The power to the garage door opener should be supplied by AWG 12/2 wire on a 20 amp breaker protected by a GFCI. The wire going to the sensors on each side of the door can be door bell wire.
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The size of the wire is stated by its gauge under American Wire Gauge. Six gauge wire is size 6 AWG.
You listed no gauge wire. This is the required breakers.14 gauge - 15 amp12 gauge - 20 amp10 gauge - 30 amp8 gauge - 40 amp
No - the two have no relationship to each other at all.
The American Wire Gauge (AWG) is a shorthand way to identify key characteristics of the size of a wire and pertinent specifications associated with the size. So you will often hear someone ask what gauge wire do I need for a 15 Amp circuit in my home. The shorthand answer in this case would be 14 AWG for a typical residential wiring job.
The wire size depends on how much current it will conduct.