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Experiments used to observe particle spin properties (such as Stern-Gerlach) rely on a varied magnetic field and a dipole-like reaction in the particle, deflecting it in one direction or another.

In the case of a point-particle such as an electron, how is it explained that a dipole with spatially separated poles exists in a single point?

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Classically a non-pointlike spinning charged object possesses a magnetic dipole moment due to the fact that charged particles in the object are spinning around some axis. In contrast, the electron has a dipole moment that arises from its intrinsic spin angular momentum. As you point out, the electron has no internal structure, so the spin does not refer to actual physical spinning. The dipole moment has spatial dimensions outside of the point where the electron exists because it arises from the quantum spin property of the electron, it's not itself a property of the electron.

The magnetic dipole moment of the electron is related to its spin in the way described here.

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In addition to joshphysics's answer, the gyromagnetic ratio of an electron which may be measured experimentally proves that it is not the rotating dipole.

Dipole momentum of an electron may also be measured experimentally. And, with a good precision is zero.

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