On the time‐dependent, generalized Coulomb and Biot–Savart laws

This paper seeks to promote a wider use of the two little‐known time‐dependent, generalized Coulomb and Biot–Savart laws in the form given by Jefimenko’s book. For some problems, these two time‐dependent laws giving the electromagnetic field from known distributions of charge and current densities without spatial differentiations provide a more convenient method of solution, but they are not well known, probably due to their difficult derivation. This paper gives an alternative derivation to these formulas using a novel ‘‘light cone transformation’’ that may be more appealing to some readers than the derivation given in the book by Jefimenko. The paper then shows that together with the law of conservation of charge these two laws can give back Maxwell’s system of four equations when the medium is infinite, homogeneous, linear, and isotropic; consequently, the time‐dependent, generalized Coulomb, Biot–Savart, and charge conservation laws can be used for such a case instead of Maxwell’s four equations. Three elegant examples, which have been traditionally solved in lengthy ways, or using the postulates of the special relativity theory, are then solved using these formulas: They are the examples of a differential antenna, of an electron in a uniform motion, and in an arbitrary motion (Feynman’s formulas). The examples also highlight the necessity of using Lienard–Wiechert’s formula in typical applications of these time‐dependent laws.