maxwell's equations in vacuum and in matter, including the polarization current, and apply the boundary conditions for \(\mathbf{D}\), \(\mathbf{B}\), \(\mathbf{E}\) and \(\mathbf{H}\) at surfaces.
- Derive and apply Poynting's theorem, compute energy flow in circuits and cables, and use the momentum density and Maxwell stress tensor to find forces and radiation pressure.
- Derive the wave equation from Maxwell's equations, describe plane waves, their energy, intensity, momentum and polarization, and compute fields and pressures of real sources.
- Derive the laws of reflection and refraction and the Fresnel equations, and use them to find reflectances, Brewster's angle and the conditions for total internal reflection.
- Describe wave propagation in conductors, compute skin depths and surface resistance, explain dispersion with the Lorentz oscillator model, and analyse plasmas, cutoff and group velocity.
- Find the modes and cutoff frequencies of rectangular waveguides, and analyse transmission lines with characteristic impedance, reflection coefficients, standing-wave ratios and quarter-wave matching.
- Express the fields through scalar and vector potentials, use gauge transformations and the Coulomb and Lorenz gauges, and compute retarded and Liénard–Wiechert potentials.
- Derive the fields and power of an oscillating dipole and the Larmor formula, and apply them to scattering, the classical atom, synchrotron radiation and antennas.
- Show that electrodynamics is consistent with special relativity, transform electric and magnetic fields between frames, use the field invariants, and write Maxwell's equations with the field tensor.
Course content
12 lessons · 4 h 43 min total · final exam
- 1 Why electrodynamics: from static fields to light 22 min · quiz
- 2 Electromotive force, Ohm's law and Faraday's law of induction 24 min · quiz
- 3 Inductance, magnetic energy and Maxwell's displacement current 23 min · quiz
- 4 Maxwell's equations in vacuum and in matter, and boundary conditions 24 min · quiz
- 5 Conservation laws: charge, energy and momentum 23 min · quiz
- 6 Electromagnetic waves in vacuum: plane waves, energy and pressure 22 min · quiz
- 7 Electromagnetic waves in matter: reflection, refraction and the Fresnel equations 22 min · quiz
- 8 Absorption, dispersion and waves in conductors and plasmas 24 min · quiz
- 9 Waveguides and transmission lines 24 min · quiz
- 10 Potentials, gauges and retarded potentials 25 min · quiz
- 11 Radiation: dipole radiation, the Larmor formula and antennas 23 min · quiz
- 12 Capstone: electrodynamics and relativity 27 min · quiz
- E Final exam 20 questions
Requirements
Recommended background. You can start any course at any time.
Description
Time-dependent electromagnetism developed from Maxwell's equations: why the static field equations are incomplete, the full Maxwell equations and the quasistatic approximation; electromotive force, Ohm's law, motional EMF, Faraday's law and Lenz's law with generators and eddy currents; mutual and self-inductance from the Neumann formula, RL and LC circuits, magnetic energy and the displacement current required by charge conservation; Maxwell's equations in vacuum and in matter with the polarization current and the boundary conditions at surfaces; the conservation of charge, energy and momentum, with the Poynting vector, energy flow in cables and resistors, the Maxwell stress tensor and radiation pressure; plane electromagnetic waves, their energy, intensity, momentum and polarization, with sunlight and solar sails; reflection and refraction at surfaces, the Fresnel equations, Brewster's angle and total internal reflection; waves in conductors and the skin effect, dispersion and absorption from the Lorentz oscillator model, plasmas, the ionosphere and group velocity; rectangular waveguides, their modes and cutoff frequencies, and transmission lines with characteristic impedance, reflection, standing waves and impedance matching; scalar and vector potentials, gauge transformations, the Coulomb and Lorenz gauges and the retarded and Liénard–Wiechert potentials; radiation from oscillating dipoles and accelerated charges, the Larmor formula, Rayleigh scattering, synchrotron light and antennas; and a capstone showing that electrodynamics is consistent with special relativity, with the transformation of fields between frames, the field invariants and the field tensor, all supported by runnable Python examples.
Part of the Science stream, typically taken in year 3.