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How can you use the relaxation method to model negative dielectrics?

The relaxation method is usually used to model electrostatics problems but negative dielectrics are only see in dynamic systems.

EDIT: To my knowledge at frequency, $f=0$, no materials act like they have a negative dielectric properties.

The university I attend offers a numerical electrodynamics course, every year as part of a homework they ask for a a negative dielectric be put into the relaxation method simulation that the students have programmed through the course. A negative dielectric slab in a parallel plate would be an example. I assume even though all of the voltages are static in time domain, that a f!=0 system is being simulated.

What I am interested in is what systems can be modelled in this fashion? How do I correlate the results of the simulation to a physical system. Or a example of a physical system that can be simulated in the above fashion.

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What do you mean that negative dielectrics are only seen in "dynamic systems"? Could you explain a bit more what level of modeling you'd like to do, i.e. at the level of some metamaterial which might have negative dielectric constant in some frequency range, or just see what happens to waves in the presence of negative dielectric optical elements, etc.? – j.c. Nov 9 '10 at 23:59

I would ask your instructor for clarification, but I'd bet that they just want you to model some hypothetical "static negative index material", even though none are known to exist. I don't know how to apply relaxation to the full time-dependent Maxwell's equations...

For examples of applications of negative index materials, a recently popular idea is "cloaking", which you can read about here. In general the website "Physics" has had a few articles on recent research on negative index materials, for instance here and here.

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Thanks thought it was a long shot but it was something I wanted to know and I thought it would help the site grow. – Davorak Nov 11 '10 at 2:39

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