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# Past brilliant physics problem about Euler's Characteristic?

Someone please help me understand this problem. I thought there are no well-explained answers from the user and also brilliant staff did not give any single comment on that problem. Thanks for your help :)

Note by Nabila Nida Rafida
3 years, 11 months ago

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While discussions about specific problems should go in solution discussions, Nabila is correct that the solutions given in the discussions may not be very enlightening to the layperson. Hence feel free to discuss it here. If no answer is forthcoming I will answer in a day or so. Staff · 3 years, 11 months ago

Thanks, I have read some references but unfortunately did not come up with $$?$$ as the answer. · 3 years, 11 months ago

If I'm not mistaken, this seems to be an application of the Gauss-Bonnet theorem - the fact that the test charge accelerates away at a point tells us something about the curvature there.

It feels a little strange to include this problem for Brilliant. · 3 years, 11 months ago

same here · 3 years, 11 months ago

Sorry, I think I gave the wrong one. But the staff has fixed it :) · 3 years, 11 months ago

My intuition on that problem is as follows: We know that charges accelerate when subjected to an electric field, but the only field source in this universe is your charge dq. Therefore, it must somehow be subject to its own field in a non-symmetrical way. The only way this could be is if the electric field came back upon dq as it went to infinity, implying that we have to be dealing with a shape with 1 face (if it has two faces, a similar self-mapping occurs, but it will always have a zero point, see the Hairy Ball Theorem). Since all topological spaces with 1 face (mobius strip, klein bottle, etc) have characteristic 0, we have our solution. · 3 years, 11 months ago

Hi Nabila,

Thanks Peter, for your help :) · 3 years, 11 months ago

Yes, Nabila is talking about that problem. The link Nabila posted works for me. · 3 years, 11 months ago

You must share the problem and then post the link of the problem to let us see the problem. · 3 years, 11 months ago