Calculating EM field using only crosssection of past light cone?by JesseM Tags: cone, crosssection, field, light 

#1
Mar1413, 11:20 PM

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If you want to know the value of the electromagnetic field at some point in space P at time t_{1}, I assume that since EM is a relativistic theory, it should be possible to derive it using only the value of the field (along with charges, but let's say we are dealing with fields in free space) at an earlier time t_{0}, in a sphere of radius c*(t_{1}  t_{0}), with the field outside the sphere unknown. Does anyone know of any books or papers that show what the equations for this would look like, and how they would be derived, preferably at a nottooadvanced level? In my EM class we did study the "retarded potential" method which is based on the past light cone, but it requires looking at the contribution from every charge whose path crosses the past light cone at any point in the arbitrarily distant past, whereas I'm thinking of a method that only looks at a crosssection of the light cone at a particular earlier time t_{0}, and which deals with the field at that time rather than looking exclusively at the sources of the field (charges).
Another question related to this: if you know the initial conditions I in the sphere at time t_{0} (both the E and B fields, and I think you'd need their instantaneous timederivatives to determine their later evolution too), I wonder if you could use the following approximation to calculate the timeevolution, an approximation which would hopefully approach the exact solution in the limit considered in step 4: 1. Divide the whole volume of the sphere up into a lattice of cubes (or any other spacefilling volume), and find the average value of the fields in I within each cube. 2. Create a new set of initial conditions I' where the fields have uniform values within each individual cube (with discontinuities at the boundaries between cubes), equal to the averages found in step 1. 3. Compute the timeevolution of [i]I'[/b] using the superposition principlethe total value of the field at a later time is just the sum of the contributions from a large number of separate initial conditions, each one consists of a single cube filled with uniform fields surrounded by an initial field of 0 everywhere outside that cube. 4. Then, consider the limit as the size of the cubes goes to zero. I would guess that in this limit, the contribution from each cube might approach some fairly simple form, perhaps identical to the contribution from a proportionally small spherical region of uniform field with an initial field of 0 everywhere outside the small spherical region. Probably the timeevolution of such an initial condition would just be a sphere of nonzero field expanding at the speed of light, though I don't know whether the field would be uniform in the expanding sphere or concentrated at the surface. Could this work as a way of deriving the timeevolution, or is there some problem I haven't considered? 



#2
Mar1513, 02:39 PM

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Perhaps you are thinking of something like Huygens' Principle. Given the field distribution across a closed surface, you can then calculate the fields of the propagated field. You're asking about projecting forward knowing the fields on a given surface in, what, the Minkowski space I assume? This seems to be equivalent to saying that you know the field distribution of a wavefront at a specific point in time. So Huygens' Principle is perfectly valid in this situation.




#3
Mar1513, 03:32 PM

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Also, I realized there was a mistake in the second part of my question, about approximating the initial conditions: it shouldn't actually be possible to have an initial condition in free space consisting of a cube filled with uniform E and B fields, and 0 field outside, because in this case the E field lines wouldn't be able to terminate on a charge. 



#4
Mar1613, 05:24 AM

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Calculating EM field using only crosssection of past light cone?http://en.wikipedia.org/wiki/Kirchhoff_integral_theorem In this theorem, only the fields on the surface of the sphere in the past time t0 are needed. Similar equation should be valid in which fields at future time 2t1  t0 are used (due to symmetry of the wave equation). Regarding the second part, I think some book on numerical methods for EM field may help you. A friend of mine found this book very useful: Alain Bossavit: Computational electromagnetism http://butler.cc.tut.fi/~bossavit/ (scroll down to the section "Books", the line "This one, out of print, ... " 



#5
Mar1613, 12:39 PM

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#6
Mar1713, 03:37 AM

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The theorem itself is exact  it is derived from the wave equation. Usually it is stated for scalar function, but since each component of the field separately obeys the same wave equation, the theorem is valid for the whole vector field.
The Kirchhoff theory of diffraction is further development of this idea, which however is only approximate. The difficulty is that we never know the field on the boundary surface exactly, so Kirchhoff made some further assumptions and these make his resulting theory approximate. 



#7
Mar1713, 03:49 AM

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JesseM, what you are looking for are Jefimenko's equations:
http://en.wikipedia.org/wiki/Jefimenko%27s_equations These are derived directly from Maxwell's equations, and show explicitly the dependence on values within the past lightcone only. In fact, since EM waves travel at the speed of light, one sees that the EM field now depends only on disturbances strictly on the past lightcone. 



#8
Mar1713, 01:09 PM

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#9
Mar1713, 01:16 PM

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#10
Mar1713, 02:47 PM

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#11
Mar1813, 01:32 AM

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#12
Mar1813, 04:06 AM

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If you consider the presence of matter, this will of course affect the behavior of the field. In general the EM field is not sufficient to describe the state of matter, so the evolution of the field will be much more complicated and the Kirchhoff theorem does not apply.
One still has wave equations for the fields though, but with nonzero sources (charge and current density). The effect of these is usually considered to depend only on their values at past light cone, so again something like the Kirchhoff theorem should work, provided you include these sources among the quantities known at the boundary of the sphere. 



#13
Mar2713, 11:24 AM

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