I was going through chapter 8 of Griffiths' "Introduction to Electrodynamics", where he introduces the Maxwell Stress Tensor, by deriving it from the Lorentz law for some charge density $\rho$,
$$\vec{F}=\int_\mathcal V(\rho\vec E+\vec J\times\vec B) \ d\tau$$
Then, he uses Maxwell's equations to rewrite this in terms of the fields only, by using:
$$\epsilon_0(\nabla\cdot\vec E)\vec E=\rho\vec E$$
and:
$$\vec J\times\vec B=(\frac{1}{\mu_0}\nabla\times\vec B-\epsilon_0\frac{\partial\vec E}{\partial t})\times\vec B$$
However, if the original fields are external (not produced by $\rho$), then the charge density should not be equal to the divergence of the field since it isn't it's source. If on the other hand the charge density is the source of these fields, then isn't this expression calculating the force the fields do on their own source? Thanks.