The Aharonov-Bohm effect is proof that there is more to electric and magnetic fields than is generally supposed. You can't calculate the size of the effect on a particle by considering just the properties of the electric and magnetic fields where the particle is. You also have to take into account the properties where it isn't.
Quantum wonders: The field that isn't there - physics-math - 08 May 2010 - New Scientist
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