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If we use Ampere's law (with a circle of radius R centered on a wire with the normal of the circle parallel to the wire) then we can get the same relationship much more easily.

However, when we use Ampere's law we are not assuming anything about the length of the wire, it can be 1 cm as long as it goes through the circle.

But if we did not integrate the Biot-Savart law from -∞ to ∞ then we would not get this result. Yet Ampere's law implies that it's true for all wires regardless of length (with a restriction on the points in space for which it applies, of course).

What am I missing?