Deriving Ampere's law in matter from vacuum form

[Arham]

Hi

We can derive equation $\nabla.D=\rho_f$ from equation $\nabla.E=\rho/\epsilon_0$. But what about Ampere's law? I tried to derive $\nabla\times{H}=J_f+\partial{D}/\partial{t}$ from $\nabla\times{B}=\mu_0J+\epsilon_0\mu_0\partial{E}/\partial{t}$ 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.

 

[mfb]

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.

 

[Arham]

$\partial{D}/\partial{t}=\epsilon_0\partial{E}/\partial{t}+\partial{P}/\partial{t}$. The second term is underivable from Ampere's law in vacuum.

 

[mfb]

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.

 

[Arham]

Dear mfb,

I think I found the solution. $\partial{P}/\partial{t}$ is some kind of current (bound charges are moving). So if we write total current density as $J=J_f+\nabla\times{M}+J_p$ where $J_p$ is polarization current density, we can solve the problem.