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Is the $\mathbf{E}$ in this equation the average macroscopic electric field? $$\epsilon_{0} \nabla \cdot \mathbf{E}=\rho=-\nabla \cdot \mathbf{P}+\rho_{f}$$

If yes, then how does the average get into this equation. Can anyone please show me how to derive it emphasising on how the average field is used to derive this equation?

I know that the average potential is given by $$V(\mathbf{r})=\frac{1}{4 \pi \epsilon_{0}} \int_{\mathcal{V}} \frac{\mathbf{P}\left(\mathbf{r}^{\prime}\right) \cdot \hat{r}}{r^{2}} d \tau^{\prime}$$ and that it can be written as $$V(\mathbf{r})=\frac{1}{4 \pi \epsilon_{0}} \oint_{\mathcal{S}} \frac{\sigma_{b}}{_{r}} d a^{\prime}+\frac{1}{4 \pi \epsilon_{0}} \int_{\mathcal{V}} \frac{\rho_{b}}{r} d \tau^{\prime}$$ where $\rho_{b} \equiv-\nabla \cdot \mathbf{P}$

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  • $\begingroup$ Can you give more context to the integrals? What is $\tau$, what is the index $b$? For now it looks like all the quantities you have are macroscopic. $\endgroup$
    – Hannes
    Apr 25 at 10:42
  • $\begingroup$ see for a very clear and thorough review Russakoff: A Derivation of the Macroscopic Maxwell Equations doi.org/10.1119/1.1976000 $\endgroup$
    – hyportnex
    Apr 25 at 22:18
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If you apply an electric field E' on a material. Then polarisation happens inside that material. This polarisation will create a new electric field that will be equal to the field caused by $\rho_b \, and \, \sigma_b$(That's why we are able to find those two factors from polarisation only.). Now if you try to measure the electric field practically what you will measure is the total electric field(which is due to the original field (E') and polarisation field). This total field which is able to measure in the lab is called E. And we know this E is a consequence of polarisation and free charges. This is what the first equation is telling you, hope that helps and sorry for the bad English.

Please note $\sigma_b$ is avoided in the equation but it's due to the fact that Gauss's law can not be applied at the surface of di-electric. Please refer Griffith's text, Introduction to Electrodynamics

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