电磁场波动方程的唯一解必然满足Maxwell方程  

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

作　　者：闫述[1] YAN Shu(School of Computer Science and Communication Engineering,Jiangsu University,Zhenjiang,Jiangsu 212013)

机构地区：[1]江苏大学计算机科学与通信工程学院,江苏镇江212013

出　　处：《物理与工程》2024年第4期11-16,共6页Physics and Engineering

基　　金：国家自然科学基金(41374129)。

摘　　要：从Maxwell方程组导出的波动方程的解,代回Maxwell方程中去是否还能得到满足,是电磁场与电磁波课程学习中有时会遇到的问题。本文给出了由自由空间的无源Maxwell方程导出电场和磁场波动方程的过程,然后根据均匀平面波的定义,求得波动方程的通解。证明这些解满足Maxwell散度方程,但由于含有与源相关的待定常数,故无法判定它们是否满足Maxwell旋度方程。验证某个矢量函数是否满足波动方程和Maxwell方程的习题,并不具有波动方程的解不一定满足Maxwell方程的含义。教材中关于满足波动方程的场量不一定满足Maxwell方程的阐述,指的是未获得波动方程唯一性解的情形;所提出的由波动方程求出一个场量后,再由Maxwell方程求出另一场量的解题方法,目的是当无源时可获得更多的信息,有源时避免复杂源项降低求解难度。可以证明,电偶极子电场和磁场波动方程的唯一解,满足Maxwell方程组中的全部方程。根据电磁场的唯一性定理和宏观电磁场时空分布的确定性,无论求解空间有源还是无源,只要能够获得波动方程的唯一解,这些解必然满足Maxwell方程。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.

关 键 词：电磁场波动方程 MAXWELL方程 解 矢量函数 满足 唯一解 唯一性定理 

分 类 号：O441[理学—电磁学] O175[理学—物理]