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ACM40010

Academic Year 2024/2025

Electrodynamics & Gauge Theory (ACM40010)

Subject:
Applied & Computational Maths
College:
Science
School:
Mathematics & Statistics
Level:
4 (Masters)
Credits:
5
Module Coordinator:
Dr Chris Kavanagh
Trimester:
Autumn
Mode of Delivery:
On Campus
Internship Module:
No
How will I be graded?
Letter grades

Curricular information is subject to change.

This module provides an introduction to gauge field theories starting with electrodynamics. We will review the theory of electromagnetic fields and introduce the Maxwell's equations in the 'modern' form of PDEs. We will examine some physical consequences including the prediction of electromagnetic waves as solutions of the Maxwell's equations, and compute the corresponding speed of wave propagation.

We will study the symmetries of the Maxwell's equations with respect to spatial rotations and Lorentz transformations, in a modern approach to Einstein's famous work ''On the electrodynamics of moving bodies'' published in 1905. This will allow us to re-formulate Maxwell's equations in relativistic form.

In this new formulation, gauge invariance will be presented and this will open the way to the formal definition of gauge theories, which are relevant to fundamental theories of particles.


Course Outline:

A) Introduction: Laws of Electrodynamics:
Lorentz Force between charged particles in terms of electric and magnetic fields.
Equations of Electromagnetism in integral form (fluxes over surfaces and circulation along curves).
Equations of Electromagnetism in Differential Form: Maxwell's Equations.

B) Consequences of Maxwell's Equations:
Conservation of electric charge.
Existence of non-trivial solutions in vacuum: plane electromagnetic waves (i.e., light).
Speed of light, dimensions of measure and units of measure.
Energy and Momentum in the Electromagnetic Field: Poynting vector.

C) Gauge Invariance and Relativistic Formulation of Electromagnetism:
Scalar and vector potentials.
Symmetries of Maxwell's equations under spatial rotations.
Symmetries of Maxwell's equations under Lorentz transformations.
Special relativity: different inertial observers in relative motion, will measure the same value for the speed of light in their respective reference frames.
4-vectors and space-time metric. 4-dimensional representation of Maxwell's equations: Electromagnetic Field Strength Tensor.
Relativistic form of Lorentz Force equation. Momentum 4-vector.
Energy-Momentum tensor of the Electromagnetic Field.

D) Applications:
Radiation from bounded sources (e.g., antennas).
Radiation from moving point charges: Liénard-Wiechert potentials.

E) Gauge Theories:
Electromagnetism as a Gauge theory.
Action Principle for Electrodynamics and invariances thereof.
Other Gauge theories.

About this Module

Learning Outcomes:

Solve Maxwell's equations under a number of conditions including plane waves and radiation from bounded sources.
Master the relativistic formulation of Maxwell's equation.
Explain the concept of a gauge theory, in terms of continuous symmetries of fundamental interacting fields.

Applications include:
Interaction of electromagnetic waves with matter (radiation pressure, energy flux, linear/angular momentum conservation, charge conservation).
Lorentz transformations of electromagnetic fields between observers in relative motion: infinitesimal generators (boosts and rotations) in the 4x4 matrix representation, and exponential map.
Lorentz invariants and Doppler effect.
Parabolic Lorentz transformations.
Electromagnetic field due to a continuous distribution of charges and currents.
Action principles for the electromagnetic field.
Gauge theories for electromagnetic field coupled with complex scalar fields.
Interaction with the wavefunction of a quantum non-relativistic particle.
The double-slit experiment for electron diffraction and the Aharonov-Bohm effect.

Student Effort Hours:
Student Effort Type Hours
Lectures

30
Specified Learning Activities

20
Autonomous Student Learning

60
Total

110

Approaches to Teaching and Learning:
Lectures, tutorials, enquiry and problem-based learning.

Requirements, Exclusions and Recommendations
Learning Requirements:

Students are required to know vector integral and differential calculus (ACM20150 or equivalent), and they must have some familiarity with electrostatics or electrodynamics, at least from the most basic instances of Maxwell's equations.

Learning Recommendations:

It is recommended that students have a firm background in vector calculus.
It would be desirable that students had a knowledge of action principles, through courses such as ACM30010 or equivalent.
Finally, some familiarity with Fourier transforms and/or the theory of distributions (e.g., the Dirac delta distribution) would be useful.


Module Requisites and Incompatibles
Equivalents:
Electrodynamics & Gauge Theory (MAPH40010)


 

Assessment Strategy
Description Timing Component Scale Must Pass Component % of Final Grade In Module Component Repeat Offered
Assignment(Including Essay): Take home written assignments. Week 4, Week 6, Week 11 Standard conversion grade scale 40% No
30
 No
Exam (In-person): Class test. Week 9 Standard conversion grade scale 40% No
10
 No
Exam (In-person): Final Exam. End of trimester
Duration:
2 hr(s)
 Standard conversion grade scale 40% No
60
 No

Carry forward of passed components
No
 

Resit In Terminal Exam
Spring Yes - 2 Hour
Please see Student Jargon Buster for more information about remediation types and timing. 

Feedback Strategy/Strategies

• Group/class feedback, post-assessment

How will my Feedback be Delivered?

Not yet recorded.

Timetabling information is displayed only for guidance purposes, relates to the current Academic Year only and is subject to change.
Autumn Lecture Offering 1 Week(s) - Autumn: All Weeks Fri 13:00 - 13:50
Autumn Lecture Offering 1 Week(s) - Autumn: All Weeks Tues 11:00 - 11:50
Autumn Lecture Offering 1 Week(s) - Autumn: All Weeks Wed 13:00 - 13:50
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