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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:

  1. 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.

  2. 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!

Abhinav
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stoic-santiago
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  • Do you mind providing a link to the question, where we can verify that it's not a current olympiad question? I apologise for having to ask this, I'm definitely not casting doubt on you in any way, but this should be checked. – Melvin Apr 25 '18 at 06:08
  • Don't have the link, just found some bits and pieces of old, worn out paper while cleaning my room. That's where I found this. – stoic-santiago Apr 25 '18 at 08:27
  • Related? https://physics.stackexchange.com/questions/176747/electric-field-due-to-a-uniformly-charged-finite-rectangular-plate – Farcher Apr 25 '18 at 09:00
  • Yeah, it's related - but it doesn't answer all parts of the question, it addresses only mathematical rigour, not physical elegance (or Gauss' Law, for that matter) – stoic-santiago Apr 25 '18 at 12:30
  • Thanks for highlighting the possibility, but the fact that it's not a duplicate has already been addressed in the comments section. – stoic-santiago Apr 25 '18 at 14:49
  • It tells you how to do the calculation for one face. Repeat 6 times to get the total. This is not a problem-solving site, so requests for elegant methods are not appropriate. How do you know this was not the solution intended? The centre of the cube is on the axis of all 6 faces. This makes the calculation easier than for any off-centre point. – sammy gerbil Apr 25 '18 at 14:57
  • @sammy gerbil, I understand what you're trying to say, however, what you're doing will only inhibit further discussion of the question and prevent the community from discussing other possibilities. The right question is one that I should ask you, "How do you know that this is the intended solution?" After all, assuming that it is the intended solution does no good, while looking for more and exploring always does. – stoic-santiago Apr 25 '18 at 15:03
  • This is not a discussion site. Questions are expected to have one definite answer, not many equally valid answers. If you wish to discuss problems you should post in a ChatRoom, eg Problem Solving Strategies or the hBar. – sammy gerbil Apr 25 '18 at 15:12
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    @sammygerbil Nope, there is supposed to be a simple elementary way to do this one. It's not the same kind of question. – knzhou Apr 26 '18 at 10:34
  • You are close to a solution when you write ... we could've probably approximated the field to be that between oppositely charged capacitor plates, and found the net field at the centre. – sammy gerbil Apr 30 '18 at 00:38
  • Also related: What is the electric field flux through the base of a cube from a point charge infinitesimally close to a vertex?. – Emilio Pisanty May 04 '18 at 21:49

1 Answers1

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If you draw a line from the vertex with the sides carrying positive charge to the opposite vertex with the sides carrying all negative charges it would pass through the center of the cube. Now you can rotate your cube along this axis and it should not affect the value of the electric field in the center of the cube. Now you can draw a thin cylinder along this line and apply Gauss's law

Rafik Fayzulin
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