Abstract
Classical electromagnetic theory uses axial pseudo-vectors to describe magnetic interactions. It is impossible to explain adequately magnetic interaction at the micro level (elements of conductors and the magnetic interactions of charges) by axial vectors. As a result, the correct form of interactions in classical electrodynamics is only an integral one. The differential formulas for magnetic interactions violate the third Newton’s law.
 In the paper, we use polar vectors (real physical vectors) to describe magnetic interactions. On this way, we show that the real physical magnetic field, in contrast to the solenoidal field of the axial vector magnetic induction B, has two components: a potential field with nonvanishing divergence and a solenoidal field with vanishing divergence. These two fields act separately and independently and have different models of interactions.
 Doing so, we can write differential form for the Ampere’s law obtaining correct formula for the magnetic interactions and adequate interpretation of the Biot-Savart law.
Highlights
Faraday, Ampere, Biot and other scientists started to study electromagnetic interactions nearly two hundred years ago
Almost two hundred years later, the electrodynamics cannot explain Ampere’s and Faraday’s experiments demonstrating axial magnetic interaction of current-carrying conductors: the Biot-Savart law controverts any probability of such interactions
Ampere’s formula conflicts with some experiments made by Ampere himself, namely, the action of a constant magnet field on a conductor when the magnetic force is not central: it is orthogonal to the conductor axis
Summary
Ampere, Biot and other scientists started to study electromagnetic interactions nearly two hundred years ago. In 1826, Ampere published his “Electrodynamics” [2] with a subtitle “Theory of electrodynamic phenomena deduced exclusively from experiments”. Ampere described his own experiments, cited Faraday experiments, and suggested the formula for conductors elements interaction following from these experiments. Recent ten years radio-amateurs use radio communication with the so called E-H aerials, that emit and receive radio waves in the axial direction of antenna conductors. The existence of this radio communication cannot be explained theoretically. We analyze the causes of these problems in electromagnetism and the correctness of the modern magnetic field theory
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