I am using a wire that is rated 10Amps at 120v ACwith a car headlight and a SLA battery. About how many amps could this wire carry before getting warm or melting of DC current at 13.8volts?
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Unless you're working with RF (e.g. high-frequencies, > ~100 Khz), or really, really large wire (cross sections in many inches), Amperes are Amperes are Amperes.
As such, a wire rated to handle 10 amps can handle 10 amps, independent of the voltage.
The voltage rating of wire generally relates to the breakdown voltage of the insulation. Basically, with wire rated to 120V, you can be confident that normal handling of the wire while it's energized with 120V will be safe.
The short answer is the same current rating is used for low voltage based on temperature rise. But if you can afford heavier cable for lower power loss, you gain with bettter %load regulation.
The Ampacity of wire is a function of temperature rise for sustained maximum current. This can vary depending if the wire is exposed to air, in a plenum or buried cable. AWG size varies depending on copper, Aluminum or Aluminum with a steel core for towers.
Power being transferred over high voltages and long distance, Utilities can waste 10% of the power to save on wire costs and allow a higher temperature rise such as 45'C rise or in some cases a 60'C rise.
In your case, I assume you selected 10A wire rated at 120Vac, which I believe is based on a 30'C temperature rise. The national electric code (NEC) specifies that the over current protection device not exceed 30A for 10 AGW wire, 20A for 12 AGW wire and 15A for 14 AWG wire. You should know the length of wire to determine the desired % voltage drop and pick a number between 2~10%. Car lights, I read are around 4% drop 1 way in wire.
Here is a chart for 2 wire 10% voltage loss - Wire Length vs Amperage @ 12Vdc.
The current carrying capacity of a wire is related to the size of the wire and not to the applied voltage. You can determine the size of your wire and then reference a standard wire table to get the maximum current carrying capacity for your wire. There are a variety of wire tables on the web and they are easy to find via Google search. Here is one that I have bookmarked in my browser.