If a trace can carry 3A for a temperature rise of 20°C, will adding forced air over the board increase current capability (by any factor)?
Is the current limit defined by the heat or is it the properties of the material (geometry, type, etc..)?
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Yes (forced air increases current capacity), and yes (the current limit is due to power dissipation).
Note that this is from the board house point of view. From the EE point of view, voltage drop across a trace can also be important. Air cooling doesn't do much for that.
I'm going to throw in my $0.02 and say that increased temp => higher resistance => higher voltage drop, and correspondingly cooling (via airflow or any other mechanism) should result in a lower resistance and therefore either less voltage drop or higher current capacity.
I initially thought that the difference would be trivial, but according to a quick google search - [sarcasm]100% guaranteed to be correct![/sarcasm] - changing temp from 40C to 20C should reduce copper resistance by ~8%. That's not huge, but if true then it's way too large to be ignored either.