User:Kmarinas86/Electromagnetics

Conventional Electromagnetics
Tenet: The fundamental fields are the E and B fields


 * {| class="wikitable" style="text-align: center;"

! scope="col" | Formulation ! scope="col" style="width: 15em;" | Name ! scope="col" | "Microscopic" equations ! scope="col" | "Macroscopic" equations ! scope="row" rowspan="4" | Integral ! scope="row" | Gauss's law ! scope="row" | Gauss's law for magnetism ! scope="row" | Maxwell–Faraday equation (Faraday's law of induction) ! scope="row" | Ampère's circuital law (with Maxwell's correction) ! scope="row" rowspan="4" | Differential ! scope="row" | Gauss's law ! scope="row" | Gauss's law for magnetism ! scope="row" | Maxwell–Faraday equation (Faraday's law of induction) ! scope="row" | Ampère's circuital law (with Maxwell's correction)
 * + Formulations
 * Same as microscopic
 * Same as microscopic
 * $$\oint_{\partial \Sigma} \mathbf{E} \cdot \mathrm{d}\boldsymbol{\ell} = - \frac{d}{dt} \iint_{\Sigma} \mathbf B \cdot \mathrm{d}\mathbf{S} $$
 * Same as microscopic
 * $$\oint_{\partial \Sigma} \mathbf{B} \cdot \mathrm{d}\boldsymbol{\ell} = \mu_0 I + \mu_0 \varepsilon_0 \iint_{\Sigma} \frac{\partial \mathbf E}{\partial t} \cdot \mathrm{d}\mathbf{S}$$
 * $$\oint_{\partial \Sigma} \mathbf{H} \cdot \mathrm{d}\boldsymbol{\ell} = I_f + \iint_{\Sigma} \frac{\partial \mathbf D}{\partial t} \cdot \mathrm{d}\mathbf{S} $$
 * $$\nabla \cdot \mathbf{E} = \frac {\rho} {\varepsilon_0}$$
 * $$\nabla \cdot \mathbf{D} = \rho_f$$
 * $$\nabla \cdot \mathbf{B} = 0$$
 * Same as microscopic
 * $$\nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}} {\partial t}$$
 * Same as microscopic
 * $$\nabla \times \mathbf{B} = \mu_0\mathbf{J} + \mu_0 \varepsilon_0 \frac{\partial \mathbf{E}} {\partial t}\ $$
 * $$\nabla \times \mathbf{H} = \mathbf{J}_f + \frac{\partial \mathbf{D}} {\partial t}$$
 * }

K. Marinas' Reformed Electromagnetics
Tenet: The fundamental fields are the D and B fields

Second Correction to Ampere's Circuital Law
Ampere's Circuital Law without Maxwell's Correction:


 * $$\nabla \times \mathbf{B} = \mu_0 \mathbf{J}_{net}$$

Ampere's Circuital Law with Maxwell's Correction:


 * $$\nabla \times \mathbf{B} = \mu_0 \left (\mathbf{J}_{net} + \frac{\partial \mathbf{D}} {\partial t} \right)$$

Experimental evidence suggests that Maxwell's Correction to Ampere's Circuital Law is not sufficient when the electric field of one plate does not fully "connect" to the other plate. The time derivative of the electric field actually consists of two terms.


 * $$ \frac{\partial \mathbf{E}} {\partial t} = - \nabla\frac{\partial \mathbf{\varphi}} {\partial t} - \varepsilon_0\frac {\partial^2\mathbf{A}}{\partial t^2} \ ,$$

However, the magnetic field can be seen as due to the relativistic correction of the electric scalar potential in the frame where charge is moving. In that case, the first term on the right, based on the rate change of the gradient of the electric scalar potential contributes to the magnetic field, while the term based on the second derivative of the magnetic vector potential does not.

