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The formulae for the force on an electric dipole and the force on a magnetic dipole seem to be closely related:

$$\mathbf{F}_{e} = (\mathbf{p} \cdot \mathbf{\nabla}) \mathbf{E}, $$

$$ \mathbf{F}_{m} = \mathbf{\nabla}(\mathbf{m} \cdot \mathbf{B}) $$

where $\mathbf{p}$ is the electric dipole moment and $\mathbf{m}$ is the magnetic dipole moment.

However, we cannot rewrite $\mathbf{F}_e$ generally as $\mathbf{\nabla} (\mathbf{p} \cdot \mathbf{E})$, so these two formulae are not analogous. Why must the $\mathbf{m}$ be operated on by $\nabla$, but the $\mathbf{p}$ must not? What fundamental difference between $\mathbf{p}$ and $\mathbf{m}$ causes this?

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  • $\begingroup$ Remark: $\nabla$ does not operate on $\vec m$ either, as $\vec m$ is not a function of space. $\endgroup$ Nov 23, 2015 at 17:30

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They are the same (at least in the static limit). Using the rule $\vec a \times (\vec b \times \vec c) = \vec b (\vec a \vec c) - \vec c (\vec a \vec b)$ one can write: $$\vec F_m = \nabla (\vec m \cdot \vec B) = \vec m \times (\nabla \times \vec B) + (\vec m \cdot \nabla)\vec B.$$ The first term is zero, as $\nabla \times \vec B = \frac 1 {c^2} \partial_t \vec E + \mu_0 \vec j$ and

  1. These equations only hold in the quasi-static case anyway (that is, we ignore radiation effects, which amounts to approximating $\frac 1 {c^2} \partial_t \vec E \approx 0$),

  2. The $\vec B$ in this equation is actually the external magnetic field (not including the field of the dipole which is divergent at the dipole's position), therefore the source term $\vec j$ will be zero.

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