In Griffith's Introduction to Electrodynamics we find a derivation of the relation $$\mathbf f + \frac{\partial \mathbf g}{\partial t} = \nabla \cdot \overleftrightarrow{\mathbf T} ,$$ where $\mathbf f$ is the Lorentz force per unit volume, $\mathbf g$ is the electromagnetic linear momentum per unit volume, and $\overleftrightarrow{\mathbf T}$ is Maxwell's stress tensor. However, this derivation employs Maxwell's equations in a vacuum, or at least in media that exhibit no polarization or magnetization.

My aim is to derive a similar relation for Maxwell's tensor, but starting from the more general set of equations that include the fields $\mathbf D$, $\mathbf E$, $\mathbf B$, and $\mathbf H$, with the general relations $\mathbf D = \varepsilon_0 \mathbf E + \mathbf P$ and $\mathbf B = \mu_0(\mathbf H + \mathbf M)$. Starting again from the local expression of the Lorentz force on free charges and current densities $$\mathbf f = \rho_0 \mathbf E + \mathbf J_0 \times\mathbf B, $$ I've managed to show that $$ \mathbf f = -\frac{\partial \mathbf g}{\partial t} + (\nabla \cdot \mathbf D)\mathbf E + (\nabla \cdot \mathbf B) \mathbf H - \mathbf D \times (\nabla \times \mathbf E) - \mathbf B \times (\nabla \times \mathbf H), $$ but I cannot seem to be able to continue. Intuitively, I think I should get something of the form $$T_{ij} = E_i D_j + H_i B_j - \frac 1 2 \delta_{ij} (\mathbf E \cdot \mathbf D + \mathbf H \cdot \mathbf B), $$ which does revert to Griffith's $$T_{ij} = \varepsilon_0 E_i E_j + \frac 1 {\mu_0} B_i B_j - \frac 1 2 \delta_{ij} \left( \varepsilon_0 E^2 + \frac 1 {\mu_0} B^2\right) $$ as soon as $\mathbf P$ and $\mathbf M$ both vanish, and also happens to contain the expression for the electromagnetic energy density $$u = \frac 1 2 \mathbf E \cdot \mathbf D + \frac 1 2 \mathbf H \cdot \mathbf B. $$ Maybe I'm missing some relevant vector calculus identities, or relations between the fields. Otherwise, I guess the only way out is to express $\mathbf D$ and $\mathbf B$ in terms of $\mathbf E$ and $\mathbf H$ respectively, through the usual relations (valid in all homogeneous isotropic linear media) $$\mathbf D = \varepsilon \mathbf E, \qquad \mathbf B = \mu \mathbf H, $$ with $\varepsilon = \varepsilon_0 \varepsilon_r = \varepsilon_0 (1+\chi_e)$ and $\mu = \mu_0 \mu_r = \mu_0(1+\chi_m)$, which would yield $$T_{ij} = \varepsilon E_i E_j + \frac 1 {\mu} B_i B_j - \frac 1 2 \delta_{ij} \left( \varepsilon E^2 + \frac 1 {\mu} B^2\right), $$ but this means we lose generality, and we won't be able to treat e.g. ferroelectric or ferromagnetic materials, or anisotropic linear media.

Any thoughts?


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