User talk:Mpatel/sandbox/Electromagnetic field

Rationale

This sandbox version has been created to replace the current article which is, IMHO, awful:

• Intro. - yuck ! Need a much more intuitive, less technical intro.

• Maybe I read the article too fast, but there is hardly any mention of the 'wave property' of EM fields. There is some mention, but it is scant.

• There should definitely be mathematical descriptions of the EM field (vector fields, tensors, differential forms, etc...) and the corresponding theories which use these mathematical structures.

• The hydrodynamic interpretation is waaaaaaaaay too lengthy; it looks as though that's the main feature of the article - it should be considerably shortened (or, more drastically, eliminated - but at least given a mention).

• After the flaws in this hydrodynamic approach, there is mention of the 'photon-field' interpretation. Good, but there is no conclusion: this is where the problems arise in conceptualising the EM field. Perhaps a section each on the 'wavy' and 'particly' attributes of the EM field would be better, then a description of the current state of affairs would serve as a conclusion. Then advantages and disadvantages of the present model would indicate it's limitations. Maybe pushing the boat out here, but perhaps a brief mention of what string theory has to say could also be appropriate.

I invite comments. MP (talk) 14:37, 4 December 2005 (UTC)

Start in major revision

Started changing the article to make it more respectable. Intro. sorted (but need more) + removed 1 section + added sections that I believe are relevant and important to mention. MP (talk) 09:33, 11 December 2005 (UTC)

After mulling over it, I've decided to eliminate a load of (what I think) is junk (for this article). Wrote some more about properties of the field - the word 'determined' is red-linked as I plan to create an article in which various derivations of the mentioned result are given. MP (talk) 10:09, 11 December 2005 (UTC)

Tried to spice up the article by stating Maxwell's equations and giving a few examples of the use of some of the equations (which is linked to the dynamical nature of the EM field, this latter property also providing a nice lead in to the link with light as an EM disturbance). A lot more needs to be done in this article. I feel that a section on comparing the electromagnetic field with the other '3 fundamental fields' may be interesting - but I'll leave the majority of that section for others to write. MP (talk) 11:11, 11 December 2005 (UTC)

Introduced subsections on structure of the EM field - continuous model versus discrete model. Will work on these later - too tired right now... MP (talk) 14:53, 11 December 2005 (UTC)

Hi (Chris ?). Removed a section (on light and EM waves, photons) that is now redundant and made a few other changes. Further plans for this article (which we can discuss):

• Discuss everyday applications of the EM field (plenty of good things in here and can relate these directly to properties of the field).

• Comparison with other fields (gravitational, strong and weak nuclear) - could even create a new article on this, if there is enough content.

• Maybe need a section on relativity (special) and EM.

MP (talk) 11:32, 18 December 2005 (UTC)

Not sure whether/where to add this

The standard vector field formulation of the source-free Maxwell's equations can be written out in terms of cartesian coordinate components as a system of eight coupled linear partial differential equations, as follows:

${\frac {\partial B^{x}}{\partial x}}+{\frac {\partial B^{y}}{\partial y}}+{\frac {\partial B^{z}}{\partial z}}=0$

${\frac {\partial B^{x}}{\partial t}}+{\frac {\partial E^{z}}{\partial y}}-{\frac {\partial E^{y}}{\partial z}}=0,\;\;{\frac {\partial B^{y}}{\partial t}}+{\frac {\partial E^{x}}{\partial z}}-{\frac {\partial E^{z}}{\partial x}}=0,\;\;{\frac {\partial B^{z}}{\partial t}}+{\frac {\partial E^{y}}{\partial x}}-{\frac {\partial E^{x}}{\partial y}}=0$

${\frac {\partial E^{x}}{\partial x}}+{\frac {\partial E^{y}}{\partial y}}+{\frac {\partial E^{z}}{\partial z}}=0$

${\frac {\partial E^{x}}{\partial t}}-{\frac {\partial B^{z}}{\partial y}}+{\frac {\partial B^{y}}{\partial z}}=0,\;\;{\frac {\partial E^{y}}{\partial t}}-{\frac {\partial B^{x}}{\partial z}}+{\frac {\partial B^{z}}{\partial x}}=0,\;\;{\frac {\partial E^{z}}{\partial t}}-{\frac {\partial B^{y}}{\partial x}}+{\frac {\partial B^{x}}{\partial y}}=0$

