# Magnetic effect of current

Corresponding Wikipedia articles: Ampère's circuital law, Biot–Savart law, Inductance

## Introduction

A charge carrier, which moves with respect to other bodies, interacts with them even if the average electric and magnetic fields are zero, because the magnetic field actually exists wherever there is the matter. This interaction is defined by the equation ("Aetheric vortex", 6): $\mathbf{\overrightarrow{j}}=q\mathbf{\overrightarrow{v}}=curl\mathbf{\overrightarrow{H}}\tag{1 - SI}$ $\mathbf{\overrightarrow{j}}=q\mathbf{\overrightarrow{v}}=\frac{c}{4\pi}curl\mathbf{\overrightarrow{H}}\tag{1 - CGS}$ $\mathbf{\overrightarrow{j}}=q\mathbf{\overrightarrow{v}}=c^2curl\mathbf{\overrightarrow{H}}\tag{1 - Sim.}$ where $$\mathbf{\overrightarrow{j}}$$ is the current density;
$$\mathbf{\overrightarrow{v}}$$ is a velocity of the charge carriers with respect to the body.

## Ampere's circuital law

The Ampere's circuital law follows directly from (1) by ignoring the limit in the curl definition, because the field is uniform: $\oint{\mathbf{\overrightarrow{H}}\mathrm{d}\mathbf{\overrightarrow{l}}}=\mathbf{\overrightarrow{S}}curl\mathbf{\overrightarrow{H}}=\mathbf{\overrightarrow{j}}\mathbf{\overrightarrow{S}}=J\tag{2 - SI}$ $\oint{\mathbf{\overrightarrow{H}}\mathrm{d}\mathbf{\overrightarrow{l}}}=\mathbf{\overrightarrow{S}}curl\mathbf{\overrightarrow{H}}=\frac{4\pi}{c}\mathbf{\overrightarrow{j}}\mathbf{\overrightarrow{S}}=\frac{4\pi}{c}J\tag{2 - CGS}$ $\oint{\mathbf{\overrightarrow{H}}\mathrm{d}\mathbf{\overrightarrow{l}}}=\mathbf{\overrightarrow{S}}curl\mathbf{\overrightarrow{H}}=\frac{1}{c^2}\mathbf{\overrightarrow{j}}\mathbf{\overrightarrow{S}}=\frac{1}{c^2}J\tag{2 - Sim.}$

## Biot–Savart law

The Biot-Savart law can be written in a form: $\mathbf{\overrightarrow{H}}=\mathbf{\overrightarrow{v}}\times\mathbf{\overrightarrow{D}}\tag{3 - SI}$ $\mathbf{\overrightarrow{H}}=4\pi\frac{\mathbf{\overrightarrow{v}}}{c}\times\mathbf{\overrightarrow{D}}\tag{3 - CGS}$ $\mathbf{\overrightarrow{H}}=\frac{1}{c^2}\mathbf{\overrightarrow{v}}\times\mathbf{\overrightarrow{D}}\tag{3 - Sim.}$ Since the field $$\mathbf{\overrightarrow{H}}$$ is vortical, the equality (3) can be replaced by the equality of curls, and the vector calculus identity can be applied to it: $curl\;\mathbf{\overrightarrow{H}}=curl(\mathbf{\overrightarrow{v}}\times\mathbf{\overrightarrow{D}})=(div\;\mathbf{\overrightarrow{D}}+\mathbf{\overrightarrow{D}}\nabla)\mathbf{\overrightarrow{v}}-(div\mathbf{\overrightarrow{v}}+\mathbf{\overrightarrow{v}}\nabla)\mathbf{\overrightarrow{D}}\tag{4 - SI}$ The vector $$\mathbf{\overrightarrow{D}}$$ is perpendicular to the vector $$\mathbf{\overrightarrow{v}}$$ according to the electric field definition, so: $\mathbf{\overrightarrow{v}}\cdot\mathbf{\overrightarrow{D}}=(\mathbf{\overrightarrow{v}}\cdot\nabla)\mathbf{\overrightarrow{D}}=(\mathbf{\overrightarrow{D}}\cdot\nabla)\mathbf{\overrightarrow{v}}=0\tag{5}$ Since also the velocity field consists of the vortical and/or uniform components ($$div\mathbf{\overrightarrow{v}}=0$$), and $$div\;\mathbf{\overrightarrow{D}}=q$$, the equation (4) becomes (1) and proves (3).

