Topological Structures of Electromagnetic Field Lines in Photons

Maxwell's equations are one of the most fundamental equations of physics and have widespread applications. These equations were discovered by many brilliant scientists over hundreds of years and finally completed by Maxwell. The decouple solutions of these equations were used by Maxwell to predict that light is an electromagnetic wave which were confirmed by subsequent experiments by Hertz in 1886. Further experimental and theoretical work with electromagnetic fields, over a century, had done little to illuminate the fundamental nature of light. Here, we have used the divergence conditions for electric and magnetic fields in Maxwell's equations, to show that only possible topological structures of electric and magnetic field lines, which can exist in free space are of the form of closed loops or knots. These solutions must evolve smoothly (due to nature of field lines) in free space and are similar to the solutions discovered by Rañada in 1989[1]. Rañada's topological solutions were based on Hopf fibration and evolve with time. The time evolution of these electric and magnetic field lines was found to preserve the topological invariants [2]. The preservation of the topological structure of the field lines in these solutions has previously been ascribed to the fact that the electric and magnetic helicities, a measure of the degree of linking and knotting between field lines, are conserved [1]. Another work [3] suggested that the smooth evolution of the field lines is due to the stricter condition that the electric and magnetic fields be everywhere orthogonal . But, using very simple divergence arguments, we will show that the topological invariants of the electromagnetic knots is due to Maxwell's divergence conditions.