A version of Ampère's law

Solution 1:

Consider a wire of length $2R$ with a semicircular return loop of radius $R$. You can prove rigorously that as $R \to \infty$ the field at a point near the middle converges as $C + O(1/R)$ to some $C$ (the contribution of distant parts of the wire become negligible). This limit can be taken to be the mathematical definition of what we mean by an "infinite wire". This definition is fine for any physics experiment.

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