This system has four independent variables and six dependent variables. The point symmetries of this system form a Lie group of transformations of the ten dimensional space of variables which preserves the form of the system of equations just given. We can determine the Lie algebra of this Lie group by giving vector fields generating the Lie algebra, which can be computed more or less mechanically from the system following the prescription given by Sophus Lie himself. (The notions of Lie groups and Lie algebras were originally developed by Lie in the context of explaining what is going on in such symmetry computations.) These computations can be very tedious if carried out by hand, which no doubt artificially retarded the developoment of Lie's ideas. Fortunately, they are very well suited to computer algebra computations involving the notions of Gröbner bases and differential rings. The result is the following list of vector fields:

• Four translations

$\partial _{t},\;\partial _{x},\;\partial _{y},\;\partial _{z}$

• Three rotations, such as

$-y\,\partial _{x}+x\,\partial _{y}-E^{y}\,\partial _{E^{x}}+E^{x}\,\partial _{E^{y}}-B^{y}\,\partial _{B^{x}}+B^{x}\,\partial _{B^{y}}$

• Three boosts, such as

$x\,\partial _{t}+t\,\partial _{x}+B^{z}\,\partial _{E^{y}}-B^{y}\,\partial _{E^{z}}-E^{z}\,\partial _{B^{y}}+E^{y}\,\partial _{B^{z}}$

• One dilation

$t\,\partial _{t}+x\,\partial _{x}+y\,\partial _{y}+z\,\partial _{z}$

• A conformal transformation

${\frac {t^{2}+x^{2}+y^{2}+z^{2}}{2}}\,\partial _{t}+t\;\left(x\partial _{x}+y\,\partial _{y}+z\,\partial _{z}\right)$

$\quad +\left(z\,B^{y}-y\,B^{z}-2t\,E^{x}\right)\partial _{E^{x}}+\left(x\,B^{z}-z\,B^{x}-2t\,E^{y}\right)\partial _{E^{y}}+\left(y\,B^{x}-x\,B^{y}-2t\,E^{z}\right)\partial _{E^{z}}$ $\quad +\left(y\,E^{z}-z\,E^{y}-2t\,B^{x}\right)\partial _{B^{x}}+\left(z\,E^{x}-x\,E^{z}-2t\,B^{y}\right)\partial _{B^{y}}+\left(x\,E^{y}-y\,E^{x}-2t\,B^{z}\right)\partial _{B^{z}}$

• Three more conformal transformations, such as

${\frac {t^{2}+x^{2}-y^{2}-z^{2}}{2}}\,\partial _{t}+x\;\left(t\partial _{t}+y\,\partial _{y}+z\,\partial _{z}\right)$

$\quad +\left(-y\,E^{y}-z\,E^{z}-2x\,E^{x}\right)\partial _{E^{x}}+\left(y\,E^{x}+t\,B^{z}-2x\,E^{y}\right)\partial _{E^{y}}+\left(z\,E^{x}-y\,B^{y}-2x\,E^{z}\right)\partial _{E^{z}}$ $\quad +\left(-y\,B^{y}-z\,B^{z}-2x\,B^{x}\right)\partial _{B^{x}}+\left(y\,B^{x}-t\,E^{z}-2x\,B^{y}\right)\partial _{B^{y}}+\left(z\,B^{x}+t\,E^{y}-2x\,B^{z}\right)\partial _{B^{z}}$

• One electromagnetic duality "rotation"

$B^{x}\,\partial _{E^{x}}+B^{y}\,\partial _{E^{y}}+B^{z}\,\partial _{E^{z}}-E^{x}\,\partial _{B^{x}}-E^{y}\,\partial _{B^{y}}-E^{z}\,\partial _{B^{z}}$

• One scalar multiplication of the dependent variables

$E^{x}\,\partial _{E^{x}}+E^{y}\,\partial _{E^{y}}+E^{z}\,\partial _{E^{z}}+B^{x}\,\partial _{B^{x}}+B^{y}\,\partial _{B^{y}}+B^{z}\,\partial _{B^{z}}$

• adding another solution of the system

$(E^{\prime })^{x}\,\partial _{E^{x}}+(E^{\prime })^{y}\,\partial _{E^{y}}+(E^{\prime })^{z}\,\partial _{E^{z}}+(B^{\prime })^{x}\,\partial _{B^{x}}-(B^{\prime })^{y}\,\partial _{B^{y}}-(B^{\prime })^{z}\,\partial _{B^{z}}$

where the primed quantities are a solution of the original system of eight coupled linear PDEs.

Note that

• some of these transform only the four independent variables; we can consider these as active motions in the underlying spacetime or as passive coordinate transformations,
• some transform only the dependent variables; these represent physical changes in the fields to a new solution,
• the rest transform both independent and dependent variables; these can be considered as transformations of the underlying spacetime accompanied by the correct corresponding transformation of the fields.

