# Abstract

The force law of Maxwell’s classical electrodynamics does not agree with
Newton’s third law of motion (N3LM), in case of open circuit magnetostatics. Initially, a generalized magnetostatics theory is presented that includes two additional physical fields B_Φ and B_l, defined by scalar functions. The scalar magnetic field B_l mediates a longitudinal Ampère force that balances the transverse Ampère force (aka the magnetic field force), such that the sum of the two forces agrees with N3LM for all stationary current distributions. Secondary field induction laws are derived; a secondary curl free electric field E_l is induced by a time varying scalar magnetic field B_l, which isn’t described by Maxwell’s electrodynamics.
The Helmholtz’ decomposition is applied to exclude E_l from the total
electric field E, resulting into a more simple Maxwell theory. Decoupled
inhomogeneous potential equations and its solutions follow directly from
this theory, without having to apply a gauge condition. Field expressions
are derived from the potential functions that are simpler and far field consistent with respect to the Jefimenko fields. However, our simple version of Maxwell’s theory does not satisfy N3LM. Therefore we combine the
generalized magnetostatics with the simple version of Maxwell’s electrodynamics, via the generalization of Maxwell’s speculative displacement current. The resulting electrodynamics describes three types of vacuum waves: the Φ wave, the longitudinal electromagnetic (LEM) wave and the transverse electromagnetic (TEM) wave, with phase velocities respectively a, b and c. Power- and force theorems are derived, and the force law agrees with Newton’s third law only if the phase velocities satisfy the following condition: a >> b and b = c. The retarded potential functions can be found without gauge conditions, and four retarded field expressions are derived that have three near field terms and six far field terms. All six far field terms are explained as the mutual induction of two free fields. Our theory supports Rutherford’s solution of the 4/3 problem of electromagnetic mass, which requires an extra longitudinal electromagnetic momentum. Our generalized classical electrodynamics might spawn new physics experiments and electrical engineering, such as new photoelectric effects based on Φ- or LEM radiation, and the conversion of natural Φ- or LEM radiation into useful electricity, in the footsteps of dr. N. Tesla and dr. T.H. Moray.