# Mathematically, where does polarization current come from?

Mathematically speaking, where does polarization current in a material (due to time variant polarization) come from.

Griffith's introduces the concepts of bound charges and bound currents first as a mathematical trick, and then argues their physicality. But when it comes to the polarization current $J_p$, he postulates it based on physical arguments.

Is this the only way at arriving at them or is there some mathematical procedure that brings them out, analogous to the bound charges and currents which come about by the application of integral theorems of vector calculus?

Assuming the polarization is not strong enough to ionize your material, the overall bound charge is neutral. Let's exploit this fact by integrating over the volume and surface of the bound charges,

$$\int_V\rho_b\:d^3r + \int_{\partial V}\sigma_b\:d^2r = 0,$$

which, by the divergence theorem, is satisfied if

$$\rho_b = -\nabla\cdot \vec{P}, \:\:\:\: \sigma_b = \vec{P}\cdot\hat{n}.$$

We also know, on physical grounds, that this bound charge density due to the polarization must satisfy its own continuity equation, so

$$\frac{\partial \rho_b}{\partial t}+\nabla\cdot \vec{J}_p = 0 \\ \\ \frac{\partial}{\partial t}(-\nabla\cdot\vec{P})+\nabla\cdot\vec{J}_p = \nabla\cdot(\vec{J}_p-\frac{\partial\vec{P}}{\partial t}) = 0 \:\:\:\Rightarrow \:\:\: \boxed{\vec{J}_p = \frac{\partial \vec{P}}{\partial t}}$$

Physically, the polarization current comes about from the rate at which the material internally responds to the external electric field. Mathematically, this is found by simultaneously solving for overall bound charge neutrality with continuity imposed. Griffiths deduces $$\vec{J}_p$$ on physical grounds, and then checks that the continuity equation is satisfied, but you can just as well impose continuity to determine $$\vec{J}_p$$.