I'd like to share an interesting physics problem with the Physics SE community, one I found in a Russian Physics Olympiad Paper (without solution)
An insulating hollow cube of edge length $L$ has six charged faces - three adjacent faces (with a common vertex) have uniform charge density $+{\sigma}$ and the other three faces have uniform charge density $-{\sigma}$. Find the electric field at the center of the cube.
I have tried approaching it in several ways, that is, using straightforward integration and Gauss' Law. However, I feel there may be other methods and ideas to approach the problem - discussing the same will help us learn more.
If the cube's dimensions were negligibly small, we could've probably approximated the field to be that between oppositely charged capacitor plates, and found the net field at the centre. This problem is different, as there's nothing mentioned which may allow us to work with approximations.
I think this problem is worth asking and discussing on Physics SE because:
It is not entirely specific, it addresses a much larger query, i.e. how to find fields in cases where infinite plate approximations can't be made. I believe such procedures are mathematically rigourous, ones that I may not be familiar with - and would like to learn how it's done.
If we do arrive at the solution, then we can put L→0 and the answer should reduce to the one found using the approximation of infinite plates (the capacitor analogy). This will offer more conceptual clarity, as observing a rather general result boil down to an approximate one is much better than accepting the latter in the first place - it'd help me understand to what extent we're accurate while working under the approximation.
P.S. There's probably some reason why they've asked field at the centre of the cube - symmetry, in my opinion. However, I don't understand how to exploit this symmetry to arrive at the desired result. Please help.
Thanks!
