I want detailed explaining with equation equations

Usually goes by Wire Gauge, not diameter.
Basically you Start with your load and see what it needs, then you size the wire accordingly and a breaker to protect the wire from burning.
#14 for 15 Amps
#12 for 20 Amps
#10 for 30 Amps
#8 For 40 Amps and so on, and is called Ampacity.
Wire size may need to be increased due to voltage drop if it is a long run, insulation is another factor.
There are Exceptions as well.

Diameter doesn't decide the current rating but the cross section and composition of wire whether stranded,single,flexible as well as its insulation,ambient temperature etc