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# Department of Mathematical Sciences

# MATH2657 Special Relativity and Electromagnetism II

Electromagnetism and special relativity lie at the foundation of much of our current understanding of the physical world. Maxwell's development of electromagnetism provided the first modern understanding of the interactions between materials, and provided the seeds for Einstein's development of special relativity. Special relativity demonstrates the profound importance of symmetry in physics.

This course will begin by developing electromagnetism, which also applies the vector calculus ideas learned in AMV. We will briefly survey basic phenomena of electricity and magnetism and how their unification leads to the wave equation for light in which the speed of light is a universal constant. We discuss the conceptual problem this creates and its resolution by special relativity. We will see how several common-sense ideas must be abandoned in special relativity: for example the absolute character of time and of energy. We introduce a mathematical setting appropriate for the description of the kinematics and dynamics of special relativity, and make an incursion into the world of 4-dimensional vectors and tensors. We discuss simple collision processes as a first illustration of physics in a special-relativistic world.

## Outline of Course

Aim: To appreciate the conceptual and mathematical structure of Special Relativity. To appreciate classical electromagnetism as an example of a relativistic theory.

• Electro- and magnetostatics: Charge and current density and conservation. Electric fields due to static charges. Electric field E expressed in terms of electrostatic potential ϕ. Electrostatic energy. Magnetic field B due to steady currents. Magnetic field in terms of vector potential A.
• Time-dependent Fields and Maxwell's Equations: Faraday's law. Maxwell's equations in vacuum. Potentials and gauge invariance. Wave equation. Energy and momentum conservation. Plane waves. Speed of light.
• Inertial Frames: Lorentz transformation. Four vectors: Minkowski spacetime. Lorentz and Poincaré groups. Space-time vectors and tensors.
• Relativistic formulation of Maxwell's Equations: Maxwell equations with microscopic sources expressed as tensor equations in Minkowski spacetime. Relativistic equation of motion for a charged particle in an external electromagnetic field.
• Spacetime: Time dilation and length contraction. Applications and false paradoxes. Lorentz transformations. Standard Lorentz boosts. Composition of velocities. Group structure of standard Lorentz boosts.
• Systems of free particles: Conservation of 4-momentum. Centre-of-mass frame. Collision processes.

### Prerequisites

For details of prerequisites, corequisites, excluded combinations, teaching methods, and assessment details, please see the Faculty Handbook.