Accelerating electromagnetic system

It will not. The force is from the device back onto itself. Since it isn't exerting a force on anything not part of itself, and it isn't ejecting any mass as propellent, it will not accelerate. It's like connecting a magnet to the end of a stick and holding in front of a wagon, you aren't going to accelerate.

The electromagnetic field can be used to transfer momentum. Could this explain what is happening here?

For example, a solar sail uses momentum from photons (electromagnetic disturbances) to accelerate.

Could this be what is happening here?

Edit: Also what is the source of the force that counteracts the force on the wire? Where is it felt?

No, a solar sail works because the Sun has already emitted the light. The recoil from emission is felt on the Sun, and the photons transfer momentum into the sail through absorption and reflection.



On the magnets. The coil feels a force in one direction, and the magnets feel the force in the opposite direction. They cancel out.

magnetic forces always come in equal and opposite pairs, just like gravity and electrostatic forces. It can be less obvious, because often, we don't specify the exact form of the magnet.

Can you link me to a website that has information about an equal but opposite force to the Lorentz force? I've never heard of this before.

Newton's third law.
http://en.wikipedia.org/wiki/Newtons_laws

Then where is the momentum for the coil coming from? It's got to come from somewhere, it can't just appear. The EM field can transfer momentum but it cannot create it.
Per the third law, the coils feel a force due to the magnetic field of the magnets. In turn the magnets feel a force from the magnetic field of the coil in the opposite direction.

There is no change in total momentum. The system gets some momentum, the electromagnetic field gets some momentum, and everybody is happy and Newton's third law is satisfied.

I'm sorry, momentum CAN be created, but it is conserved. Not sure what I was thinking.
However, the force exerted on the coil MUST have an equal and opposite force exerted on the magnet.

That is the electrostatic force, static being the key word here. It implicitly assumes an action at a distance (i.e., infinitely fast) kind of force and it is not a complete description of the electromagnetic interactions when the particles are in motion.

Newton's Third Law can fail in a number of cases:

• There is a time delay in the equations of motion, such as is the case for electrodynamics (as opposed to electrostatics). What is happening here is that the field that mediates the interaction is itself storing momentum. There is no room for such in Newton's 3rd. As mentioned before, this can be reconciled by observing that momentum is still conserved. Newton's 3rd law is conservation of momentum in the special case that forces are instantaneous and central in nature.
  
  
• The force is not central in nature, which once again is the case for electrodynamics. In the strong form of Newton's third law, third law force pairs must be equal but opposite in nature and the force must be directed along or against the line connecting the pair of particle. This form of Newton's third law conserves both translational and angular momentum. Translational and angular momentum can still be conserved in the case of non-central forces if the mediating field stores these momenta, but Newton's third does not apply in such cases.
  
  
• The underlying interaction inherently involves three or more particles. Newton's third demands that forces be resolvable down to pairs of particles. There are some multi-body interactions in quantum mechanics where the interactions only appears when three or more particles are present. These interactions cannot be isolated down to pairs, and once again Newton's third law fails.

In more advanced physics, it is the conservation laws that reign supreme. Newton's third law derives from the conservation laws with the assumption that forces act in pairs, act instantaneously, and act along the line connecting particle pairs. Drop those assumptions and you have to drop Newton's third law. You do not have to drop the conservation laws, however. In even higher level physics, the conservation laws themselves can be derived from the very nature of space and time.

In this case, and in many of the simplest EM problems, magnetic forces come in 'near-instantaneous' pairs.