Relativistic Electrodynamics

Full course description

Electrodynamics is the theory that describes all (non-quantum) aspects of electric and magnetic fields and their interaction with charged matter; most notably is describes the dynamics of these fields in time. The basic rules of Electrodynamics are laid out by the famous Maxwell Equations, that were covered in the prerequisite course Electromagnetism (PHY2004).

Maxwell’s Equations reveal, when written in the appropriate mathematical language of scalar and vector potentials, that the theory of Special Relativity is fully embedded in Electrodynamics from the get-go, without having to artificially build this in. In fact, it can be shown that Electrodynamics would be mathematically inconsistent if the laws of physics had not obeyed the rules of Special Relativity. In this course, the goal is to make this intimate connection between Electrodynamics and Special Relativity explicitly clear.

The course will start with an overview of Maxwell’s Equations and their qualitative meaning, starting from a few experimental facts (Gauss’ Law and Biot-Savart’s Law). Taking Maxwell’s Equations as the foundation of the rest of the course, we will reformulate them in terms of scalar and vector potentials and show that there is a mathematical freedom in choosing these potentials without affecting the resulting physics. We will then find the equations that the potentials obey, and write down the general solution to them. It will next be discussed how one can take into account the time delay that occurs when sources and particles on which they act are a sizeable distance away. Finally, after an overview of 4-vectors, Lorentz-transformations, Minkowski-spacetime and tensors has been provided, the theory will     be cast into the language of 4-vectors to make explicit the deep connection Electrodynamics shares with  Special Relativity. The course ends with the introduction of the Principle of Least Action and the derivation of the entire classical field theory in covariant form from this principle. The concept of gauge transformation and gauge invariance is explained in application to electromagnetic fields and charged particles.