I am trying to power a 3 Watt light-bulb. I would like to calculate how long that light would run on a triple A battery versus a AA battery.
How do I do that?
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A single battery? Not very long. You can look up battery datasheets. 3W at 1.5V is 2A.
Even with an AA battery, a 2A discharge current is off the chart. Maybe constant power you could get a few minutes. AAA would be worse.
Two fresh top-quality alkaline cells might be able to buy you 1/2 hour operating if you allowed the voltage to drop to 1V (that's constant current discharge, so much of the time you'd actually be getting between 3W and 2W. Constant power or AAA would be worse.
A decent quality AA battery will have about 2400mAh capacity. A similar quality AAA battery will have about 1200mAh capacity. So, for a given load, the AA will last about twice as long.
Your 3W light running off a single 1.5V battery will consume 3 / 1.5 = 2A of current, so will (nominally) last about 2.4/2 = 1.2 hours for the AA and 0.6 hours for the AAA. However, given that 2A is a very high current for these batteries, I wouldn't be surprised if you got 3/4 of that time or less. Another problem is that the voltage will drop quite quickly under that kind of load, so the light will dim quickly as well.
As to how you do it, there are some clues above. You should also look for a datasheet for your particular battery and examine the graphs provided. Decent battery manufacturers will give you useful graphs about how their batteries respond to different loads.