Multiple stranded cable is used in transmission lines because the high voltage potential of each line causes the electrons to tend to run only on the surface of the line due to repulsion. You want to maximize the surface area of the line, hence the multiple stranded lines, so that you maximize the efficiency of the line. In fact, some transmission lines have a non-conductive core, adding strength while maintaining low weight.
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The minimum size wire that engineers spec for industrial and commercial installations is #12 and it usually is stranded not solid wire.
Either one can be used. Electricians prefer to use a solid wire and use the screw terminals rather that the pressure terminals. An eyelet can be formed with solid wire where as using stranded wire takes a bit more preparation to use. The stranded wire has to be wound tighter at the termination end. This is so that when the stranded wire is tightened under the screw terminal the wire does not splay into individual strands some of which do not get under the terminal screw. With some of the strands not tightened under the screw, this is the same as using a smaller size wire. The breaker feeding the receptacle is matched to a specific wire size. The now smaller wire size has a breaker greater than the ampacity of the wire. This could lead to overheating of the wire under full load and eventually cause a breakdown of the insulation. Once the insulation has broken down, the breaker will trip due to a short circuit. This wire will have to be replaced. This could involve opening the wall board to remove the burnt wiring should this happen.Most residential wiring is done with solid copper wire. In industrial applications, stranded THHN wire is used because many wires are run through conduits from service boxes and many different colors and gauges can be run to different locations. I personally would not use stranded in a residential application.
They should be the same but have different purposes. Stranded wire will flex easier than solid wire. And if one strand breaks there are sufficient numbers of other strands to continue the electrical current. When a solid wire breaks all connection is lost.
Need to know the wire # size to compute wire fill in conduits
Flexibility is the main advantage of stranded wires; however for high frequency AC (i.e. radio frequencies), stranded wires increase the surface area, minimizing the skin effect (i.e. high frequency current avoiding the core of a wire, increasing its resistance at that frequency). This is not an issue at typical power frequencies (e.g. 60Hz) as the skin effect is still insignificant at those frequencies.