User:Kmarinas86/Electromagnetics

Conventional Electromagnetics

Tenet: The fundamental fields are the E and B fields

Formulations

Formulation	Name	"Microscopic" equations	"Macroscopic" equations
Integral	Gauss's law	${\scriptstyle \partial \Omega }$ $\mathbf {E} \cdot \mathrm {d} \mathbf {S} ={\frac {Q_{encl}(V)}{\varepsilon _{0}}}$	${\scriptstyle \partial \Omega }$ $\mathbf {D} \cdot \mathrm {d} \mathbf {S} =Q_{f,encl}(V)$
	Gauss's law for magnetism	${\scriptstyle \partial \Omega }$ $\mathbf {B} \cdot \mathrm {d} \mathbf {S} =0$	Same as microscopic
	Maxwell–Faraday equation (Faraday's law of induction)	$\oint _{\partial \Sigma }\mathbf {E} \cdot \mathrm {d} {\boldsymbol {\ell }}=-{\frac {d}{dt}}\iint _{\Sigma }\mathbf {B} \cdot \mathrm {d} \mathbf {S}$	Same as microscopic
	Ampère's circuital law (with Maxwell's correction)	$\oint _{\partial \Sigma }\mathbf {B} \cdot \mathrm {d} {\boldsymbol {\ell }}=\mu _{0}I+\mu _{0}\varepsilon _{0}\iint _{\Sigma }{\frac {\partial \mathbf {E} }{\partial t}}\cdot \mathrm {d} \mathbf {S}$	$\oint _{\partial \Sigma }\mathbf {H} \cdot \mathrm {d} {\boldsymbol {\ell }}=I_{f}+\iint _{\Sigma }{\frac {\partial \mathbf {D} }{\partial t}}\cdot \mathrm {d} \mathbf {S}$
Differential	Gauss's law	$\nabla \cdot \mathbf {E} ={\frac {\rho }{\varepsilon _{0}}}$	$\nabla \cdot \mathbf {D} =\rho _{f}$
	Gauss's law for magnetism	$\nabla \cdot \mathbf {B} =0$	Same as microscopic
	Maxwell–Faraday equation (Faraday's law of induction)	$\nabla \times \mathbf {E} =-{\frac {\partial \mathbf {B} }{\partial t}}$	Same as microscopic
	Ampère's circuital law (with Maxwell's correction)	$\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

Supplementary information

A highly symmetrized set of field equations can be derived by noting the following (From the article on Magnetic susceptibility): Sometimes(http://info.ee.surrey.ac.uk/Workshop/advice/coils/mu/#itns) an auxiliary quantity, called intensity of magnetization (also referred to as magnetic polarisation J) and measured in teslas, is defined as $\mathbf {I} =\mu _{0}\mathbf {M} \,$ . This allows an alternative description of all magnetization phenomena in terms of the quantities I and B, as opposed to the commonly used M and H.

Poynting vector In Poynting's original paper and in many textbooks, it is usually denoted by S or N, and defined as:[1][2] $\mathbf {S} =\mathbf {E} \times \mathbf {H} ,$ which is often called the Abraham form; where E is the electric field and H the magnetic field.[3][4] (All bold letters represent vectors.) Occasionally an alternative definition in terms of electric field E and the magnetic flux density B is used. It is even possible to combine the displacement field D with the magnetic flux density B to get the Minkowski form of the Poynting vector, or use D and H to construct another.[5] The choice has been controversial: Pfeifer et al.[6] summarize the century-long dispute between proponents of the Abraham and Minkowski forms.

The near field (or near-field) and far field (or far-field) and the transition zone are regions of time varying electromagnetic field around any object that serves as a source for the field. The different terms for these regions describe the way characteristics of an electromagnetic (EM) field change with distance from the charges and currents in the object that are the sources of the changing EM field. The more distant parts of the far-field are identified with classical electromagnetic radiation.

