Understanding the meaning of (and using) Maxwell Stress Tensor From reading Griffiths, I understand that the total EM force on a set of charges in volume $\mathcal{V}$ can be found as
$$ \textbf{F}=\oint_{\mathcal{S}} \overleftrightarrow{\textbf{T}} \cdot d\textbf{a}-\epsilon_0\mu_0\frac{d}{dt}\int_{\mathcal{V}}\textbf{S}\ d\tau. $$
where $\mathcal{S}$ is the boundary of $\mathcal{V}$. $\overleftrightarrow{\textbf{T}}$ is the Maxwell stress tensor, and $\textbf{S}$ is the Poynting vector. I have the following questions:


*

*The stress tensor physically represents the force per unit area acting on the surface. Thus, its appearance here is understandable. But, how can I interpret the second term? It seems to be the rate at which energy flows out of the volume, but what does that have to do with force?

*How is it that the EM force on charges is the same as the EM force on the surface of any volume that encloses the charges? How can I understand this intuitively without the equations? Someone asked a similar question here (Maxwell's Stress Tensor) but I didn't quite understand the answers.

*To me, it seems like you consider both the $\textbf{E}$ and $\textbf{B}$ fields that pass through the surface $\mathcal{S}$. Looking at problem 8.3 in Griffiths (a spherical shell of charge density $\sigma$ rotating at $\omega$, we want to know the magnetic force on the upper hemisphere), I would use a volume that encloses the upper bowl and the base disk. In this case, I have magnetic fields inside and outside, and an electric field outside that I have to care for. However, when I look at the solutions, Griffiths only considers the inside and outside $\textbf{B}$ fields, ignoring $\textbf{E}$. Is this because he asked for the magnetic force, and not the total force? Am I right in thinking that the $\textbf{E}$ should too be accounted for if one is to find the total force on the upper hemisphere?  

 A: Field momentum:
$\mu_0 \epsilon_0 \vec{S}$ is the momentum density of the field.  Integrating this about a volume, finds the TOTAL field momentum in that volume. The rate of change of this, is the rate of change of
field momentum within a volume.
Let's look closely, how can there be a change in field momentum within a volume?
Mechanical momentum:
f - this by definition is the rate at which mechanical momentum increases. If there is an increase in mechanical momentum, the total field momentum decreases ( hence the minus sign)
Momentum entering through the surface boundary:
The other term is $\iint T \cdot \vec{da}$
As you pointed out, this represents the force acting on the surface
Another way of putting it, is that this term represents the rate at which momentum is carried into that volume, by the fields. Much like in poyntings theorem, the poynting vector represents the rate at which energy flows out of that volume.
This makes sense, the stress tensor is dotted with the da vector, analogous to flow directly through of that volume.
It is also pleasing that the rate at which momentum flows through the surface, IS BY DEFINITION, the force acting on the surface.
When there is no change in mechanical momentum:
$$\textbf{F} = 0$$
$$\oint_{\mathcal{S}} \overleftrightarrow{\textbf{T}} \cdot d\textbf{a}=\epsilon_0\mu_0\frac{d}{dt}\int_{\mathcal{V}}\textbf{S}\ d\tau. $$
$$ \int_{\mathcal{V}} \nabla \cdot  \overleftrightarrow{\textbf{T}} d\tau=\epsilon_0\mu_0\frac{d}{dt}\int_{\mathcal{V}}\textbf{S}\ d\tau. $$
$$  \nabla \cdot  \overleftrightarrow{\textbf{T}} = \epsilon_0\mu_0\frac{d\textbf{S}}{dt}\ $$
Which is a continuity equation for field momentum.
Also your point (2) is incorrect, you are ignoring the actual change of field momentum. There are 2 terms determining the force on charges, not 1
A: 

*

*But, how can I interpret the second term? It seems to be the rate
at which energy flows out of the volume,
but what does that have to do with force?


You are right, the Poynting vector
$$\mathbf{S}=\frac{1}{\mu_0}\mathbf{E}\times\mathbf{B}$$
is the energy flow density (i.e. energy per time and area) of the EM field.
But incidentally, the vector
$$\mathbf{g}=\epsilon_0\mu_0\mathbf{S}=\epsilon_0\mathbf{E}\times\mathbf{B}$$
is also the momentum density (i.e. momentum per volume) of the EM field.
See for example "Resource Letter EM-1: Electromagnetic Momentum"
by D.J. Griffiths.
Hence the term
$$-\epsilon_0\mu_0\frac{d}{dt}\int_{\mathcal{V}}\textbf{S}\ d\tau$$
or $$-\frac{d}{dt}\int_{\mathcal{V}}\textbf{g}\ d\tau$$
is the decrease rate of the momentum contained in the EM field in the volume.
And this relates well to the force $\mathbf{F}$.
