Flux through a Mobius strip

I was sent here from mathoverflow, hoping for a complete answer to this:

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A friend of mine asked me what is the flux of the electric field (or any vector field like $$\vec r=(x,y,z)\mapsto \frac{\vec r}{|r|^3}$$ where $|r|=(x^2+y^2+z^2)^{1/2}$) through a Mobius strip. It seems to me there are no way to compute it in the "standard" way because the strip is not orientable, but if I think about the fact that such a strip can indeed be built (for example using a thin metal layer), I also think that an answer must be mathematically expressible.

Searching on wikipedia I found that:

A Möbius resistor is an electrical component made up of two conductive surfaces separated by a dielectric material, twisted 180° and connected to form a Möbius strip. It provides a resistor which has no residual self-inductance, meaning that it can resist the flow of electricity without causing magnetic interference at the same time.

How can I relate the highlighted phrase to some known differential geometry (physics, analysis?) theorem?

Thanks a lot!

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Now, I'm convinced that there are no way to apply Gauss law since there are no ways to bound a portion of space with a Mobius strip. Nonetheless I would like to "see" some equations showing me that "[the mobius resistor] can resist the flow of electricity without causing magnetic interference at the same time".

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Hello tetrapharmakon and welcome to the site. Imho this is a nice question since one has to think about some fundamental concepts. You might also want to look at prb.aps.org/pdf/PRB/v79/i3/e035321 for another application of the Möbius strip :) – Robert Filter Jan 5 '11 at 10:41
This is a great question! – user346 Jan 5 '11 at 11:13

Assuming the material of the strip is a good conductor any charges, and currents, on the strip must end up on the boundary. This is true at least in the limit of small applied e.m.f So its not really Gauss', but Ampere's law you're looking for here:

$$\int_{\partial M} I \cdot \mathrm dl = \Phi_B$$

i.e. the line integral of the current $I$ around the boundary $\partial M$ of the Mobius strip $M$. This quantity is actually a topological invariant of the manifold (in this case $M$) under question. For the Mobius strip it is as can be seen to be zero by starting at any given point on the boundary and evaluating the integral as one moves along the edge for one complete cycle.

Note: Some of the arrows are incorrect. Think of the strip as a freeway with the vertical axis running through its center, where cars drive only on the edges. You can divide the edge into parts "upper" and "lower", then the current should be circulating one way in the "upper" half and the opposite way in the "lower" half, changing directions at the place where the "interchange" is located. Label accordingly.

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The currents in the picture are correct. The "inner" part is in contact with the + , the "outer" with the -. At the twist, there is a switch between "inner" and "outer". – Raskolnikov Jan 5 '11 at 12:37
The arrows are incorrect if you take the current in the strip to be running along the edges, instead of in the interior of the strip itself. So, I guess they are correct in the original context of the figure, but incorrect for my purposes. – user346 Jan 5 '11 at 13:47
I didn't understand if you're considering the currents going through the boundary of the strip or along the z-axis (I suppose that in that picture -I found it on wikipedia too- the central circle of the strip lies on the plane xy, and its center is in the origin (0,0,0)) – Fosco Loregian Jan 5 '11 at 14:11
I mean, along the axis through an infinite line crossed by current – Fosco Loregian Jan 5 '11 at 14:12
@Tetrapharmakon yes that is the orientation of the strip. The strip lies in the xy plane and the z-axis is the vertical direction. – user346 Jan 5 '11 at 14:22