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The Aharonov-Bohm effect is often presented via a region in space where the magnetic field $\textbf B=0$, but the vector potential $\textbf A \neq 0$. Usually, this is motivated via an infinitely long solenoid, which is said to have a vanishing magnetic field outside the coil.

However, if the solenoid becomes infinitely long, it appears to be more and more like an infinitely long wire, which does have a magnetic field surrounding it. So my question is: How can we really prepare a region where $\textbf B=0$, but $\textbf A \neq 0$, considering that an infinitely long solenoid coil looks like a wire?

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Experimentally, you're never going to get field that's exactly zero. However, it would help to have 2 coaxial solenoids, one wound right on top of the other. Current flows up through one and back down through the other. The solenoid-like field is the same for both, but the wire-like field is opposite and cancels out.

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