Formulating Initial and Boundary Effects for Maxwell Equations

Abstract

In classical treatment of Maxwell equations, the initial and boundary conditions are introduced by mathematical consideration rather than strictly using the Maxwell equations. As a result, the initial and boundary conditions are not logic consistent with the Maxwell equations. In fact, the boundary problem of classical Maxwell equations is not solvable for general cases, if the whole boundary fields are given. This problem is caused by implying the divergence and the curl of a vector have no relations during introducing the boundary condition. This research introduces new ideas and formulations to solve this problem. The research shows that based on Chen S-R decomposition of a rank-two tensor, this logical un-consistency can be discarded and, as a consequence, the classical Maxwell equations are reformulated. According to the reformulated Maxwell equations, the initial condition and boundary condition have global effects. As an example, the London equations for superconductors are deduced.