a low battery that wont start a car will also screw up an alternator after you jump the car. The alternator is designed to put out 13.5 amps to maintain a battery charge after cranking the car. It actually takes about 7 miles of travel to recharge the battery to what was lost cranking the car. However if a battery is very low on charge and you jump the car to start it the battery isn't what is keeping the car running but rather the alternator is trying to do both keeping the car running and trying to charge a dead battery. This causes the internal voltage regulator of the alternator to burn up inside the alternator as it is charging beyond 13.5 amps. Sounds like you have a bad alternator due to a poorly charged battery. Both must be good . You can have the battery charged and alternator tested. Or Battery charged and remove the positive from the positive post while it is running to see if it dies. If it does the alternator is bad. Keep in mind if it doesn't die reconnect the positive quickly as this is heating up the internal voltage regulator and will burn it up just as a dead battery will. If it doesn't die the alternator is good. If you don't know why the battery is loosing power find out why by doing a draw test if the battery is good.