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1. No, when you're plugged in and the battery's charged, the computer is powered off of the plug.

2. It's not a very good idea to leave your notebook plugged in ALL the time, so you should calibrate it (run out its battery and charge it back up again) about every month or two. But it won't hurt for it to stay plugged in while you're at home, as long as it's not plugged in for months on end, so your battery's fine, you don't need to take it out. :)
 
yeah, everyonce in a while i drain it all the way and give it a full charge but i was just wondering whats really happening when it is fully charged and just sitting there...
 
For those that run their machine plugged in for long periods of time (days/weeks), I recommend removing the battery and setting it aside somewhere until you need it. I have noticed with my PowerBook, as well as a few Dell laptops I've owned in the past, this can help retain the capacity level of the battery. I've done experiments using coconutBattery to confirme this.

Over a period of about a month, while leaving the battery in the machine but never actually running the machine off the battery (read: always plugged in) I noticed a decrease in capacity of about 4-5%.

This is of course assuming you never calibrate the battery. Calibrating the battery (usually) restores most, if not all of the battery's original capacity.
 
As long as you're calibrating regularly, you should not remove your battery while plugged in. All it does is open the innards of your computer to dust and potential physical damage.
 
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