Appreciating the Symmetry of Electromagnetism Through the Perspective of Topology

This paper explores the unity of Gauss' s law and Ampère's circuital law in electromagnetism from the perspective of topology, providing an elegant framework for revealing the integrality of electromagnetic phenomena. Gauss' s law for electrostatic fields describes the relationship between charge distribution and electric flux, which can be regarded as the representation of the topological properties of charge by the curvature integral. Ampère's circuital law for steady magnetic fields, which describes the relationship between current and magnetic field, corresponds to the constraint of linking numbers between loops. Both can be unified in terms of the first Chern number, a topological invariant whose value depends only on the overall topological structure of the manifold or the way the links are intertwined, and is independent of the details of the local field strength. This holistic description breaks through the limitations of the local description of traditional differential equations and highlights the symmetry between electricity and magnetism. The study points out that the perspective of topology can provide a new theoretical framework for physicists to understand the laws of electromagnetism.