Or in other words, is there a limit to total battery capacity? Is the limit a physical one or is it an economical one?
Answer
Yes, batteries act like fuel tanks: the bigger the tank, the greater the total energy stored. Also note that different types of AA cells can store different amounts of energy: Lithium cells can store more energy than NiCd cells, etc.
Why does battery size determine the total energy available? Simple answer is that batteries are a type of "Fuel Cell" but with all the "fuel" contained on board. A battery is a pump for electric charges, and when its internal fuel is exhausted, the "electricity pump" stops running.
In a standard fuel cell you'd inject a certain amount of hydrogen and oxygen into the device, which then would send a certain amount of electrical energy out into the connected circuit. But what if you had small tanks of H2 and O2 on board the fuel cell? In that case the cell would only operate until the fuel was used up (and turned into water.) If you ran it at higher power, the fuel would be consumed faster. And a bigger device would provide more H2-O2 fuel in its larger tanks.
Batteries are only a bit different: they typically use solid metals as fuel, turning them into solid metal compounds. The battery weight doesn't change as the fuel gets used up. The fuel turns into solid waste. (This also lets us explain rechargeable batteries: whenever we force electricity backwards through the cell, it converts the waste products back into the metal "fuel" again, and energy is stored, because work is required to force the "pump" backwards.)
And as always, electric circuits become much more understandable once you eliminate typical physics misconceptions:
http://amasci.com/miscon/eleca.html#batt
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