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 :)

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.

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.

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.

Easy Math Editor

`*italics*`

or`_italics_`

italics`**bold**`

or`__bold__`

boldNote: you must add a full line of space before and after lists for them to show up correctlyparagraph 1

paragraph 2

`[example link](https://brilliant.org)`

`> This is a quote`

Remember to wrap math in \( ... \) or \[ ... \] to ensure proper formatting.`2 \times 3`

`2^{34}`

`a_{i-1}`

`\frac{2}{3}`

`\sqrt{2}`

`\sum_{i=1}^3`

`\sin \theta`

`\boxed{123}`

## Comments

Sort by:

TopNewestWhile 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.

Log in to reply

Please answer now David Sir. There aint any satisfactory answer posted yet.Thanks in advance...

Log in to reply

Thanks, I have read some references but unfortunately did not come up with \(?\) as the answer.

Log in to reply

your link doesn't link to a problem for me

Log in to reply

same here

Log in to reply

Sorry, I think I gave the wrong one. But the staff has fixed it :)

Log in to reply

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.

Log in to reply

You must share the problem and then post the link of the problem to let us see the problem.

Log in to reply

The link has been edited though, go check it :)

Log in to reply

Hi Nabila,

I'm not sure, but I think you might be talking about this problem here. I have edited your link.

Log in to reply

Yes, Nabila is talking about that problem. The link Nabila posted works for me.

Log in to reply

Thanks Peter, for your help :)

Log in to reply

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.

Log in to reply