# Ampere's force and Faraday's law

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Corresponding Wikipedia articles: Ampère's force law, Faraday's law of induction

The electric current $$J$$ is a value of an electric charge, which flows through the conductor cross section per one unit of time. An element of the current in the path length $$\mathrm{d}\mathbf{\overrightarrow{l}}$$ can be determined as: $\mathrm{d}\mathbf{\overrightarrow{J}}=J\mathrm{d}\mathbf{\overrightarrow{l}}=\frac{\mathrm{d}Q}{\mathrm{d}t}\mathrm{d}\mathbf{\overrightarrow{l}}=\mathbf{\overrightarrow{v}}\mathrm{d}Q\tag{1}$ The Ampere’s force for a current element is the Lorentz force for it: $\mathrm{d}\mathbf{\overrightarrow{F}}=\mathrm{d}\mathbf{\overrightarrow{J}}\times\mathbf{\overrightarrow{B}}\tag{2 – SI, Sim.}$ $\mathrm{d}\mathbf{\overrightarrow{F}}=\frac{\mathrm{d}\mathbf{\overrightarrow{J}}}{c}\times\mathbf{\overrightarrow{B}}\tag{2 – CGS}$ Ampere believed that this is a fundamental interaction force of the currents themselves rather than the moving particles. In accordance with (2), the orthogonal conductors interaction violates the Newton’s third law, requiring the so-called longitudinal Ampere’s force. The misconception also exists that the orthogonal conductors should not interact. However, this interaction is observed in many experiments [1] (see also "Railgun").

Marinov Motor

Stefan Marinov, who looked at the experiment #21 [1] of prof. G.V. Nikolaev from Tomsk (Siberia), invented the motor called "Siberian Coliu", supposedly using Nikolaev’s "scalar" magnetic field. Actually, the parameters of this motor are calculated using a conventional vector potential [2]. The operation of this motor can also be explained by a magnetic force (Magnetism, 8). The curl of a permanent magnet is directed across the currents. These currents produce a field and the magnetic force. The field strengths within the ring is higher than outside, setting the direction of a net force, which produces a torque.

The Tomilin’s generator [3] uses an inverse effect.

The Chernikov’s effect [4]. If the current-carrying conductor is enclosed by the cylindrical magnetic shield, the effect of the magnetic field on the conductor almost disappears and the force is applied to the de-energized shield. This effect is explained by the fact that all material particles (electrons, etc.) are the fields, which are propagated over the unlimited volume. Therefore, in this case, the Lorentz/Ampere force occurs in the magnetized shield under the influence of the moving electrons within the conductor.

The Ampere’s force does not depend on the conductor velocity with respect to the magnetic field source, because this is the net force of the opposite charges within the conductor.

The Lorentz force in a conductor, which is moving in a magnetic field, causes the electromotive force by the electromagnetic induction, as when the magnetic field is varying. The Faraday's law of electromagnetic induction follows from the general definition ("Electricity", 3) of the electric field by the magnetic potential: $\mathbf{\overrightarrow{E}}=-\frac{\partial\mathbf{\overrightarrow{A}}}{\partial t}\tag{3 – SI, Sim.}$ $\mathbf{\overrightarrow{E}}=-\frac{1}{c}\frac{\partial\mathbf{\overrightarrow{A}}}{\partial t}\tag{3 – CGS}$ The Maxwell’s differential form of this law is produced by applying the curl operator: $curl\mathbf{\overrightarrow{E}}=-\frac{\partial\mathbf{\overrightarrow{B}}}{\partial t}\tag{4 – SI, Sim.}$ $curl\mathbf{\overrightarrow{E}}=-\frac{1}{c}\frac{\partial\mathbf{\overrightarrow{B}}}{\partial t}\tag{4 – CGS}$ The integral form is produced by applying to (3) the circulation operator, and also the following assumption for an uniform field: $\oint{\frac{\partial\mathbf{\overrightarrow{A}}}{\partial t}\mathrm{d}\mathbf{\overrightarrow{l}}}=\frac{\partial}{\partial t}\oint{\mathbf{\overrightarrow{A}}\mathrm{d}\mathbf{\overrightarrow{l}}}\tag{5}$ Also, according to the Stokes' theorem: $\oint{\mathbf{\overrightarrow{A}}\mathrm{d}\mathbf{\overrightarrow{l}}}=\int{(curl\mathbf{\overrightarrow{A}})\mathrm{d}\mathbf{\overrightarrow{S}}}=\int{\mathbf{\overrightarrow{B}}\mathrm{d}\mathbf{\overrightarrow{S}}}\tag{6}$ So, the integral form of Faraday’s law for an uniform field is: $\oint{\mathbf{\overrightarrow{E}}\mathrm{d}\mathbf{\overrightarrow{l}}}=-\frac{\partial}{\partial t}\int{\mathbf{\overrightarrow{B}}\mathrm{d}\mathbf{\overrightarrow{S}}}\tag{7 – SI, Sim.}$ $\oint{\mathbf{\overrightarrow{E}}\mathrm{d}\mathbf{\overrightarrow{l}}}=-\frac{1}{c}\frac{\partial}{\partial t}\int{\mathbf{\overrightarrow{B}}\mathrm{d}\mathbf{\overrightarrow{S}}}\tag{7 – CGS}$

Hering’s paradox
Hering’s paradox

For an inhomogeneous field, as noticed Atsukovsky[5], the integral form of Faraday's law is not valid.

The Hering’s paradox is explained in the same way. The toroidal permanent magnet contains almost whole its magnetic field inside itself. The field inside the circuit is not uniform and even does not cross the wires, so the induction EMF is not detected.

The vortical field by the Faraday's law does not sum with the Lorentz force, because they are the forces of the same nature.

## References

1. Г.B. Николаев, Экспериментальные парадоксы электродинамики // Современная электродинамика и причины ее парадоксальности. Перспективы построения непротиворечивой электродинамики. Теории, эксперименты, парадоксы, Томск, 2003 — ISBN 5-88839-045-3
2. J. P. Wesley, The Marinov Motor, Motional Induction without a Magnetic B Field.
3. Томилин А.К. Обобщенная электродинамика. – Усть-Каменогорск: ВКГТУ, 2009. – 166 с.: ил. — ISBN 978-601-208-100-8
4. Черников В. Как я встретился с нечистой силой // Техника молодежи. — 1974. — № 1. — С. 37.
5. В.А. Ацюковский. 12 экспериментов по эфиродинамике. Г. Жуковский: изд-во «Петит», 2003, 46 с. — ISBN 5-85101-065-7