In the calculation of the forces acting on a charge/current distribution, one arrives at the Maxwell stress tensor:
$$\sigma_{ij}=\epsilon_0 E_iE_j + \frac{1}{\mu_0} B_iB_j -\frac{1}{2}\left(\epsilon_0E^2+\frac{1}{\mu_0}B^2\right)$$
In the case of electrostatics, this element of the stress tensor denotes the electromagnetic pressure acting in the $i$ direction with respect to a differential area element with its normal pointing in the $j$ direction. Equivalently, we can replace "electromagnetic pressure" with "electromagnetic momentum flux density" in order to "make sense". With this mathematical construction, assuming a static configuration, the total force acting on a bounded charge distribution $E$ is given by
$$(\mathbf{F})_i=\oint_{\partial E} \sum_{j}\sigma_{ij} da_j $$
Where $da_j$ is the area element pointing in the $j$ direction (e.g. $da_{3}=da_z=dxdy$).
What I would like to know is, what is the advantage of introducing such an object? I have yet to see a problem where this has any real utility. Sure, we can now relate the net force on a charge distribution to the E&M fields on the surface, but are there any problems where that is really better than just straight up calculating it? In an experiment, does one ever really measure the E&M fields on the boundary of an apparatus to calculate the net force?
Best Answer
There are two big advantages to having a stress tensor calculated (and I'm sure there are others):
1) If rather than just the forces, you want to calculate strains and shears on an object that has physical extent. then, you are interested in the off-diagonal terms of $\sigma_{ij}$, and these don't naturally pop out of a simple force description
2) General relativity is phrased in terms of the stress-energy tensor, the four-dimensional generalization of the stress tensor, so if you want to understand the matter terms in general relativity, you should definitely understand the stress tensor.