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Fields due to Möbius Strip

Physics Asked on January 4, 2022

Is there any way to calculate the following from a möbius strip?

(a) Electric Field :Given that the strip is an insulator and has localized charge uniformly distributed over its surface.

(b) Magnetic field generated :Given that the strip has a uniform current flowing on its surface.

And calculate equipotential surfaces/points if any?

A link to the shape of the curve
https://www.desmos.com/calculator/6zkdcha8my

I am more interested to know the method of solving such a question than the actual formula so obtained. By method, I mean that how one should consider the twists and evaluate a field integral over it.
My teacher suggested to use angle of rotation but didn’t explicitly solve the question post the suggestion. I know about the raw formulas involved but not how to specifically evaluate it. One might use curl and divergence as per their ease.

One Answer

The answer to this question can be found on https://math.stackexchange.com/a/3769396/811225 as well.

To quote it,

'The issue here is that the Möbius strip is not orientable. The only global differential of area it has is scalar, not vector, so you cannot take the dot product with respect to it.

Now you can take a double cover of the strip and have a vector differential of area over that double cover. But then the integral is always 0 since you cover every point twice, with the area vector changing signs between the two visits.

Or you can effectively cut the Möbius strip, treating it as just a twisted rectangle in space, and accepting there is a discontinuity at the cutting line. The result you get will vary on where you make that cut.'

Answered by Dorothea on January 4, 2022

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