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I'm taking an analytically simple geometry here, an infinite, straight
wire of resistance per unit length r, carrying a DC current i*.
You may
approach this situation as nearly as you wish in practice, say with wire
wrapped around the Earth's equator, or even more nearly with gedanken
materials.
I'm also taking the DC case, so of course the part about
predicting the future is nonsense, but the reification of the energy
flow is the most condemnable nonsense; that's my point.
David is correct in saying the flux does not extend to infinity in the
case of a finite circuit.