On Binding energy: Since all forms of energy exhibit rest mass within systems at "rest" (that is, in systems which have no net momentum), the question of where the missing mass of the binding energy goes, is of interest. The answer is that this mass is lost from a system which is not closed. It transforms to heat, light, higher energy states of the nucleus/atom or other forms of energy, but these types of energy also have mass, and it is necessary that they be removed from the system before its mass may decrease. The "mass deficit" from binding energy is therefore removed mass that corresponds with removed energy, according to Einstein's equation E = mc2. Mass change (decrease) in bound systems, particularly atomic nuclei, has also been termed mass defect, mass deficit, or mass packing fraction. Just as the formation of a bond may displace a certain amount of mass and energy from the newly bound system, such formation of a bond may also displace a certain amount charge from the newly bound system.

Descriptions Name Formulation Standard electrical formula $\mathbf {D} -\mathbf {P} =\varepsilon _{0}\mathbf {E}$ Standard magnetic formula $\mathbf {B} -\mathbf {I} =\mu _{0}\mathbf {H}$ Directional energy flux density (Poynting vector) Reformed Gauss's law Divergence coulombs / meter3 Surface integral coulombs Reformed Gauss's law for magnetism Divergence webers / meter3 Surface integral webers Reformed Maxwell–Faraday equation (Faraday's law of induction) Curl coulombs / meter3 Line integral coulombs / meter Reformed Ampère's circuital law (with Maxwell's correction) Curl webers / meter3 Line integral webers / meter

