I'm taking introductory, undergraduate-level E&M, for which we're following Griffiths. In his chapter on the conservation laws, he gives the following as the statement for conservation of momentum under electrodynamics:
$$\frac{d\vec{p}_{\textrm{mech}}}{dt} = \oint_S \overleftrightarrow{T}d\vec{a} -\epsilon_0 \mu_0 \frac{d}{dt} \int_V \vec{S} \; d\tau$$
Griffiths claims (without much elaboration) that the term involving Maxwell's stress tensor $\overleftrightarrow{T}$ yields the "momentum per unit time flowing through the surface" (which is believable, on a purely intuitive basis), and that the term involving $\vec{S}$ (the Poynting vector) yields the "momentum stored in the fields".
While I'm able to follow the mathematical derivation leading to this result without much trouble, I'm struggling to grasp on an intuitive level how an integral over the Poynting vector leads to a measure of some kind of "momentum stored". I'm more accustomed to thinking about the Poynting vector as a measure of energy flux (i.e. energy entering or leaving a region). It also isn't clear to me what a time derivative of such an integral "does".
Some similar questions exist, and several mention the momentum density $g$, which involves the Poynting vector $\vec{S}$, often pointing to this resource letter by Griffiths. None, however, elaborate on what precisely is meant by "momentum density", nor why this is related in any manner to the Poynting vector, except to say that "it must be" in order for conservation of momentum to work out.
In a more intuitive sense, what is going on here?
