The battery is in continuous use, putting out power while the car is on. The alternator returns more power to the battery than the battery puts out, keeping the battery charged. If you have a voltage dial on your dashboard this will show the amount of current being put out by the alternator. It should hold at about 13 and 1/2 volts. If the voltage meter is showing the correct voltage but your car will not start then the problem is with the battery. If the alternator is going bad then the battery will slowly run down; the voltage gauge will start to drop. There is also a warning light indicator for the alternator on the instrument panel. If that light comes on while the car is still running then the alternator is not working. If the light comes on dimly then gets gradually brighter and brighter then your alternator is starting to go bad.
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The alternator is driven by the alternator belt. If this belt breaks then—boom—there is a sudden loss of power as the alternator is no longer moving and will be unable to charge anything. If the battery goes totally dead and after you successfully jump start the car it still keeps working then the alternator must be good as it is keeping the battery recharged. However if the battery is bad and you keep driving on it then you are going to overwork the alternator and it will overheat and burn out. Even a brand new alternator will burn right out. Perhaps the mechanic fails to spot that the battery is no longer re-chargeable. He replaces the alternator which has been burned out trying to drive a dead battery without replacing the battery. In a very short time you once again have a busted alternator.
Note: If the battery goes dead overnight then there is a short in the system or the interior light is jammed on (or you forgot to turn off the headlights!)