Introduction to Maxwell’s equations and their symmetries

Electrodynamics is a very exciting subject to learn. Unfortunately, Maxwell’s equations are often taught in a rather dull manner, which usually involves solving a lot of boring problems without emphasizing the pivotal role symmetry principles play in the formal structure of these field equations. This may lead to the rather unpleasant feeling of not truly understanding these equations. So this course is entirely devoted in trying to understand and make sense of Maxwell’s equations through their symmetries. The lectures are suitable for advanced undergraduate students and beginning graduates. Finally, let me try to persuade you to take this course with the words of Freeman Dyson:

¨ The ultimate importance of the Maxwell theory is far greater than its immediate achievement in explaining and unifying the phenomena of electricity and magnetism. Its ultimate importance is to be the prototype for all the great triumphs of twentieth-century physics. It is the prototype for Einstein’s theories of relativity, for quantum mechanics, for the Yang-Mills theory of generalised gauge invariance, and for the unified theory of fields and particles that is known as the Standard Model of particle physics…We may hope that a deep understanding of Maxwell’s theory will result in dispersal of the fog of misunderstanding that still surrounds the interpretation of quantum mechanics. And we may hope that a deep understanding of Maxwell’s theory will help to lead the way toward further triumphs of physics in the twenty-first century.¨

Prerequisites: Enjoying the subject

Lecture Notes: Lecture Notes currently in progress!


Electrostatics and Magnetostatics 

  • I  Electrostatics, magnetostatics, and the Helmholtz theorem
  • II  Potentials and gauge symmetry; magnetic monopoles and duality symmetry.


  • III Maxwell’s equations and the Helmholtz theorem for retarded fi elds.
  • IV Gauge symmetry in Maxwell’s equations.
  • V Conservation laws: charge, energy, momentum, and angular momentum.
  • VI Duality symmetry in electrodynamics: A review of Maxwell’s equations with magnetic monopoles.
  • VII The Dirac monopole and the quantization condition. 

Electrodynamics in spacetime

  • VIII Review of special relativity.
  • IX Covariant formulation of Electrodynamics.
  • X Covariant formulation of Electrodynamics with magnetic monopoles.

How to “discover” Maxwell’s Equations? 

  • XI A heuristic derivation of Maxwell’s equations from charge conservation.
  • XII A heuristic derivation of Maxwell’s equations from quantum mechanics.

Action Principle in Electrodynamics

  • XIII Review on the action principle.
  • XIV Lagrangian formulation of electrodynamics.
  • XV Conservation laws and Noether’s theorem.
  • XVI Proca electrodynamics= Massive photons!


Useful textbooks

  • J. D. Jackson, 1999, Classical Electrodynamics, 3rd edn (New York: Wiley)
  • A. Zangwill, 2012, Modern Electrodynamics (Cambridge University Press: Cambridge)
  • J. Schwinger et al., 1998,  Classical Electrodynamics (Westview press)
  • F. E. Low, 1997, Classical Field Theory: Electromagnetism and Gravitation (Wiley)

Recommended Articles

  • R. Heras, The Helmholtz theorem and retarded fields, Eur. J. Phys, 37, 065204 (2016) [PDF]
  • R. Heras,  Lorentz transformations and the wave equation, Eur. J. Phys. 37, 025603 (2016) [PDF]
  • F. Dyson, Why is Maxwell’s Theory so Hard to Understand?, [PDF]
  • J. D. Jackson, From Lorenz to Coulomb and other explicit gauge transformations, Am.  Jour. Phys. 70, 917 (2002) [PDF]
  • D. H. Kobe, Helmholtz’s theorem revisited,  Am. J. Phys. 54, 552 (1986) [PDF]
  • D. H. Kobe, Helmholtz theorem for antisymmetric second-rank tensor fields and electromagnetism
    with magnetic monopoles,  Am. J. Phys. 52, 354 (1984)[PDF]
  • D. A. Woodside, Three-vector and scalar field identities and uniqueness theorems in Euclidean and
    Minkowski spaces, Am. J. Phys. 77, 438 (2009) [PDF]
  • C-N Yang, The conceptual origins of Maxwell’s equations and gauge theory, Physics Today 67 (11),
    45 (2014) [PDF]

Suggested papers on Symmetry

Brief non-technical overview on symmetries in Physics

Gauge Symmetry 

Duality Symmetry