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Normally, no. But at really high frequencies they do. As we increase the frequency in AC (alternating current), the current begins to shift from an "evenly distributed" flow to something else. The center of the conductor will begin to carry fewer electrons and the outer surface or "skin" of the conductor will begin to carry more. It isn't surprising that we call this phenomenon skin effect and we see it in high frequency applications. In really high frequency circuits, particularly high power circuits, we see "wire" being substituted with a wave guide to carry the high frequency, high amperage currents used in applications like radar. The household microwave oven uses a waveguide to carry the power into the cavity where it heats what you put in. For anything the AC power grid, which works at about 60 cycles per second (60 Hertz) in America, there isn't a substantial problem with this phenomenon. A link is provided to the Wikipedia article on skin effect.

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