Definition of electric polarization

[Manchot]

I recently realized that I have never really seen a rigorous definition of the electric polarization field in matter (and for that matter, magnetization). On the one hand, I know what its physical meaning is, but on the other, I don't believe that I'll really trust it until I come up with one. Depending on how I proceed, I run into certain issues. If I define it as a density of individual dipole moments, I do not have very much "calculatory" power, and I cannot even reproduce basic results like the fact that its divergence should be the opposite of the bound charge density and the time derivative should be the bound current density.

I also tried defining it as the (decaying) quantity whose divergence is the opposite of the bound charge density, and whose curl is zero. Though this gives me more calculatory power, and also reproduces the dipole density approach, it runs into such problems such as the fact that it requires the current density to be curl-free in order to correctly reproduce the bound current density. Does anyone have any other suggestions?

 

[Manchot]

Well, I figured out what I think is a good way to make the definition. (It's always funny how I can agonize over something for weeks, but once I sit down to articulate my problem on PF, it often comes to me within hours.) In case anyone's wondering (or should come upon this thread sometime in the future), I'll tell you what I did. First, I defined the charge distribution of the "infinitesimal dipole" as follows:
$$\rho(\vec r,t) = -\vec p(t) \cdot \nabla \delta^3(\vec r)$$

In the above definition, p is the dipole moment of the dipole. If you perform a calculation of the dipole moment of the above expression, you find that it is indeed p. While this is just a mathematical abstraction, it can be easily made to accommodate physical dipoles (e.g., a hydrogen atom) by approximating them as infinitesimal dipoles.

Next, I defined the polarization density due to one infinitesimal dipole as the following:
$$\vec P(\vec r,t)=\delta^3(\vec r) \vec p(t)$$

Taking the divergence of the above expression almost immediately yields the relationship $\nabla \cdot \vec P=-\rho$. By differentiating the above with respect to time, using the continuity equation, and performing some additional manipulation with delta functions, I found that $\frac{\partial P}{\partial t}=J$, where J is the current associated with the changing dipole. Since everything here is linear, they can be fairly easily extended to include multiple dipoles, and I'm fairly certain that something similar can be done for the magnetization density as well. If anyone has any thoughts on this, let me know.

 

[Meir Achuz]

The electric polarization P is simply the dipole moment per unit volume in a dielcetric.
M is the magnetic moment per unit volume.
What you are giving are proterties.