# SPECIAL RELATIVITY AND ELECTRODYNAMICS - 2022/3

Module code: MAT3047

## Module Overview

This module is designed to be an introduction to the ideas, techniques and applications of classical electromagnetism and Einstein’s theory of special relativity. The natural relationship between these two key components of mathematical physics is of particular importance throughout. It will be demonstrated how the whole structure of electromagnetism is most naturally treated in a unified and consistent way using special relativity. Therefore, equal emphasis is placed on electromagnetism and special relativity within the module.

Beginning with an introduction to the Maxwell equations in their classic non-relativistic form, elementary properties of these equations, such as conservation laws, will be investigated. Standard examples, including electrostatics and other examples, as well as some techniques for constructing solutions, will also be considered.

After discussing why Galilean relativity is not sufficient to describe electromagnetism in a consistent way, the course moves to a detailed treatment of special relativity. The properties of Lorentz transformations are examined, and the measurement of distance and time is considered. The natural development from Newtonian particle mechanics to relativistic particle mechanics is described, with appropriate examples.

The last part of the module consists of a rewriting of the non-relativistic Maxwell equations into a manifestly relativistic formalism. Key properties of the Maxwell equations, such as gauge invariance, as well as the dynamics of particles moving in electric and magnetic fields, will be re-interpreted in this way. Applications and examples, such as relativistic plane wave solutions, will be considered.

### Module provider

Mathematics

GUTOWSKI Jan (Maths)

### Module cap (Maximum number of students): N/A

Independent Learning Hours: 117

Lecture Hours: 33

Semester 2

## Prerequisites / Co-requisites

MAT1036 (Classical Dynamics)

## Module content

Indicative content includes:

1. Introduction to electrostatics, boundary value problems
in electrostatics, multipoles, magnetostatics. Discussion of time varying fields, the (non-relativistic) Maxwell equations, and conservation laws. The Lorentz force law; discussion of the inadequacy of Galilean relativity
for describing electromagnetic phenomena. [2.5 Weeks]

2. Constancy of the speed of light. Lorentz transformations and their mathematical properties. The invariance of the wave operator. Time dilation, length contraction, the relativistic Doppler effect. The barn-door paradox. [2 weeks]

3. Four-vectors; four-velocity and four-momentum; equivalence of mass and energy, comparison of massive vs massless particles. Particle collisions and four-momentum conservation, with examples including Compton scattering. Four-acceleration and four-force, with examples. [3 Weeks]

4. Scalar and vector potentials; the four-potential. Gauge invariance; Maxwell's equations in Lorentz gauge. Plane waves, polarization. Energy density and the Poynting vector. [2.5 Weeks]

The electromagnetic field tensor, the transformation law for the electric and magnetic field. The Lorentz four-force law. [1 Week]

## Assessment pattern

Assessment type Unit of assessment Weighting
School-timetabled exam/test In-semester test 20
Examination Examination 80

n/a

## Assessment Strategy

The assessment strategy is designed to provide students with the opportunity to demonstrate

• The ability to understand and formulate statements about: properties of the Maxwell equations, the key concepts of Special Relativity with classical examples,
and how electromagnetism is described using Special Relativity.

• A knowledge of the subject, important definitions, and concepts

• The ability to apply this knowledge to the analysis of unseen problems in the tests and the examination.

Thus, the summative assessment for this module consists of:

• One two hour examination (the best three answers out of four questions contribute to the exam mark) at the end of Semester 2; worth 80% of the module mark.

• One in-semester test, worth 20% of the module mark.

Formative assessment and feedback

There is feedback from marked coursework assignments over an 11 week period.
Verbal feedback is provided by the lecturer during seminars, and also in office hours.

## Module aims

• Introduce students to the Maxwell equations, the properties of their solutions and applications to key examples.
• Introduce the students to the concepts of special relativity, and describe how Special relativity modifies intuitive notions of space and time, with applications to relativistic particle mechanics.
• Unify the treatment of electromagnetic phenomena by re-interpreting the theory of electromagnetism in the context of special relativity.

## Learning outcomes

 Attributes Developed 001 A) On successful completion of this module, students should understand the fundamental properties of the Maxwell equations, such as conservation laws and gauge invariance; the key concepts in special relativity including Lorentz transformations and their properties, four-vectors, four-momenta and relativistic particle mechanics; and the relativistic formulation of classical electromagnetism. KC 002 B) On successful completion of the module the students should have a clear idea of how to implement the above theory in examples KCP

Attributes Developed

C - Cognitive/analytical

K - Subject knowledge

T - Transferable skills

P - Professional/Practical skills

## Methods of Teaching / Learning

The learning and teaching strategy is designed to:

1. Be an introduction to the key ideas in Classical Electromagnetism, concentrating on an introduction to properties of the Maxwell equations and their solutions, with illustrative examples. [Part I]

2. Develop the formalism of Special Relativity including Lorentz transformations, length contraction & time dilation, and conservation of 4-momentum with applications to relativistic particle mechanics. [Parts II, III]

3. Show how Classical Electromagnetism can be further developed, and entirely written in a manifestly Lorentz covariant fashion, with appropriate applications and examples. [Parts IV, V]

The learning and teaching methods consist of:

2x1 and 1x1 hour lectures per week, for 11 weeks, with typeset notes to complement the course, and Q+A opportunities for students.

Indicated Lecture Hours (which may also include seminars, tutorials, workshops and other contact time) are approximate and may include in-class tests where one or more of these are an assessment on the module. In-class tests are scheduled/organised separately to taught content and will be published on to student personal timetables, where they apply to taken modules, as soon as they are finalised by central administration. This will usually be after the initial publication of the teaching timetable for the relevant semester.