In that case, the correct formula for the curl of the magnetic field is:

$$\nabla \times \mathbf{B} = \mu_0 \left ( \mathbf{J}_{net} + \frac{\partial \mathbf{P}} {\partial t} - \nabla\frac{\partial \mathbf{\varphi}} {\partial t} \right)$$

Or equivalently:

$$\nabla \times \mathbf{B} = \mu_0 \left ( \mathbf{J}_{net}+ \frac{\partial \mathbf{D}} {\partial t} + \varepsilon_0\frac {\partial^2\mathbf{A}}{\partial t^2} \right)$$

A Second Alternative to the Second Correction to Ampere's Circuital Law
The magnetic field is already the curl magnetic vector potential $$\mathbf{A}$$, so for it to be dependent on the second time derivative of $$\mathbf{A}$$ is to say that the curl of the curl of the vector potential is dependent on its second time derivative. In cases where there is no scalar potential, it would imply that:


 * $$\nabla \times \nabla \times \mathbf{A} = - \mu_0 \varepsilon_0 \frac {\partial^2\mathbf{A}}{\partial t^2} = - \frac{1}{c^2} \frac {\partial^2\mathbf{A}}{\partial t^2}$$

This would mean a cylindrical magnetic field due to line current (with its bundle of parallel vector potential field vectors) would have an amendment to it that would depend on the changing magnitude of vector potential. As the vector potential of a moving charge depends in proportion to its velocity, the second derivative of the vector potential of such charge depends on its jerk. However, if current doesn't increase quadratically (or faster) with time, then such jerk cannot be sustained for very long. Ensuring this term to be non-zero for most of the time therefore inevitably involves an oscillation of back and forth changes in acceleration.

If the frequency of this oscillation is significantly higher than the frequency of the current in the surrounding plates, then the effect of magnetic induction of this changing field into an inductive pick-up coil should be significantly less, to the point of being undetectable by all inductive pick-up coils, save for those with a very high natural frequency. One way for the oscillation frequency to be much higher would be for it to be dependent on the acceleration or deceleration of individual source charges as they stochastically vibrate in and out of the capacitor plates, rather than the bulk behavior of the source alternating current into and out of the plates that the change of the scalar potential gradient depends on.

Since the discoherent oscillations of the charges are the primary contributors to the above equation (especially for a slowly-changing current), coupled with the fact such vibrations occur at many orders of magnitude smaller wavelength than the size of your typical capacitor, their contributions to the magnetics of, say, a 60hz electric motor, are essentially irrelevant (excluding heating effects).

Selected articles on Electromagnetics from Knowino.org
Knowino is a volunteer wiki-based project dedicated to creating a free general compendium. It is built on two main principles: anyone can contribute, and experts are encouraged to review articles for factual accuracy.


 * Electric Displacement: http://knowino.org/wiki/Electric_displacement

In the special case of a parallel-plate capacitor, often used to study and exemplify problems in electrostatics, the electric displacement D has an interesting interpretation. In that case D (the magnitude of vector D) is equal to the true surface charge density &sigma;true   (the surface density on the plates of the  right-hand capacitor in the figure).The nomenclature of the several surface charge distributions is not standardized. Here we will follow by and large R. Kronig, Textbook of physics, Pergamon Press London, New York (1959). (English translation from the Dutch Leerboek der Natuurkunde)


 * Displacement current: http://knowino.org/wiki/Displacement_current
 * Ampere's equation: http://knowino.org/wiki/Ampere%27s_equation

Knowino is an online Wiki encyclopedia with contributions from volunteers. Most of the following articles are modified versions of articles written earlier for Citizendium and some of the articles contain contributions from other authors. If authorship is of importance, the history of the corresponding Citizendium article may be consulted. - ''http://www.theochem.ru.nl/~pwormer/Wiki_articles.php ''

Selected articles on Electromagnetics from Citizendium.org
Citizendium, a wiki for providing free knowledge where authors use their real, verified names. We welcome anyone who wants to share their knowledge by writing and improving articles on virtually any subject. Expert authors can be recognized with a special role, but membership is open to all.


 * Electric Displacement: http://en.citizendium.org/wiki/Electric_displacement
 * Displacement current: http://en.citizendium.org/wiki/Displacement_current
 * Ampere's equation: http://en.citizendium.org/wiki/Ampere%27s_equation