You can run it as far as you want, knowing that the farther you go, the more voltage drop will occur under load. If you have a very small load, you can run 14 AWG for hundreds of feet.
For instance, 14 AWG copper THHN 90C in conduit with 3 percent voltage drop would get you over 600 feet at 1 amp, but only about 50 feet at 15 amps.
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10 AWG has a resistance of .1 ohms per 100 feet. As a rule of thumb you want less than a 10% drop in voltage because of wire. You use Ohm's Law to calculate the allowable drop. Volts = Amps x Resistance. The maximum amps for 10 AWG is 30 for typical household applications. So for example a 120 volt circuit could drop only 12 volts because of wire length (this is worse case) so R = 12/30 = .4 ohms. So dividing the .1 per 100 ft the result is 400 feet. For most applications I would recommend only about 250 ft.
Around 75 feet is all I would run it. The NEC recommends you never run the wire so that you have a voltage drop of over a 3.6 volts. If the run is over 75 feet then I would switch to 12 gauge on a 20 amp breaker just in case of a brownout of heavy load applied to that circuit. You cannot go wrong running 12 gauge but you can running 14 gauge.
Depends on voltage and the length of the wire. The longer the wire, the more resistance, the more heat generated, etc.
Yes, you can run electrical wires in a load bearing wall.
14-3 Is the standard wire use for residental smoke detectors.
10 AWG.
According to the 2008 National Electrical Code, the smallest wire that can be run in parallel is a 1/0 AWG (American Wire Gauge). If the wire you are using is smaller than a 1/0 AWG, you will need to increase the size of the wire to handle the amperage (Table 310.16 of the NEC).
12 gauge underground wire or if you think you will ever add any devices to this circuit use 10 gauge.