We can integrate the flows corresponding to these first order linear partial differential operators to obtain the unipotent transformations of our ten dimensional space.

Our boost gives rise to the following unipotent subgroup (with parameter $\alpha$):

$t\rightarrow t\,\cosh \alpha +x\,\sinh \alpha ,\;x\,\rightarrow x\cosh \alpha +t\,\sinh \alpha ,$

$E^{y}\rightarrow E^{y}\,\cosh \alpha +B^{z}\,\sinh \alpha ,\;B^{y}\rightarrow B^{y}\,\cosh \alpha -E^{z}\,\sinh \alpha ,$

$E^{z}\,\rightarrow E^{z}\cosh \alpha -B^{y}\,\sinh \alpha ,\;B^{z}\,\rightarrow E^{z}\cosh \alpha -B^{y}\,\sinh \alpha$

This shows that boosting an electrostatic field in some transverse direction will in general "create" a magnetic field, which strongly suggests that the electromagnetic field is better treated as another kind of mathematical quantity (the appropriate choice turns out to be an antisymmetric second rank tensor field) than as two vector fields.

Electromagnetic duality gives rise to the following unipotent subgroup (with parameter $\theta$):

$E^{x}\rightarrow E^{x}\,\cos \theta +B^{x}\,\sin \theta ,\;\;E^{y}\rightarrow E^{y}\,\cos \theta +B^{y}\,\sin \theta ,\;\;E^{z}\rightarrow E^{z}\,\cos \theta +B^{z}\,\sin \theta$

$B^{x}\rightarrow B^{x}\,\cos \theta -E^{x}\,\sin \theta ,\;\;B^{y}\rightarrow B^{y}\,\cos \theta -E^{y}\,\sin \theta ,\;\;B^{z}\rightarrow B^{z}\,\cos \theta -E^{z}\,\sin \theta$

The three rotations and the three translations together generate the Lie algebra of the six-dimensional Lorentz group; throwing in the four translations gives the Poincaré group. Thus, we have directly verified that the fundamental Poincaré symmetry of electrodynamics arises entirely from the mathematical form of the electromagnetic field equations. However, the full symmetry group is in fact much larger; in its action on the spacetime it gives the full fifteen-dimensional conformal group of Minkowski spacetime. Since the concept of null geodesics is invariant under conformal transformations of Minkowski spacetime, this is consistent with the idea that the wave vector field of an electromagnetic wave should be a null vector field, and therefore consistent with the standard interpretation of the light cone of Minkowski spacetime. In addition to the conformal transformations, there are two transformations which act only on the six dependent variables. Strictly speaking, the point symmetry group is infinite-dimensional because of the last generator, but it is probably better to think of it as the fifteen dimensional conformal group augmented by electromagnetic duality and the various transformations which take account of the linearity of Maxwell's equations.

Our computation reveals some important mathematical properties of the Maxwell equations, but it can also be useful in finding exact solutions from a symmetry Ansatz. This technique is sometimes helpful even when dealing with a linear system of PDEs. For example, the invariants of the unipotent subgroup of electromagnetic duality are (ignoring the independent variables, which are unaffected by this subgroup):

$\|E^{x}\|^{2}+\|B^{x}\|^{2}=c_{1},\;\;\|E^{y}\|^{2}+\|B^{y}\|^{2}=c_{2},\;\;\|E^{z}\|^{2}+\|B^{z}\|^{2}=c_{3}$

From this we obtain a symmetry Ansatz which we can use to search for self-dual electromagnetic fields.

We can perform a similar symmetry analysis for the potential form of the source-free Maxwell equations, which gives four coupled PDEs in four independent and four dependent variables, or for the full Maxwell equations (augmented by the equation of charge conservation, relating the charge density scalar and the current density vector), which gives nine coupled PDEs in four independent variables and ten dependent variables. The differences between the various symmetry groups gives further insight into electromagnetism.

Self note

Plans for major revision:

• After intro., discuss generation/production of the field (should have some diagrams for this).

• Move discrete/cts. structure further down.

More to follow... MP ^{(talk•contribs)} 22:55, 2 November 2007 (UTC)

• Merge 'Dynamics of the field' and 'The EM field as a feedback loop' sections.

• Include diagram or picture or movie at start of article to capture interest. MP ^{(talk•contribs)} 20:07, 3 November 2007 (UTC)

Light as an electromagnetic disturbance

The picture for this little subsection is the best that I could find; technically, it shows how an electromagnetic field is generated by an oscillating charge and doesn't depict 'light as an electromagnetic wave', as I wrote. This needs to be sorted. MP ^{(talk•contribs)} 19:23, 9 November 2007 (UTC)