Consider the simple case of electromagnetic irradiation of a homogeneous isotropic dielectric, neglecting the dispersion of the refractive index. Assuming a transparent medium, the spatial density of forces acting on the dielectric in a static external electromagnetic field can be given as
$$\mathbf{f} = - \nabla p - \nabla \epsilon \dfrac{\langle \mathbf{E}^2 \rangle}{8 \pi} - \nabla \mu \dfrac{\langle \mathbf{H}^2 \rangle}{8 \pi} + \nabla \left[ \left( \rho \dfrac{\partial{\epsilon}}{\partial{\rho}} \right)_T \dfrac{\langle \mathbf{E}^2 \rangle}{8 \pi} + \left( \rho \dfrac{\partial{\mu}}{\partial{\rho}} \right)_T \dfrac{\langle \mathbf{H}^2 \rangle}{8 \pi} \right] + \dfrac{\epsilon \mu - 1}{4 \pi c} \dfrac{\partial}{\partial{t}}\langle [ \mathbf{E} \times \mathbf{H}] \rangle.$$
$p$ is the pressure in the medium (for a given density $\rho$ and temperature $T$ in zero field.
$\epsilon$ and $\mu$ are the permittivity and magnetic permeability.
$c$ is the speed of light.
The angular brackets denote averaging over a time period far greater than the characteristic alternation period of light.
It is said that, by expressing $\langle E^2 \rangle$ through $I$ (the light intensity) and introducing the refractive index $n = \sqrt{\epsilon}$, we can transform the striction force equation to
$$\mathbf{f}_{\text{str}} = \nabla \left[ \left( \rho \dfrac{\partial{\epsilon}}{\partial{\rho}} \right)_T \dfrac{\langle \mathbf{E}^2 \rangle}{8 \pi} \right] = \nabla \left[ \left( \rho \dfrac{\partial{n}}{\partial{\rho}} \right)_T \dfrac{I}{c} \right].$$
I'm trying to understand how exactly we get $\nabla \left[ \left( \rho \dfrac{\partial{\epsilon}}{\partial{\rho}} \right)_T \dfrac{\langle \mathbf{E}^2 \rangle}{8 \pi} \right] = \nabla \left[ \left( \rho \dfrac{\partial{n}}{\partial{\rho}} \right)_T \dfrac{I}{c} \right]$. I've been doing a lot of research to try and understand this, but I'm stuck.
My best attempt is as follows. As said here, in optics, the time-averaged value of the radiated flux is technically known as the irradiance, more often simply referred to as the intensity. The Wikipedia article for intensity says that, if $I$ is the local intensity (I'm not completely sure if this is the correct assumption for our case), then we have that $I = \dfrac{cn \epsilon_0}{2}|E|^2$, where $\epsilon_0$ is the vacuum permittivity. And so, if we assume that $\langle \mathbf{E}^2 \rangle = |E|^2$ (which seems to be true, given the answer here), then we get that $|E|^2 = \dfrac{2I}{cn \epsilon_0}$, and so $\nabla \left[ \left( \rho \dfrac{\partial{\epsilon}}{\partial{\rho}} \right)_T \dfrac{\langle \mathbf{E}^2 \rangle}{8 \pi} \right] = \nabla \left[ \left( \rho \dfrac{\partial{n^2}}{\partial{\rho}} \right)_T \dfrac{I}{4 \pi c n \epsilon_0} \right]$. But it is not clear how one proceeds from here.
Some other potentially relevant facts that I found during my research are as follows:
- According to the article on irradiance (different from the article on intensity), $E_{{\mathrm {e}}}={\frac {n}{2\mu _{0}{\mathrm {c}}}}E_{{\mathrm {m}}}^{2}\cos \alpha ={\frac {n\varepsilon _{0}{\mathrm {c}}}{2}}E_{{\mathrm {m}}}^{2}\cos \alpha$. If we let that $\cos(\alpha) = 1$ for our case, then this might be relevant.
- The article on vacuum permittivity states that $\varepsilon _{0}={\frac {1}{\mu _{0}c^{2}}}$, where $\mu_0$ is the vacuum permeability.
- This page on "energy density, flux and power" has numerous relevant-looking facts that include $E$ and time-averaged values, and look like they could potentially cancel the necessary factors, such as $4\pi$ or $8\pi$, somehow.
I would greatly appreciate it if people would please take the time to explain exactly how we get from $\nabla \left[ \left( \rho \dfrac{\partial{\epsilon}}{\partial{\rho}} \right)_T \dfrac{\langle \mathbf{E}^2 \rangle}{8 \pi} \right]$ to $\nabla \left[ \left( \rho \dfrac{\partial{n}}{\partial{\rho}} \right)_T \dfrac{I}{c} \right]$.
