- #1

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I looked on Google but it's 2D and uses a Lorentz boost wheteas I would consider it classically.

Thanks in advance for any link or reference.

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- Thread starter jk22
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- #1

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I looked on Google but it's 2D and uses a Lorentz boost wheteas I would consider it classically.

Thanks in advance for any link or reference.

- #2

Cryo

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Now lets return to what, for example, polarization is. It is a volume density of induced electric dipoles. But electric dipole only appears as such if it is stationary. If you have a pair of separated opposite charges (i.e. electric dipole) moving relative to you, you will observe it as electric & magnetic dipoles. This is due to Lorentz boosts. This also applies to polarization and magnetization, but even more so since they are densities (so there are additionnal transformations). Basically if you see a material with polarization moving, it will appear to have both polarization and magnetization. This will affect the susceptibilities and, in turn, the refractive index etc.

So, I think, you have to use Lorentz boosts.

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- #4

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The usual way to do non-relativistic approximation is to Taylor expand the relativistic expression in ##v## and see under what circumstances (if any) you can neglect higher order terms. The last time you asked a similar question PAllen linked to a mathpages page that gives the exact relativistic version of Snell's Law - expand it and see if it matches what you got.

I'll just note that Fizeau's experiment was initially interpreted as supporting partial ether dragging because naive Galilean velocity addition did not match his results. So you may find that the domain of validity of your approximation is extremely narrow.

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- #6

Cryo

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at a classical level

You keep saying 'classical'='not relativistic', but there is no non-relativistic theory of electromagnetism. You can do full relativistic treatment and then take the limit of low velocity, sure, but you still have to go through relativity first. Maxwell's equations is what is commonly understood to be classical electromagnetism, and they are not invariant under Gallilei transforms, so there is no sane way to do electromagnetism 'non-relativistically'.

Thanks for pointing out that this has already been addressed.

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I would expect this to be the low v, extremely high n (in both media), limit of the relativistic formula.

- #8

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Thanks in advance for any link or reference.

This problem may be trivial when v << c, but I don't think the general problem is simple at all. My go-to reference for this is:

Paul Penfield, Hermann A. Haus

M.I.T. Press, 1967 - Electrodynamics - 276 pages

In it, there is a claim that using the Minkowski formulation correctly predicts Snell's law (and Cerenkov radiation), but as I said, the topic is highly non-trivial.

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