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If I, for example, connect a wire across the ends of a battery (short out the circuit) would the electric field going through this wire be uniform all the way through?

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Yes.

There isn't much of an electric field within a wire, but wires do have a small amount of resistance, so you can think of the wire as a bunch of equal-length short sections, each section of which has the same (very small) resistance. The current through all the sections is the same, so the voltage across each section is the same, and dividing that by the section length gives that the magnitude of the electric field is the same through each section. (Obviously the direction of the electric field will change from place to place if the wire is curved.)

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As measured along the length of the wire, yes. If there's 1A at one end, it will be the same at the other, and at any (macroscopic) point along the way.

As measured along the cross-section of the wire, maybe.

In the case of DC like a battery, I believe it will be fairly uniform, but I can't say I've really looked.

For AC, however, it is decidedly non-uniform. Every time the current reverses direction, 50 or 60 times a second in most cases, there's a time delay that causes eddy currents to form. The action of these currents creates a magnetic field that presses the current out to the surface of the wire.

This is known as the skin effect, and it has real practical outcomes. For instance, most high-voltage wires consist of an inner section of steel cable and then an outer layer of aluminum. The skin effect ensures the current stays inside the highly conductive aluminum, which is rather handy (and cheaper).

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