# CLASSIC ELECTRODYNAMICS

**Academic Year 2023/2024**- Teacher:

**GABRIELE COCI**

## Expected Learning Outcomes

The objective of this course is to provide students with the basic knowledge of classical electrodynamics (in both static and dynamic conditions) and special relativity. Although the course is inherently theoretical due to the large number of discussed applications, it is suitable not only for students aiming to specialize in theoretical physics but also as a foundation for those intending to specialize in nuclear and subnuclear physics (both theoretical and experimental), astrophysics, or those planning to work in radiation protection.

The course duration is 42 hours, equivalent to 6 ECTS credits. Points 1 to 8 of the curriculum (21 hours) will be taught by Prof. Ruggieri, while the remaining points (21 hours) will be taught by Prof. Coci.

In more detail, the learning outcomes are as follows:

*1. Knowledge and Understanding:*

- Demonstrate a solid understanding of the fundamental principles of classical electrodynamics.

- Describe concepts related to electromagnetic fields, retarded potentials, and electromagnetic radiation.

- Explain Maxwell's equations and their significance.

**2. Application of Knowledge and Understanding:**

- Apply the laws of electrodynamics to solve complex problems.

- Use the formalism of special relativity to analyze physical situations.

- Apply the principles of classical electrodynamics to applications in nuclear physics, compact stellar object astrophysics, and relativistic nuclear collisions.

*3. Drawing Conclusions:*

- Perform advanced analyses to deduce significant physical results from Maxwell's equations and special relativity.

- Interpret the results of simulations and experiments related to the topics covered.

*4. Communicative Skills:*

- Present fundamental concepts and solutions to problems clearly and effectively through oral and written reports.

- Actively participate in class discussions and presentations on topics related to classical electrodynamics and its applications.

*5. Learning Skills:*

- Demonstrate the ability to learn independently, acquiring new knowledge and further exploring the topics covered in the course.

- Be able to connect classical electrodynamics concepts to recent developments in theoretical and experimental physics.

## Course Structure

The course will be conducted through frontal lessons.

## Required Prerequisites

To successfully follow the course, students should already have a solid foundation in classical physics, particularly in mechanics, electrostatics, and magnetostatics, as well as mathematical analysis (vector analysis and multiple-dimensional integral theorems are important) and algebra (vector operations). These topics will, however, be briefly reviewed at the beginning of the course or whenever they are needed for the development of electrodynamics. Where necessary, references will be made to analytical mechanics (Lagrangian and Hamiltonian formulations). The relativistic formulation in terms of four-tensors, on the other hand, is not required as a prerequisite and will be presented in detail during the course.

## Attendance of Lessons

Attendance is mandatory.

## Detailed Course Content

**Mathematical Preamble**

Dirac delta, vector analysis, differential operators, coordinate systems

**Recap of Electrostatics and Magnetostatics**

Electrostatic potential, electric dipole, electrostatic energy, continuity equation, vector potential, far-field and multipole expansion, applications to the calculation of atomic nuclei magnetic moments

**Time-Dependent Fields**

Maxwell's equations for time-dependent sources, potentials for time-dependent fields, magnetic energy, conservation of energy and Poynting vector, conservation of momentum and stress-energy tensor of the electromagnetic field, plane waves, spherical waves from point sources, retarded potentials

**Electromagnetic Radiation**

Retarded electromagnetic fields, electromagnetic radiation, radiation from electric dipoles, radiation from magnetic dipoles. Fields of moving charges: Lienard-Wiechert potentials, fields of a charge in uniform motion, radiation from accelerated charges, bremsstrahlung, synchrotron radiation, applications to Nuclear Physics and Astrophysics

**Relativistic Mechanics**

Postulates of Special Relativity, four-dimensional interval, Lorentz transformations, analytical mechanics of a relativistic free particle, 4-vectors and 4-tensors, covariance of the laws of nature, invariant relativistic measurements

**Covariant Formulation of Electrodynamics**

Analytical mechanics of a point charge, Lorentz transformations of the electromagnetic field, covariant form of Maxwell's equations, action of the electromagnetic field

## Textbook Information

1. D. J. Griffiths, *Introduction to Electrodynamics (Fourth Edition)*, Cambridge University Press (2017)

2. L. D. Landau and E. M. Lifsits, *Fisica Teorica 2: Teoria dei Campi*, Editori Riuniti Univ. Press (2010)

3. Provided teaching materials

**Other texts suggested for consultation**

A. Zangwill, *Modern Electrodynamics*, Cambridge University Press (2013)

J. D. Jackson, *Classical Electrodynamics International Adaption** (Third Edition)*, John Wiley & Sons (2021)

M. Schwartz, *Principles of Electrodynamics*, Dover Publications (1987)

Author | Title | Publisher | Year | ISBN |
---|---|---|---|---|

D.J. Griffiths | Introduction to Electrodynamics (Fourth Edition) | Cambridge University Press | 2017 | 978-1108420419 |

L. D. Landau and E. M. Lifsits | Fisica Teorica 2: Teoria dei Campi | Editori Riuniti Univ. Press | 2010 | 978-8864732077 |

## Course Planning

Subjects | Text References | |
---|---|---|

1 | Dirac Delta, vectorial analysis, differential operators, coordinate systems (2 hours) | text 1, 3 |

2 | Electrostatic potential, electric dipole, electrostatic energy (2 hours) | text 1, 3 |

3 | Continuity equation, vector potential of magnetic field (2 hours) | text 1, 3 |

4 | Electrostatic and magnetostatic field at large distance, multipole expansion, applications to calculation of nuclear and atomic magnetic moments (4 hours) | text 1, 3 |

5 | Maxwell's equations for time-dependent sources, potentials for time-dependent fields (2 hours) | text 1, 3 |

6 | Magnetic energy, energy conservation and Poynting vector, conservation of momentum and electromagnetic stress–energy tensor(2 hours) | text 1, 3 |

7 | Plane waves, spherical waves produced by pointlike sources, retarded potentials (2 hours) | text 1, 3 |

8 | Retarded electromagnetic fields, electromagnetic radiation, electric dipole radiation, magnetic dipole radiation (5 hours) | text 1, 3 |

9 | Lienard-Wiechert potentials, fields produced by uniformly moving charge, radiation from accelerated charges, Bremsstrahlung, syncrotron radiation, applications to nuclear physics and astrophysics (6 hours) | text 1, 3 |

10 | Postulates of Special Relativity, spacetime interval, Lorentz transformations, analytic mechanics of relativistic free particle, 4-vectors and 4-tensors, covariance of nature's laws, invariant relativistic measurements (6 hours) | text 2, 3 |

11 | Analytic mechanics of pointlike particle, Lorentz transformations of electromagnetic fields, covariant form of Maxwell's equations (9 hours) | text 2, 3 |

## Learning Assessment

### Learning Assessment Procedures

The evaluation will take place through an oral examination, consisting of four questions, two of which will be related to time-dependent fields and electromagnetic radiation, and two on special relativity. Among the criteria for determining the final grade, correctness of the answers will be assessed, as well as the student's ability to communicate using appropriate technical language and to make connections with other topics covered in the course.

### Examples of frequently asked questions and / or exercises

Maxwell's Equations for Time-Dependent Sources

Retarded Potentials

Fields Produced by a Charge in Uniform Rectilinear Motion

Radiation Produced by a Charge in Uniform Circular Motion

Energy and Momentum of the Electromagnetic Field

Action of the Electromagnetic Field

Maxwell's Equations in Covariant Form

Relevant 4-Vectors and 4-Tensors in Electrodynamics