Can Ampere's Law be written in terms of magnetization current desnity and polarization current density?

Question

I know Ampere's Law can be described like:

I am aware of how to use this law; however, as I was reading through Maxwell's equations I came across magnetization current density and polarization current density. The above law seems to be defined in terms of the electric current density vector, I was wondering how this law would be different if it were to involve magnetization current density or polarization current density vectors?

Can someone please help explain, I really want to ensure a good understanding of this equation and would appreciate the help.

Thank you and sorry for any formatting issues/missing tags, please feel free to edit as needed.

Answer 1

In matter (not in vaccuum, as written in your question) the "Ampere-Maxwell" law is $$\nabla \times \mathbf H = \mathbf J + \frac {\partial \mathbf P}{\partial t} + \epsilon_0 \frac {\partial \mathbf E}{\partial t}.$$ Here $\mathbf P$ is the dipole moment density of the material (it is obvioulsy zero in vacuum) and satisfies the "Gauss-Maxwell" equation $\nabla \cdot \mathbf P + \epsilon_0 \nabla \cdot \mathbf E = \rho $ that are conventionally written with the for the field vector $\mathbf D = \mathbf P + \epsilon_0 \mathbf E$ as $$\nabla \times \mathbf H = \mathbf J + \frac {\partial \mathbf D}{\partial t}.$$ $$\nabla \cdot \mathbf D = \rho .$$ In the Ampere-Maxwell law, $ \frac {\partial \mathbf P}{\partial t}$ is the polarization current and $\epsilon_0 \nabla \cdot \mathbf E $ is the displacement current. Regarding "magnetization current", are you talking about this modification of Faraday's law? If not then what do you mean by it?