Formulas [More formulas] Jump Right Jump Left [Formulas] Net Field EM (cause) Minus $-$ Near-field EM (cause) Equals $=$ Far-field EM (cause) bounded and unbounded charge/current bounded charge/current unbounded charge/current $(\mu ,\epsilon )$ $(\mu _{0}\chi _{v},\epsilon _{0}\chi _{\text{e}})$ $(\mu _{0},\epsilon _{0})$ $\mathbf {D}$ $=$ Electric displacement field $\mathbf {P}$ $=$ Polarization field $\varepsilon _{0}\mathbf {E}$ $=$ Electric field $\mathbf {B}$ $=$ Magnetic field $\mathbf {I}$ $=$ Magnetization $\mu _{0}\mathbf {H}$ $=$ Magnetizing field $\mathbf {S} _{Minkowski}c^{2}$ $=$ $\left(\mathbf {D} \times \mathbf {B} \right)c^{2},$ $\mathbf {S} _{Abraham}$ $=$ $\mathbf {E} \times \mathbf {H} ,$ Below: The left-column expressions below are derived by using the vector calculus identities for distributive properties of the curl and divergence operations. $\nabla \cdot \mathbf {D}$ $=$ $\rho _{net}$ $\nabla \cdot \mathbf {P}$ $=$ $\rho _{defect}$ $\varepsilon _{0}\nabla \cdot \mathbf {E}$ $=$ $\rho _{free}$ ${\scriptstyle \partial \Omega }$ $\mathbf {D} \cdot \mathrm {d} \mathbf {S}$ $=$ $Q_{net,encl}(V)$ ${\scriptstyle \partial \Omega }$ $\mathbf {P} \cdot \mathrm {d} \mathbf {S}$ $=$ $Q_{defect,encl}(V)$ ${\scriptstyle \partial \Omega }$ $\varepsilon _{0}\mathbf {E} \cdot \mathrm {d} \mathbf {S}$ $=$ $Q_{free,encl}(V)$ $\nabla \cdot \mathbf {B}$ $=$ $0$ $\nabla \cdot \mathbf {I}$ $=$ $-\mu _{0}\nabla \cdot \mathbf {H}$ $\mu _{0}\nabla \cdot \mathbf {H}$ $=$ $-\nabla \cdot \mathbf {I}$ ${\scriptstyle \partial \Omega }$ $\mathbf {B} \cdot \mathrm {d} \mathbf {S}$ $=$ $0$ ${\scriptstyle \partial \Omega }$ $\mathbf {I} \cdot \mathrm {d} \mathbf {S}$ $=$ $-$ ${\scriptstyle \partial \Omega }$ $\mu _{0}\mathbf {H} \cdot \mathrm {d} \mathbf {S}$ ${\scriptstyle \partial \Omega }$ $\mu _{0}\mathbf {H} \cdot \mathrm {d} \mathbf {S}$ $=$ $-$ ${\scriptstyle \partial \Omega }$ $\mathbf {I} \cdot \mathrm {d} \mathbf {S}$ $\nabla \times \mathbf {D}$ $=$ $-\varepsilon _{0}{\frac {\partial \mathbf {B} }{\partial t}}$ $\nabla \times \mathbf {P}$ $=$ $-\varepsilon _{0}{\frac {\partial \mathbf {I} }{\partial t}}$ $\varepsilon _{0}\nabla \times \mathbf {E}$ $=$ $-\mu _{0}\varepsilon _{0}{\frac {\partial \mathbf {H} }{\partial t}}$ $\oint _{\partial \Sigma }\mathbf {D} \cdot \mathrm {d} {\boldsymbol {\ell }}$ $=$ $-\varepsilon _{0}\int \!\!\!\!\int _{\Sigma }{\frac {\partial \mathbf {B} }{\partial t}}\cdot \mathrm {d} \mathbf {S}$ $\oint _{\partial \Sigma }\mathbf {P} \cdot \mathrm {d} {\boldsymbol {\ell }}$ $=$ $-\varepsilon _{0}\int \!\!\!\!\int _{\Sigma }{\frac {\partial \mathbf {I} }{\partial t}}\cdot \mathrm {d} \mathbf {S}$ $\oint _{\partial \Sigma }\varepsilon _{0}\mathbf {E} \cdot \mathrm {d} {\boldsymbol {\ell }}$ $=$ $-\mu _{0}\varepsilon _{0}\int \!\!\!\!\int _{\Sigma }{\frac {\partial \mathbf {H} }{\partial t}}\cdot \mathrm {d} \mathbf {S}$ $\nabla \times \mathbf {B}$ $=$ $\mu _{0}\left(\mathbf {J} _{net}+{\frac {\partial \mathbf {D} }{\partial t}}+\varepsilon _{0}{\frac {\partial ^{2}\mathbf {A} }{\partial t^{2}}}\right)$ $\nabla \times \mathbf {I}$ $=$ $\mu _{0}\left(\mathbf {J} _{defect}+{\frac {\partial \mathbf {P} }{\partial t}}\right)$ $\mu _{0}\nabla \times \mathbf {H}$ $=$ $\mu _{0}\left(\mathbf {J} _{free}+\varepsilon _{0}{\frac {\partial \mathbf {E} }{\partial t}}+\varepsilon _{0}{\frac {\partial ^{2}\mathbf {A} }{\partial t^{2}}}\right)$ $\oint _{\partial \Sigma }\mathbf {B} \cdot \mathrm {d} {\boldsymbol {\ell }}$ $=$ $\mu _{0}\int \!\!\!\!\int _{\Sigma }\left(\mathbf {J} _{net}+{\frac {\partial \mathbf {D} }{\partial t}}+\varepsilon _{0}{\frac {\partial ^{2}\mathbf {A} }{\partial t^{2}}}\right)\cdot \mathrm {d} \mathbf {S}$ $\oint _{\partial \Sigma }\mathbf {I} \cdot \mathrm {d} {\boldsymbol {\ell }}$ $=$ $\mu _{0}\int \!\!\!\!\int _{\Sigma }\left(\mathbf {J} _{defect}+{\frac {\partial \mathbf {P} }{\partial t}}\right)\cdot \mathrm {d} \mathbf {S}$ $\oint _{\partial \Sigma }\mu _{0}\mathbf {H} \cdot \mathrm {d} {\boldsymbol {\ell }}$ $=$ $\mu _{0}\int \!\!\!\!\int _{\Sigma }\left(\mathbf {J} _{free}+\varepsilon _{0}{\frac {\partial \mathbf {E} }{\partial t}}+\varepsilon _{0}{\frac {\partial ^{2}\mathbf {A} }{\partial t^{2}}}\right)\cdot \mathrm {d} \mathbf {S}$ Net Field EM (cause) Minus $-$ Near-field EM (cause) Equals $=$ Far-field EM (cause) bounded and unbounded charge/current bounded charge/current unbounded charge/current $(\mu ,\epsilon )$ $(\mu _{0}\chi _{v},\epsilon _{0}\chi _{\text{e}})$ $(\mu _{0},\epsilon _{0})$

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  σ_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

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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