THE UNIQUE SOLUTION OF WAVE EQUATIONS OF THE ELECTROMAGNETIC FIELDS MUST SATISFY MAXWELL'S EQUATIONS

Whether the solutions of the wave equations derived from Maxwell's equations can be satisfied by substituting them back into the Maxwell's equations is a question asked in the study of electromagnetic fields and waves course. This paper gives the process of deriving the wave equations of electric and magnetic fields from Maxwell's equations without sources in a free space, and then, by the definition of uniform plane wave, the general solutions of wave equations are obtained. And it is proved that these solutions satisfy the Maxwell's divergence equations. Therefore, due to the presence of undetermined constants relative to the source, it is impossible to determine whether they satisfy the Maxwell's curl equations. The exercise to verify whether a vector function satisfies wave equation and Maxwell's equation does not mean that the solution of wave equation does not necessarily satisfy the Maxwell's equation. A description in the textbook that the field satisfying wave equation does not satisfy the Maxwell's equation refers to the situation where the unique solution of the wave equation is not obtained. The purpose of the method to solve a field from wave equation first and then to solve another field from Maxwell's equation is to obtain more information when the source does not exist, and to avoid complex source term to reduce the difficulty of solving the problem when the source exists. It can be proved that the unique solutions of the wave equations of the electric field and magnetic field generated by an electric dipole satisfy all the equations in the Maxwell equations. According to the uniqueness theorem of electromagnetic field and determinacy of space-time distribution of macroscopic electromagnetic field, whether the solving space with the source or without the source, as long as the unique solution of the wave equation can be obtained, these solutions must satisfy Maxwell's equations.