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

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.

It is possible to show that the total momentum of any static system is zero in an inertial frame where nothing is moving. This does not mean that the momenta associated with various components of that system are individually zero. As you point out, there can be finite electromagnetic momentum associated with static charge distributions. Even though there is no obvious motion in the system, the momentum associated with the matter distribution is actually nonzero. It is equal and opposite to the electromagnetic momentum.

This is often referred to as the hidden mechanical momentum. It is a special case of a much more general result that the net momentum of an extended object need not be parallel to its center of mass velocity.

Electrodynamics books like Griffiths or Jackson have a nice microscopic interpretation for this effect in the simple case of a magnetic dipole placed near a charge. Internally, the dipole may be thought of as containing a current loop. The charges in this current loop accelerate and decelerate in response to the external electric field. One may show that this gives them a net momentum that is exactly equal and opposite to the electromagnetic momentum. Note that this is an intrinsically relativistic effect. It does not arise if Lorentz factors are neglected when computing the momenta of the circulating charges.

Nonzero Momentum in static field configuration is actually a good thing. Consider a coaxial cable carrying DC current and voltage. Internally it has a constant E and H field. The Poynting flux is nonzero and shows that there is energy transport. The energy flow is indeed in the direction ExH.

The fact that the field configuration shows no movement is irrelevant. Momentum is moving energy, not something else moving!

-- Jos