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consider a car sitting overnight with the dome light on. wonder how long it would take for the alternator to fully charge that with youre running the air and headlights.
Depends on how many amps the dome light takes and how many hours it was left on..it's a pretty simple problem to figure out once you know that.
Now, let me tell you about the time I couldn't start my 1988 Mustang GT because it had been sitting in my garage for a while, like a couple of months, without being driven. It simply wouldn't turn over. It wasn't that the battery was totally dead, it had just been sitting way too long without being driven.
I didn't have jumper cables long enough to reach and I didn't want to have to remove the battery from my other car to jump it.
I also didn't have a battery charger other than a 1.5amp maintainer.
So I tried connecting a 5-amp 13.8V power supply to the battery. I measured the current it was putting into the battery--it was about 3 amps.
Left it hooked up for about 15 minutes and tried starting the car. It fired right up.
It really didn't take much to get that battery to the point where it would start the car...3 amps over 15 minutes.
As for your other point, some cars do have barely-adequate alternators. The worst of them won't even charge the battery at idle with too many accessories turned on, because the idle output of the alternator is very low.
These are mainly older cars, like my 1988 Mustang GT, that do not have electric cooling fans. This came with a 75-amp 2G alternator. If I turned everything on at idle the voltage would drop as low as 13V. At 2000RPM it was the normal 13.8V.
Ford's replacement for the 2G alternator, the 3G, was designed for much improved output at idle. The 130-amp version puts out 80 amps at idle. (I've never found a specification for what the 75-amp 2G puts out at idle but I suspect it's around 20-30 amps) This sort of output at idle is important for cars with electric cooling fans, since those fans use quite a bit of power and are most likely to come on at idle.
I upgraded the Mustang GT with a 3G alternator when the 2G failed since it's an easy upgrade. Now I can turn everything on at idle and the voltage never dips below 13.6V.
Incidentally, I had a rewound 130-amp 2G alternator on that car for a while and the output at idle on that one was even worse than the 75-amp 2G I replaced it with. So the conclusion I came to is that an alternator needs to be specifically designed for high output at idle. The 3G is physically larger than 2G so it may be that the 2G simply isn't big enough to ever have good output at idle.