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We know that the skin effect is the tendency of current to flow mainly near the surface of a conductor that happens when it is driven by a time-varying current. This situation is usually explained in case of a cylindrical conductor, like that shown in the following picture at left.

enter image description here

But in some situations, our conductor is so thin to be approximated as a foil (2D), for instance in case of strip lines, patches etc. Let's consider a rectangular foil as that shown in the picture at the middle. Since it's a foil, it does not have (ideally) a cross-section. Will the AC current flow uniformly along all the area of the foil, or in this case does the skin mean that AC current will flow in the edges of the foil (which are segments)?

I'd say that current will flow uniformly on all the area of the foil. But someone told me that in this case, the current on the edges is more important.

Picture taken from here.

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Yes, as you suggest, edge current density is greater than center-of-foil current density.

Skin effect comes from the magnetic field that a current generates, and which impinges on adjacent conductive material. Conductive materials are often diamagnetic (resist the penetration of magnetic field) and at high AC frequencies, are very much diamagnetic.

The magnetic field generated by an edge-of-foil current only impinges on adjacent metal on ONE SIDE of the current, while the magnetic field generated by a center-of-foil current impinges on two sides. The foil-center current is lessened by this, because the magnetic field generates an opposing electromotive force. It isn't as simple as a 'skin depth' situation, because the highly symmetric effect on a cylinder conductor can be approximated easily, while the strip of foil has no such symmetry.

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