Calculating force on charges using Maxwell stress tensor

Question

In Griffiths' Introduction to electrodynamics, chapter 8.2.2, if you want to calculate the force on charge bounded by V volume, it is said to choose an area that encloses that volume (surface area of that volume), or more properly that charge (from Stokes theorem). It is derived assuming like this. in Problem 8.4, we are asked to calculate the force- two point charges act on each other. We have to calculate using Maxwell stress tensor, and it is written that we have to calculate this over the area of a plane which is equidistant from both charges. My question is that the area we need to calculate the force doesn't enclose charges at all (even it is not a boundary of a specific volume) and that ruins our assumption of the surface area of volume.

Answer 1

Yes that plane does not enclose the particle. But the result will be correct if the field is electrostatic, here is why.

Add 5 additional planes perpendicular to the first one and each other, so that the particle is inside a cuboid defined by those planes. The tensor integral over this closed space is function of position of the 5 additional planes.

Now take the limit where the 5 additional planes get infinitely far from the particle. Since the field falls off as $1/r^2$, all contributions to the integral from the additional planes vanish. The only non-zero contribution is due to the first plane between the particles.