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I know that the magnetic moment of a particle is given by:

$\vec{\mu} = \frac{gq}{2mc}\vec{S}$

I know that in the Stern-Gerlach experiment, neutral silver atoms were used. Additionally, the deflection in this experiment was due to the force $F = \nabla (\vec{\mu} \cdot \vec{B})$.

How is a nonzero force experienced, given that $\vec{\mu}$ is dependent on charge $q$, which is zero for silver atoms.

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You're really asking how there can be a magnetic moment $\vec{\mu}$ when the atom has zero charge.

Moments are, in general, about separation. For example, consider two forces that are equal in magnitude and opposite in direction: The net force is zero. But if they're applied at different points, they still provide a moment, i.e. torque, and can cause something to spin.

An electric dipole is two separated charges, usually opposite charges. They sum to zero charge, but because they're separated they have some moment.

Same for a magnetic dipole. The classical form is a current loop. The total system is net neutral (the wire has moving charges and opposite fixed charges so as to be neutral), but the motion around the extended loop creates a dipole moment.

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The Stern-Gerlach experiment is famous because it verified quantization of angular momentum in quantum mechanics. However, your question is really a question about classical electromagnetism. For example, you can replace the silver atom with a loop of wire carrying a current, and the question is the same: why does the loop experience a nonzero force, when its charge is zero?

The answer is that when you have a mixture of particles with different charge-to-mass ratios, the relation between the magnetic moment and the angular momentum doesn't have to hold. It holds only for each type of particle, not for the aggregate.

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