When I write about the various methods to actuate an air valve I use the term actuator to refer to the electrical or mechanical device that shifts the internal workings of an air valve. I use the term operator to refer to the person that moves the lever or pushes the button to actuate the valve. Here is some background information on air valves on this site if you are interested.
The other is covered on this page on this site - the Air Pilot Solenoid actuator.
Motion in what? The item the solenoid magnetic field moves is a tiny valve to control air to an actuator on an air valve.
More information follows as we first talk about what a direct acting solenoid is.
In the drawing above, item #1 is the coil, and inside the black exterior are the coil windings are depicted by the gold color.
Item #2 is the compressed air supply to this valve. It is a 2/2 normally closed valve. This is a link to information on the 2/2 air valve on this site. For purposes of this page, the 2/2 refers to a two position, two ported valve.
Note how the compressed air enters the valve through the supply port, and is blocked from flowing through the valve by a poppet (purple color). The poppet is sitting on an "O" ring, thus sealing the incoming air stream from flowing through the valve and out the other port.
The coil on this valve drawing is de-energized, meaning that electricity is not flowing to it, and the internal valve actuator, the spring, is in control at this time, shutting off the air flow.
The solenoid actuator, when it is energized with electricity, creates a magnetic field. That electrical field will move the pole piece (item #3 - brown color) inside the coil housing. In this drawing the pole piece will move to the right. Since it is attached to the poppet inside the valve, when the coil is energized and the pole piece is moved by the magnetic field, the poppet will move along as well.
Item #4 depicts the outer valve body. The arrow is pointing towards the internal valve actuator, the coil spring.
When the solenoid energized the coil creates a magnetic field pulling the pole piece to the right as shown in the drawing just above.
The poppet, connected directly to the pole piece, also shifts (#3), moving off one seat and onto another, now allowing the compressed air to flow through the valve.
Item #2 depicts the inner valve actuator, the coil spring, which has now been compressed by the force generated by the coil, and the movement of the pole piece and the poppet.
When this direct acting solenoid valve is de-energized, the return spring shifts the poppet and the pole piece the other way inside the valve, and the compressed air is, once again, blocked.
But, why not just use solenoids to do work instead of air cylinders?
While a solenoid coil can generate sufficient force to move a small, very light object over a short distance, the movement and the size of the item being moved is limited by the relatively limited force from the coil.
It would take a huge electric coil to move any meaningfully sized device over a distance greater than about one-tenth of an inch. In order to have a coil actuator that would do larger physical work, like moving some heavy tooling 18" for example, an air cylinder using the power of compressed air can easily do so. A solenoid coil needed to generate that force over that distance would completely dwarf the tooling and maybe even the size of the machine it was situated on. Electrical solenoids for moving larger items over longer distances are not at all practical, and would be very, very expensive.
Having said that, here's how we move larger objects over longer distances, using a solenoid. See direct acting solenoids on this page.