# 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:

1. 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?
2. 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.
3. 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?
• For Q1, my guess is that force acting on charged particles is a result of (1): static EM field (Maxwell stress tensor) and (2): dynamically, the rate of change of momentum of EM field (Poynting vector term) – K_inverse Sep 12 '18 at 3:37
• Ok I think that makes sense. Does this mean that in a static case, the force calculated using the tensor is the same as the force calculated using Columbs law? If yes, then this brings me to Q2: how is it that the integral over surface gives me a quantity that I can find from considering the direct interaction between charges. – Ptheguy Sep 12 '18 at 3:47
• It is the result of divergence theorem. Maybe it is better to review the derivation en.wikipedia.org/wiki/Maxwell_stress_tensor#Motivation. – K_inverse Sep 12 '18 at 3:56
• Integral of poyting vector is total momentum. The time derivative of momentum is force (think about newtons law in this way) – lalala Sep 12 '18 at 4:52
• Yes, but isn't the Poynting vector energy per unit time per unit area? Why integrate over volume? – Ptheguy Sep 13 '18 at 5:04