10 mm2 cross section should be sufficient (#6 wire?)
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10 kw or 10,000 watts divided by 240 volts equals 41.6666 amps. 8 gauge is borderline so you will need 6 gauge wire to be safe.
A 10 mm cable is most commonly used for a 500 KVA transformer, 240 volt, 3 phrase. The size of the wire that is used in a transformer is based on the voltage.
The question is irrelevant if you don't include the amperage.
Yes <<>> In North America, a three wire 120/240 volt system uses a neutral wire. For 240 volts two "hot" wires are used with no neutral.
A 1/0 aluminum conductor will limit the voltage drop to 3% or less when supplying 30 amps for 500 feet on a 240 volt system.
AWG #3 copper.