The electrostatic field, which is produced by an infinitesimally small charge $$\mathrm{d}Q$$ at distance $$\mathbf{\overrightarrow{r}}$$ from the observation point, is: $\mathrm{d}\mathbf{\overrightarrow{D}}=-\frac{\mathrm{d}Q}{4\pi r^2}\frac{\mathbf{\overrightarrow{r}}}{|\mathbf{\overrightarrow{r}}|}\tag{6 - SI, Sim.}$ $\mathrm{d}\mathbf{\overrightarrow{D}}=-\frac{\mathrm{d}Q}{r^2}\frac{\mathbf{\overrightarrow{r}}}{|\mathbf{\overrightarrow{r}}|}\tag{6 - CGS}$ The Biot-Savart law in its usual vector form by substituting (6) into (3): $\mathrm{d}\mathbf{\overrightarrow{B}}=\mu_0\mu\mathrm{d}\mathbf{\overrightarrow{H}}=\mu_0\mu\mathbf{\overrightarrow{v}}\times\mathrm{d}\mathbf{\overrightarrow{D}}=-\mu_0\mu\frac{\mathrm{d}Q}{4\pi r^2}\frac{\mathbf{\overrightarrow{v}}\times\mathbf{\overrightarrow{r}}}{|\mathbf{\overrightarrow{r}}|}=\frac{\mu_0\mu}{4\pi r^2}\frac{\mathbf{\overrightarrow{r}}\times\mathrm{d}\mathbf{\overrightarrow{J}}}{|\mathbf{\overrightarrow{r}}|}\tag{7 - SI}$ $\mathrm{d}\mathbf{\overrightarrow{B}}=\mu\mathrm{d}\mathbf{\overrightarrow{H}}=4\pi\mu\frac{\mathbf{\overrightarrow{v}}}{c}\times\mathrm{d}\mathbf{\overrightarrow{D}}=-\mu\frac{\mathrm{d}Q}{cr^2}\frac{\mathbf{\overrightarrow{v}}\times\mathbf{\overrightarrow{r}}}{|\mathbf{\overrightarrow{r}}|}=\frac{\mu}{cr^2}\frac{\mathbf{\overrightarrow{r}}\times\mathrm{d}\mathbf{\overrightarrow{J}}}{|\mathbf{\overrightarrow{r}}|}\tag{7 - CGS}$ $\mathrm{d}\mathbf{\overrightarrow{B}}=\mu\mathrm{d}\mathbf{\overrightarrow{H}}=\frac{\mu}{c^2}\mathbf{\overrightarrow{v}}\times\mathrm{d}\mathbf{\overrightarrow{D}}=-\mu\frac{\mathrm{d}Q}{4\pi c^2r^2}\frac{\mathbf{\overrightarrow{v}}\times\mathbf{\overrightarrow{r}}}{|\mathbf{\overrightarrow{r}}|}=\frac{\mu}{4\pi c^2r^2}\frac{\mathbf{\overrightarrow{r}}\times\mathrm{d}\mathbf{\overrightarrow{J}}}{|\mathbf{\overrightarrow{r}}|}\tag{7 - Sim.}$ The scalar form of this law for an angle $$\alpha$$ between a direction of the current and the direction from a conductor point to a point with given magnetic field: $\mathrm{d}B=\frac{\mu_0\mu}{4\pi}\frac{J\mathrm{d}l}{r^2}\sin\alpha\tag{8 - SI}$ $\mathrm{d}B=\frac{\mu}{c}\frac{J\mathrm{d}l}{r^2}\sin\alpha\tag{8 - CGS}$ $\mathrm{d}B=\frac{\mu}{4\pi с^2}\frac{J\mathrm{d}l}{r^2}\sin\alpha\tag{8 - Sim.}$

## Inductance

The inductance is a quantity that links the magnetic flux (flux linkage) with a current, which produce it. The inductance depends on the conductor geometry and the environmental permeability: $LJ=\int{\mathbf{\overrightarrow{B}}\mathrm{d}\mathbf{\overrightarrow{S}}}\tag{9 - SI}$ $LJ=c\int{\mathbf{\overrightarrow{B}}\mathrm{d}\mathbf{\overrightarrow{S}}}\tag{9 - CGS}$ $LJ=c^2\int{\mathbf{\overrightarrow{B}}\mathrm{d}\mathbf{\overrightarrow{S}}}\tag{9 - Sim.}$ The magnetic energy of the conductor is determined by its volumetric density within 3-dimensional space: $W=\frac{1}{2}\int{\mathbf{\overrightarrow{H}}\mathbf{\overrightarrow{B}}\mathrm{d}V}=\frac{1}{2}\int{\oint{\mathbf{\overrightarrow{H}}\mathbf{\overrightarrow{B}}\mathrm{d}\mathbf{\overrightarrow{S}}}\mathrm{d}\mathbf{\overrightarrow{l}}}\tag{10 - SI}$ $W=\frac{1}{8\pi}\int{\mathbf{\overrightarrow{H}}\mathbf{\overrightarrow{B}}\mathrm{d}V}=\frac{1}{8\pi}\int{\oint{\mathbf{\overrightarrow{H}}\mathbf{\overrightarrow{B}}\mathrm{d}\mathbf{\overrightarrow{S}}}\mathrm{d}\mathbf{\overrightarrow{l}}}\tag{10 - CGS}$ $W=\frac{c^2}{2}\int{\mathbf{\overrightarrow{H}}\mathbf{\overrightarrow{B}}\mathrm{d}V}=\frac{c^2}{2}\int{\oint{\mathbf{\overrightarrow{H}}\mathbf{\overrightarrow{B}}\mathrm{d}\mathbf{\overrightarrow{S}}}\mathrm{d}\mathbf{\overrightarrow{l}}}\tag{10 - Sim.}$ This integral (10) contains a constant value of the magnetic field strength circulation, so: $W=\frac{LJ^2}{2}\tag{11 - SI}$ $W=\frac{LJ^2}{2c^2}\tag{11 - CGS, Sim.}$ The Faraday's law can be written using the inductance: $\oint{\mathbf{\overrightarrow{E}}\mathrm{d}\mathbf{\overrightarrow{l}}}=-\frac{\partial(LJ)}{\partial t}\tag{12 - SI}$ $c^2\oint{\mathbf{\overrightarrow{E}}\mathrm{d}\mathbf{\overrightarrow{l}}}=-\frac{\partial(LJ)}{\partial t}\tag{12 - CGS, Sim.}$