Difference between revisions of "Electricity"

Revision as of 09:35, 28 May 2017

Corresponding Wikipedia article: Electric field

Definition

The electric field is not a basic parameter of the aether state, but this is a force field, which exists relatively to the affected objects.

The electric force is separated from the magnetic force ("Magnetism", 6) by separating its fraction from the total magnetic field: \[\mathbf{\overrightarrow{B}}=curl\;k\mathbf{\overrightarrow{v}}\;\;\;\;\;\;k=const\] Under this condition, the vectors \(\mathbf{\overrightarrow{B}}\) and \((curl\mathbf{\overrightarrow{B}})\), are collinear according to a property ("Magnetism", 1), so the magnetic force is zero. The value and sign of electric charge depends on the \(k\) value. In order to determine the \(k\) and force (energy) values, the mass density\(\rho\) should be included into the equation. The equations, which are consistent with the known electromagnetism laws, are the following:

\(\mathbf{\overrightarrow{A}}\) is a magnetic potential vector;
\(q\) is a volumetric density of electric charge;
\(\varepsilon\) is a relative permittivity of the medium (\(\varepsilon=1\) for the vacuum);
\(c\) is a speed of light in vacuum.

The magnetic potential is the only source of electric field: \[\rho\mathbf{\overrightarrow{f}}=-q\frac{\mathrm{d}\mathbf{\overrightarrow{A}}}{\mathrm{d}t}\tag{3 - SI}\] \[\rho\mathbf{\overrightarrow{f}}=-\frac{q}{c}\frac{\mathrm{d}\mathbf{\overrightarrow{A}}}{\mathrm{d}t}\tag{3 - CGS}\] \[\varepsilon\rho\mathbf{\overrightarrow{f}}=-q\frac{\mathrm{d}\mathbf{\overrightarrow{A}}}{\mathrm{d}t}\tag{3 - Sim.}\] The magnetic field derivative with respect to time is decomposed as a material derivative but with an additional term: \[\frac{\mathrm{d}\mathbf{\overrightarrow{B}}}{\mathrm{d}t}=\frac{\partial\mathbf{\overrightarrow{B}}}{\partial t}+(\mathbf{\overrightarrow{v}}\cdot\nabla)\mathbf{\overrightarrow{B}}-(\mathbf{\overrightarrow{B}}\cdot\nabla)\mathbf{\overrightarrow{v}}\tag{4}\] The term \(-(\mathbf{\overrightarrow{B}}\cdot\nabla)\mathbf{\overrightarrow{v}}\) expresses that deceleration of a beam compresses it (increases \(\mathbf{\overrightarrow{B}}\)), and acceleration rarefies it (decreases \(\mathbf{\overrightarrow{B}}\)).

The electric field is divided into two components with different causes of the forces:

Electrostatic field

The formula for the electrostatic field, which is produced by the moving charge \(q\) with respect to the source of a constant (in time) magnetic field \(\mathbf{\overrightarrow{B}}\) with a velocity \(\mathbf{\overrightarrow{v}}\) is: \[\rho\mathbf{\overrightarrow{f}}=q\mathbf{\overrightarrow{v}}\times\mathbf{\overrightarrow{B}}\tag{5 - SI}\] \[\rho\mathbf{\overrightarrow{f}}=q\frac{\mathbf{\overrightarrow{v}}}{c}\times\mathbf{\overrightarrow{B}}\tag{5 - CGS}\] \[\varepsilon\rho\mathbf{\overrightarrow{f}}=q\mathbf{\overrightarrow{v}}\times\mathbf{\overrightarrow{B}}\tag{5 - Sim.}\] The proof of (5) is reduced to the proof of an equation: \[\mathbf{\overrightarrow{v}}\times\mathbf{\overrightarrow{B}}=-\frac{\mathrm{d}\mathbf{\overrightarrow{A}}}{\mathrm{d}t}\tag{6}\] Applying of the curl operator to both sides of (6) gives: \[curl(\mathbf{\overrightarrow{v}}\times\mathbf{\overrightarrow{B}})=-curl\frac{\mathrm{d}\mathbf{\overrightarrow{A}}}{\mathrm{d}t}=-\frac{\mathrm{d}\mathbf{\overrightarrow{B}}}{\mathrm{d}t}=-(\mathbf{\overrightarrow{v}}\cdot\nabla)\mathbf{\overrightarrow{B}}+(\mathbf{\overrightarrow{B}}\cdot\nabla)\mathbf{\overrightarrow{v}}\tag{7}\] Also, according to the known identity, which is applicable to the 3-dimensional space: \[curl(\mathbf{\overrightarrow{v}}\times\mathbf{\overrightarrow{B}})=(div\mathbf{\overrightarrow{B}}+\mathbf{\overrightarrow{B}}\cdot\nabla)\mathbf{\overrightarrow{v}}-(div\mathbf{\overrightarrow{v}}+\mathbf{\overrightarrow{v}}\cdot\nabla)\mathbf{\overrightarrow{B}}\tag{8}\] Since the magnetic field is vortical (\(div\mathbf{\overrightarrow{B}}=0\)), and the velocity field consists of the vortical and/or uniform components (\(div\mathbf{\overrightarrow{v}}=0\)), the formula (5) is proven.