arXiv physics/05060542005-06-22physics.class-ph

回溯之路：从电荷守恒到麦克斯韦方程组

The way back: from charge conservation to Maxwell equations

F. De Zela

备注：: 15 pages, added references

摘要：

这篇文章的主要目的是在广大物理学家中传播一个已知的结果，这是几年来流形上微分形式的\emph{cognoscenti}的一个结果;即，电荷守恒意味着非齐次麦克斯韦方程组。这是物理学家们非常熟悉的一个命题的倒数：电荷守恒，以连续性方程的形式表示，作为麦克斯韦方程组的结果。讨论了电荷守恒蕴含麦克斯韦方程组的条件。在此强调了本构方程所起的关键作用。讨论的基础是亥姆霍兹定理，根据该定理，向量场由其散度和旋度决定。绿色的功能也显示出发挥了根本性的作用。我们提出了所有的结果，在三个矢量，以及在张量符号。我们只使用大多数物理学家熟悉的数学工具。

英文摘要：

The main purpose of this article is to disseminate among a wide audience of physicists a known result, which is available since a couple of years to the \emph{cognoscenti} of differential forms on manifolds; namely, that charge conservation implies the inhomogeneous Maxwell equations. This is the reciprocal statement of one which is very well known among physicists: charge conservation, written in the form of a continuity equation, follows as a consequence of Maxwell equations. We discuss the conditions under which charge conservation implies Maxwell equations. The key role played by the constitutive equations is hereby stressed. The discussion is based on Helmholtz theorem, according to which a vector field is determined by its divergence and its curl. Green's functions are also shown to play a fundamental role. We present all results in three-vector, as well as in tensorial notation. We employ only those mathematical tools most physicists are familiar with.

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