Classical electrodynamics

A.Y. 2020/2021
Overall hours
Learning objectives
Starting from Maxwell's equations, advanced knowledge of electromagnetism and special relativity is provided. The generation and propagation of electromagnetic waves in vacuum and in dielectric media, the covariant formalism of the electromagnetic field and the emission of radiation from accelerated charges are studied.
Expected learning outcomes
At the end of the course the student will acquire the following skills:

1. will be able to describe the phenomenology of dispersion and absorption of radiation in a linear medium;

2. will be able to describe the phenomenology of radiation emission by distributions of accelerated charges;

3. will be able to describe the relativistic dynamics of charged particles in electromagnetic fields;

4. will be able to tackle specific electrodynamic problems in different fields of physics (accelerators, particle physics, astrophysics) concerning the relativistic dynamics of charged particles and radiation emission;

5. will be able to follow field theory courses in a profitable way using the acquired knowledge of relativistic electrodynamics.
Course syllabus and organization


Lesson period
First semester
Synchronous remote lessons, using the Zoom platform. Notices, lesson recordings and additional material on the Ariel website.
Course syllabus
1) Maxwell equations, conservation laws, electromagnetic waves.

Maxwell equations. Scalar potential and vector potential. Gauge transformations; Lorenz and Coulomb gauge. Green functions for the wave equation. Retarded solutions for fields. Poynting theorem and conservation of energy, momentum and angular momentum for a system of charged particles and electromagnetic fields. Poynting theorem for harmonic fields; field definition of impedance. Electromagnetic waves. Plane waves in lossless and and in conductive/dissipative media. Brief notes on the optical properties of metamaterials. Reflection and refraction at a plane surface between two media. Total internal reflection. Dispersive characteristics of dielectrics, conductors and plasmas. Simplified model of wave propagation in the ionosphere. Group speed and spreading of a pulse in a dispersive medium. Causality in the relationship between D and E; Kramers-Kronig relations. Propagation in waveguides: TEM, TE and TM modes. Fields and radiation of a localized oscillating source: electric dipole, magnetic dipole and electric quadrupole. Thomson and Rayleigh diffusion.

2) The theory of special relativity, dynamics of relativistic particles and electromagnetic fields.

Lorentz transformations and basic kinematics of special relativity. Mathematical properties of space-time. Covariance of electrodynamics. Transformation of electromagnetic fields. Lagrangian and Hamiltonian of a particle in external electromagnetic fields. Quadratic invariants of the electromagnetic field. The fields generated by a charge in uniform straight motion. Motion of a charged particle in static and uniform electric and magnetic fields. Lagrangian of the electromagnetic field. Energy-moment tensor of the electromagnetic field and conservation laws in covariant form.

3) Emission of radiation from charged particles in accelerated motion

Liénard-Wiechert potentials and fields of a point charge. Total power radiated by an accelerated charge. Angular and frequency distribution of the radiation emitted by an accelerated charge. Energy loss by radiation in linear and circular accelerators.
Prerequisites for admission
Knowledge of the concepts and methods introduced in the Bachelor's Degree in Physics, in particular relating to Classical Mechanics, Electromagnetism and Mathematical Analysis.
Teaching methods
Lectures using blackboard and slides.
Teaching Resources
J. D. Jackson, "Classical Electrodynamics", 3rd ed., John Wiley & Sons (1999) [Italian edition: J. D. Jackson, "Elettrodinamica Classica", II ed., Zanichelli, 2001].

Notes on the Ariel platform.

Other textbooks:
L. D. Landau, E. M. Lifshitz, "The Classical Theory of Fields", 3rd ed., Pergamon, 1971
E. M. Purcell and J. D. Morin, "Electricity and magnetism", 3rd ed., Cambridge University Press, 2013
D. J. Griffiths, "Introduction to electrodynamics", 4th ed., Pearson, 2014
Assessment methods and Criteria
Oral exam with questions on the topics covered in class, to check if the teaching objectives have been achieved and the student has acquired the basic knowledge of the subject. The exam typically lasts 45 minutes, with two or more topics covered by the student.
FIS/01 - EXPERIMENTAL PHYSICS - University credits: 6
Lessons: 42 hours
Professor: Rome' Massimiliano


Lesson period
First semester
The course will be delivered entirely remotely in case of travel
restrictions due to Covid-19. In this case, the lectures will be offered
in virtual classrooms (zoom platform) in synchronous connection, with the
possibility of real-time interaction between the students and the teacher.

Course syllabus
1. Maxwell equations and their properties: Maxwell equations. Scalar and vector potentials. Wave equation and Green functions. Conservation of energy of the electromagnetic fields. The Poyntings's theorem for harmonic fields. Momentum of the electromagnetic fields. Momentum of the electromagnetic fields in a dielectric medium. Plane electromagnetic waves and linear dispersive media. Linear and circular polarization. Reflection and refraction at a plane interface between two dielectric media. Total internal reflection. Electromagnetic energy in dispersive material with losses. Metamaterials and negative refraction. The Lorentz model. Surface plasmons. Propagation in wave guides. Cylindrical wave guides. Resonant cavities. Dielectric wave guides. Radiation from localized sources. Scattering Rayleigh at long wavelengths.

2. Theory of Relativity: Foundations of Theory of Relativity. Relativistic Dynamics. Lagrangian for a particle moving in a given electromagnetic field. Equation of motion. Electromagnetic tensor. Covariant form of the Maxwell equations. Electromagnetic energy-momentum tensor. Lagrangian of a free electromagnetic field. Field emitted by a uniformly moving charge. Motion of a charged particle in assigned static and uniform fields.

3. Radiation by a moving charge: Liènard-Wiechert potentials for a point charge. Covariant expression of the Liènard-Wiechert potentials. Electric and magnetic field from a moving charge. Velocity and radiation fields. Angular distribution of the emitted power. Total radiated power. Radiative losses in linear and circular accelerators. Frequency distribution of the radiation by a charge in an ultra-relativistic motion. Radiation from an undulator.
Prerequisites for admission
No prior knowledges are required
Teaching methods
Lectures in remote, synchronized and recoredered. Connection by Zoom, with writing on a digital tablet, shared by Zoom.
Teaching Resources
J. D. Jackson, "Classical Electrodynamics", John Wiley & Sons, Third ed. (1999)
Notes available on the web platform Ariel
Assessment methods and Criteria
Oral exam on zoom to verify that the objectives of the course have been met and the student has acquired the fundamental aspects of the subjects of the course. The exam last typically 60', with three subjects treated by the student.
FIS/01 - EXPERIMENTAL PHYSICS - University credits: 6
Lessons: 42 hours
Friday, 9: 30-12: 30 (by appointment)
office at the Department of Physics