A class of non-null toroidal electromagnetic fields and its relation to the model of electromagnetic knots

Abstract

An electromagnetic knot is an electromagnetic field in vacuum in which the magnetic lines and the electric lines coincide with the level curves of a pair of complex scalar fields ϕ and θ (see equations (), ()). When electromagnetism is expressed in terms of electromagnetic knots, it includes mechanisms for the topological quantization of the electromagnetic helicity, the electric charge, the electromagnetic energy inside a cavity and the magnetic flux through a superconducting ring. In the case of electromagnetic helicity, its topological quantization depends on the linking number of the field lines, both electric and magnetic. Consequently, to find solutions of the electromagnetic knot equations with nontrivial topology of the field lines has important physical consequences. We study a new class of solutions of Maxwellʼs equations in vacuum Arrayás and Trueba (2011 arXiv:1106.1122) obtained from complex scalar fields that can be interpreted as maps , in which the topology of the field lines is that of the whole torus-knot set. Thus this class of solutions is built as electromagnetic knots at initial time. We study some properties of those fields and consider if detection based on the energy and momentum observables is possible.