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In this experiment we only consider the force at z direction, but $\vec B$ field gradient doesn't exclusively exist at z direction according to Maxwell's equations. So why don't we see the splitting in other directions?

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up vote 3 down vote accepted

The point made by Otto Stern in the original publication (german) is that the contribution of $\frac{\partial \vec{B}}{\partial x}$ and $\frac{\partial \vec{B}}{\partial y}$ can be neglected when averaging over a suitable period of time.

In that case the force, aka the negative gradient of $\vec{\mu}\cdot\vec{B}$ is only relevant along the z-axis, but technically you are then talking about an time-average of the force.

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