Classical Electrodynamics
A.Y. 2026/2027
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.
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.
Lesson period: First semester
Assessment methods: Esame
Assessment result: voto verbalizzato in trentesimi
Single course
This course cannot be attended as a single course. Please check our list of single courses to find the ones available for enrolment.
Course syllabus and organization
Group 1
Responsible
Lesson period
First semester
Course syllabus
1) Maxwell's equations, conservation laws, electromagnetic waves.
Maxwell's equations. Scalar potential and vector potential. Gauge transformations; Lorenz and Coulomb gauges. Green's functions for the wave equation. Delayed solutions for fields. Poynting's theorem and conservation of energy, momentum, and angular momentum for a system of charged particles and electromagnetic fields. Poynting's theorem for harmonic fields. Electromagnetic waves. Plane waves in lossless media and in conducting or dissipative media. Notes on the optical properties of metamaterials. Reflection and refraction on a plane surface separating two materials. Total internal reflection. Notes on surface waves. Dispersive properties of dielectrics, conductors, and plasmas. Simplified model for wave propagation in the ionosphere. Group velocity and pulse broadening in a dispersive medium. Causality in the relationship between D and E; Kramers-Kronig relations. Fields and radiation from a localized oscillating source: electric dipole, magnetic dipole, and electric quadrupole. Thomson and Rayleigh scattering.
2) Special relativity theory, dynamics of relativistic particles and electromagnetic fields.
Lorentz transformations and basic kinematics of special relativity. Mathematical properties of space-time. Covariance of electrodynamics. Transformations of electromagnetic fields. Quadratic invariants of the electromagnetic field. Motion of a charged particle in static and uniform electric and magnetic fields. Lagrangian and Hamiltonian of a particle in external electromagnetic fields. Electromagnetic field generated by a charge in uniform rectilinear motion. Lagrangian of the electromagnetic field. Energy-momentum tensor of the electromagnetic field and conservation laws in covariant form.
3) Radiation emission from accelerated charged particles
Liènard-Wiechert potentials and the electromagnetic field generated by a point charge. Total power radiated by an accelerated charge. Angular and frequency distribution of radiation emitted by an accelerated charge. Energy loss due to radiation in linear and circular accelerators.
Maxwell's equations. Scalar potential and vector potential. Gauge transformations; Lorenz and Coulomb gauges. Green's functions for the wave equation. Delayed solutions for fields. Poynting's theorem and conservation of energy, momentum, and angular momentum for a system of charged particles and electromagnetic fields. Poynting's theorem for harmonic fields. Electromagnetic waves. Plane waves in lossless media and in conducting or dissipative media. Notes on the optical properties of metamaterials. Reflection and refraction on a plane surface separating two materials. Total internal reflection. Notes on surface waves. Dispersive properties of dielectrics, conductors, and plasmas. Simplified model for wave propagation in the ionosphere. Group velocity and pulse broadening in a dispersive medium. Causality in the relationship between D and E; Kramers-Kronig relations. Fields and radiation from a localized oscillating source: electric dipole, magnetic dipole, and electric quadrupole. Thomson and Rayleigh scattering.
2) Special relativity theory, dynamics of relativistic particles and electromagnetic fields.
Lorentz transformations and basic kinematics of special relativity. Mathematical properties of space-time. Covariance of electrodynamics. Transformations of electromagnetic fields. Quadratic invariants of the electromagnetic field. Motion of a charged particle in static and uniform electric and magnetic fields. Lagrangian and Hamiltonian of a particle in external electromagnetic fields. Electromagnetic field generated by a charge in uniform rectilinear motion. Lagrangian of the electromagnetic field. Energy-momentum tensor of the electromagnetic field and conservation laws in covariant form.
3) Radiation emission from accelerated charged particles
Liènard-Wiechert potentials and the electromagnetic field generated by a point charge. Total power radiated by an accelerated charge. Angular and frequency distribution of radiation emitted by an accelerated charge. Energy loss due to 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 in the courses of Classical Mechanics, Electromagnetism and 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", 5th ed., Cambridge University Press, 2024
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", 5th ed., Cambridge University Press, 2024
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.
PHYS-01/A - Experimental Physics of Fundamental Interactions and Applications - University credits: 3
PHYS-03/A - Experimental Physics of Matter and Applications - University credits: 3
PHYS-03/A - Experimental Physics of Matter and Applications - University credits: 3
Lessons: 42 hours
Professor:
Rome' Massimiliano Gaetano
Group 2
Responsible
Lesson period
First semester
PHYS-01/A - Experimental Physics of Fundamental Interactions and Applications - University credits: 3
PHYS-03/A - Experimental Physics of Matter and Applications - University credits: 3
PHYS-03/A - Experimental Physics of Matter and Applications - University credits: 3
Lessons: 42 hours
Professor:
Piovella Nicola Umberto Cesare
Professor(s)
Reception:
Friday, 9: 30-12: 30 (by appointment)
office at the Department of Physics