The electrical circuit sounds okay, but the proper way to join (splice) two wires together is either with solder or a crimped barrel connector. You can then cover the splice with shrinkable tubing or wrap it with electrical tape. DO NOT simply wrap the bared ends of the two wires together and cover with electrical tape! Yeah, I know that will usually work, but it is not good practice and it's a bad habit to get into.
Take the time to make a "
Western Union" wrap of the ends of the two wires and apply 60/40 tin/lead rosin-core solder to the join. Then slip a sleeve of shrinkable tubing over the soldered joint and apply sufficient heat to shrink the tubing tightly around the soldered connection. The tubing should shrink to about half its original diameter with very little shrinkage along its length. If you choose tubing that is too small in diameter, it will split when you attempt to shrink it. For solderless breadboard work, the 22 AWG wire should be solid, not stranded. If you do choose to use stranded wire, the ends should be stripped and tinned so the tips do not unravel and splay when inserted in the breadboard.
Back in the day, I had to work with scientists (many of whom held PhD degrees) that knew virtually NOTHING about electricity. One of their "favorite" uses for Variacs was to "control" the speed of gear-reduced synchronous clock motors by "starving" the motors for power. Now this type of motor doesn't use much power to begin with, since it derives it's torque from eddy currents induced in a thin fast-spinning aluminum disk by stator windings, the disk serving as the motor rotor. The lightly loaded disk rotates at a synchronous speed determined by the power-line frequency... when the stator winding is properly energized with AC. These bozos discovered that by reducing the voltage to the gear-reduced synchronous clock motors, they could reduce the output shaft rotation rate. Never mind that they had no idea how fast the output shaft was actually turning since the motors were no longer operating as synchronous motors...
Typically, these motors were installed inside a vacuum chamber used for sputtering. The sputtered material would find its way in a line-of-sight-path to the target specimen and deposit as a thin film. The specimen was rotated with the gear-reduced synchronous clock motor to ensure all sides were exposed to the sputtered material. Problems arose during pump-down and later, when the chamber was brought up to atmospheric pressure. During these two intervals the gas pressure became "just right" to allow a highly conductive plasma to form as a result of the AC applied to the motor. Often the chamber was backfilled with argon after pumpdown to deliberately form a plasma suitable for sputtering. This plasma, intentional or not, caused all sorts of problems when a Variac was used to control the gear-reduced synchronous clock motor..
Since the chamber was made of stainless steel and well-grounded, the "hot" side of the AC line was shorted to ground through the plasma, blowing the fuse in the Variac and sometimes tripping the circuit breaker serving the convenience outlet that the Variac was using. The problem was solved by placing an isolation transformer in the primary of the Variac and NOT connecting the white-wire neutral OR the green-wire ground from the Variac to the deposition chamber frame, which remained grounded to the power-line ground. Thus both leads of the synchronous motor were isolated and very little current actually flowed through the plasma.