Properties of Knotted Solutions to Maxwell's Equations: An Exploration of Lipkin's Zilches in Vacuum Electromagnetic Fields, and a Multipole Expansion of the Vector Spherical Harmonics of Knotted Electromagnetic Fields

Classical Maxwell's equations have been fundamental to our understanding of physics since their inception in the 19th century. Any research which contributes to our understanding of them is of importance. In 1989, Ranada introduced a topological formalism of Maxwell's equations in vacuum - investigating a set of solutions involving a mapping between spherical spaces projected into the electric and magnetic field lines - named the Hopf fibration. This research has been furthered by Ranada and other prominent scientists to include: the family of torus knots within the field lines; a theory of the fundamental charge; and a description of the number of right and left handed photons. The Hopf fibration, embedded into charge-free electric and magnetic field lines as exact solutions to Maxwell's equations, is a central component to this body of work. Topological, classical electromagnetism is, therefore, an area that has the potential to further our understanding of Maxwell's equations. This thesis describes a project that advances the knowledge of knotted electromagnetic fi elds and electromagnetism in two, distinct areas: zilches of unusual electromagnetic fi elds; and a multipole expansion on the vector spherical harmonics of the knotted electromagnetic fi elds. Zilches are a set of little-known conserved quantities within the charge-free Maxwell's equations. They were originally posited by Lipkin as new and potentially exciting - if their physical nature could be determined. This research examines the zilches in three unusual topologically non-trivial sets of electromagnetic fi elds - starting with the knotted solutions to Maxwell's equations. The results show a profound connection between the zilches and the fi elds' topology. It conjectures that this is the case for all integrable solutions. Alongside this, it is shown that the zilches can be written in terms of other, known conserved quantities of the fi elds. The latter half of this thesis focuses on delivering a generalised multipole expansion of the vector spherical harmonics for the knotted electromagnetic fi elds, after outlining various flaws in the assumptions made by previous literature. Finally, the results from generating the generalised multipolar expansion coefficients are analysed and presented. A summary of the fi rst 3680 coefficients are given in 18 equations - after noticing patterns occurring between them. Firstly, this research further develops the area of knotted electromagnetic fields, providing a platform from which to generate them experimentally and to develop the theory further. Secondly, it sheds considerable light on the interpretation of the zilches and suggests a more central role in electromagnetism.