Question about Ampere's law in vacuum and in matter

  1. 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.
    Last edited: Nov 14, 2012
  2. jcsd
  3. mfb

    Staff: Mentor

    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.
  4. [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.
  5. mfb

    Staff: Mentor

    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.
  6. 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.
    Last edited: Nov 14, 2012
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