question about Ampere's law in vacuum and in matter
Hi
We can derive equation [itex]\nabla.D=\rho_f[/itex] from equation [itex]\nabla.E=\rho/\epsilon_0[/itex]. But what about Ampere's law? I tried to derive [itex]\nabla\times{H}=J_f+\partial{D}/\partial{t}[/itex] from [itex]\nabla\times{B}=\mu_0J+\epsilon_0\mu_0\partial{E}/\partial{t}[/itex] but I could not. This is strange because I thought that Maxwell's equations in vacuum are enough for studying electromagnetic field in any matter and that Maxwell's equations in matter are derivable from them. 
Re: question about Ampere's law in vacuum and in matter
They are, if you add some assumptions about the material  D proportional (and parallel) to E and so on.
For materials where this is not true, I don't know. 
Re: question about Ampere's law in vacuum and in matter
[itex]\partial{D}/\partial{t}=\epsilon_0\partial{E}/\partial{t}+\partial{P}/\partial{t}[/itex]. The second term is underivable from Ampere's law in vacuum.

Re: question about Ampere's law in vacuum and in matter
Add the assumption that ##D \propto E##, and it works.
In general, this can be wrong, but I don't know if the regular Maxwell equations work there at all. If ##\epsilon_r## is a tensor (or nonlinear), things can get difficult. 
Re: question about Ampere's law in vacuum and in matter
Dear mfb,
I think I found the solution. [itex]\partial{P}/\partial{t}[/itex] is some kind of current (bound charges are moving). So if we write total current density as [itex]J=J_f+\nabla\times{M}+J_p[/itex] where [itex]J_p[/itex] is polarization current density, we can solve the problem. 
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