electromagnetism - How can there be net linear momentum in a static electromagnetic field (not propagating)?

I understand from basic conservation of energy and momentum considerations, it is clear in classical electrodynamics that the fields should be able to have energy and momentum. This leads to the usual Poynting vector and energy density relations for electromagnetic fields.

However, I do not know how to interpret situations where there is a net linear momentum in a static electromagnetic field. The fields aren't propagating. It doesn't make sense to me that momentum can be divorced from motion.

As a concrete example to discuss: Consider a massless string of length $L,$ with a spherical shell on each end with a magnetic dipole moment m and positive charge q. The radius of the sphere $R\ll L,$ or alternatively, consider the dipoles to be perfect "point" dipoles. Let the string be along the $y$ axis, with one dipole at the origin and the other at $y=+L.$ If the magnetic dipole at the origin is oriented in the $-z$ direction, and the other dipole in the $+z$ direction, if you calculate the total linear momentum in the fields, the answer is:

$$p_\text{em} = m q \frac{\mu_0}{2 \pi L^2}\, \hat{x}$$

While this is an unstable equilibrium, it is an equilibrium. So classically the state can remain static with no need to evoke other external entities, interactions, etcetera. So there doesn't seem to be any of the usual potential pitfalls to save us here.

Please, can someone explain how a static field can have momentum?

Answer

This is a fairly subtle question! Griffiths recently published a paper on this.

Hidden momentum, field momentum, and electromagnetic impulse:

Electromagnetic fields carry energy, momentum, and angular momentum. The momentum density, $ϵ_{0}(E\times B)$, accounts (among other things) for the pressure of light. But even static fields can carry momentum, and this would appear to contradict a general theorem that the total momentum of a closed system is zero if its center of energy is at rest. In such cases, there must be some other (nonelectromagnetic) momenta that cancel the field momentum. What is the nature of this “hidden momentum” and what happens to it when the electromagnetic fields are turned off?

EDIT:
Free version of the above link.