Undergrad Does Poynting's Theorem only involve external fields?

[jcap]

Poynting's Theorem (https://en.wikipedia.org/wiki/Poynting's_theorem) says:

The rate of energy transfer (per unit volume) from a region of space equals the rate of work done on a charge distribution plus the energy flux leaving that region.
$$-\frac{\partial u}{\partial t}=\mathbf{J}\cdot\mathbf{E}+\nabla\cdot\mathbf{S}$$
Poynting's vector is given by
$$\mathbf{S} = \frac{1}{\mu_0} \mathbf{E} \times \mathbf{B}$$
Field energy density is given by
$$u = \frac{1}{2}\left(\epsilon_0 \mathbf{E}\cdot\mathbf{E} + \frac{1}{\mu_0}\mathbf{B}\cdot\mathbf{B}\right)$$
Does the EM field $\mathbf{E}$,$\mathbf{B}$ include the field from the charge distribution itself or is it only the external field due to external sources?

 

[Dale]

It is the field from all sources.

Note that point charges cause mathematical problems here.