A copper slug or shading ring also helps prevent buzzing in AC solenoids and relay coils.
Because the AC voltage switches from positive to negative (or crosses zero) twice per cycle, the current through the solenoid or coil goes to zero twice per cycle as the current changes direction. When the current is zero or close to zero there is not enough magnetic field in the coil to overcome the spring tension so the spring starts to pull the plunger or armature back towards its rest position. Once the current has recovered and the magnetic field strengthens the plunger or armature is pulled back towards the coil. The effect is a slapping or buzzing at twice the frequency of the AC voltage (e.g. 120 times per second in the US or 100 times per second elsewhere).
A shading ring or copper slug is designed to eliminate the buzzing. To understand how first consider a simple transformer with two windings which are just coils of wire. An AC voltage applied to the first winding will induce another AC voltage in the second winding. This happens because the applied AC voltage drives an AC current through the first winding which generates an alternating magnetic field. The magnetic field couples to the 2nd winding and induces an AC current and voltage in the 2nd winding.
The important bit about the induced voltage in this case though is that it lags or trails the input voltage in time. In other words the sine waves of the input and induced voltages don't overlap directly. The induced voltage sine wave is shifted to the right so that it is always just a little behind the input voltage. The same is true of the input and induced AC current.
That nugget is how a shading ring prevents buzzing. The main coil gets the input AC voltage and does all the heavy lifting of the solenoid or relay. But it also induces a current in the shading ring just like a transformer does. When the input AC voltage and current drop to zero there is still a small induced current in the shading ring because it is always delayed a little. That delayed induced current provides just enough magnetic field to keep the spring from pulling the plunger or armature away. By the time the induced current has dropped to near zero the main input voltage and current have recovered. So the shading ring just covers the small gaps while the main coil is effectively off.
Both AC solenoids and AC relays coils have a copper shading ring. In solenoids the shading ring is in the coil stop. In relay coils the shading ring is usually mounted to the end of the coil that attracts the relay armature.
And if you're still reading, most AC motors used in EM arcade games also have shading rings wrapped around the frame right next to the rotor. The purpose of those shading rings is to make sure that the motor always starts running in the same direction. Without the shading ring the motor could start in either direction based on whether the AC voltage was rising or falling when it was first applied to the motor.
/Mark