The wire gauge in thin headphone cables is quite small. I have read of people rewiring their headphones using 12 Gauge wire. There are commercial cables that use 11 gauge. The thicker the wire the less the resistance up to a point. Many headphones use 22 or 24 gauge and it seems to work OK.
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Standard 23 gauge wire is (0.0226 inches/0.57404 mm) thick in diameter.
In normal residential use it is 20 Amps. That depends on alot of things such as the wire insulation, ambient temperature, etc. Standard NM 12/2 (Romex or equivalent, which is likely what you are asking as it is the primary type used to wire residential) is rated for 20 amps, 14 gauge is 15 amps, 10 gauge is 30 amps.
Beats by Dr Dre..
if you know the gauge of the wire you can learn its current carrying capacity
remove it and twist a similar gauge conductive bare wire into the same shape. then install