Flux through a M\"obius strip?

TL;DR

This paper explores the implications of non-orientable surfaces like the Möbius strip on flux calculations and integral theorems in physics, highlighting conceptual and experimental considerations.

Contribution

It clarifies how flux and integral theorems apply or fail on non-orientable surfaces, providing insights into physical measurements and theoretical understanding.

Findings

Flux cannot be computed using Stokes' theorem on non-orientable surfaces.

Experimental setups can measure flux through non-orientable surfaces despite theoretical limitations.

The conceptual puzzle about flux on Möbius strips is resolved by examining the role of orientability.

Abstract

Integral theorems such as Stokes' and Gauss' are fundamental in many parts of Physics. For instance, Faraday's law allows computing the induced electric current on a closed circuit in terms of the variation of the flux of a magnetic field across the surface spanned by the circuit. The key point for applying Stokes' theorem is that this surface must be orientable. Many students wonder what happens to the flux through a surface when this is not orientable, as it happens with a M\"obius strip. On an orientable surface one can compute the flux of a solenoidal field using Stokes' theorem in terms of the circulation of the vector potential of the field along the oriented boundary of the surface. But this cannot be done if the surface is not orientable, though in principle this quantity could be measured on a laboratory. For instance, checking the induced electric current on a circuit along the boundary of a surface if the field is a variable magnetic field. We shall see that the answer to this puzzle is simple and the problem lies in the question rather than in the answer.