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I know that a magnetic dipole in a uniform magnetic field will precess.

What happens qualitatively (or quantitatively) to a magnetic dipole in a gradient magnetic field? According to Wikipedia, "In a case when the external magnetic field is non-uniform, there will be a force, proportional to the magnetic field gradient, acting on the magnetic moment itself." I'm not sure how to interpret this.

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As the article explains, the force on a magnetic dipole in a non-uniform magnetic field is

$$\vec F=\vec\nabla(\vec m \cdot \vec B)= (\vec m\cdot\vec\nabla)\vec B.$$

The final expression involves the directional derivative of the magnetic field in the direction of the dipole moment. This is what it means by “proportional to the magnetic field gradient”.

If you want, you can write this as $\vec m\cdot(\vec\nabla\vec B)$ where the parenthesized expression is a tensor which is indeed the gradient of the magnetic field. The gradient of a scalar field is a vector field; the gradient of a vector field is a (rank-2) tensor field.

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  • $\begingroup$ In an uniform magnetic field $\vec B$, is the $\nabla\vec B$ always 0? $\endgroup$
    – Loris Foe
    Commented Nov 19, 2021 at 0:40
  • $\begingroup$ @LorisFoe Yes, the gradient of a uniform vector field is always 0 everywhere, so while a non-zero spatially-uniform magnetic field does exert torque on a dipole, it does not exert force on it. $\endgroup$
    – cameronroytaylor
    Commented Dec 19, 2022 at 14:44

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