Greg Gregorich :
two simple alternator checks 1. is start the car (in a dark time dont recommend doing this in not ventilated areas like garage or small werhouse ) turn lights on find switch for defrosters fan turn high beams should show blue light on dashboard of the car then turn defroster fan on high if the lights DIMM A LOT that sign alternator problems 2. start the car remove positive or negative cable from battery if the engine shuts off you have problems with alternator
Never disconnect the battery on a car that is running, This could cause thousands of dollars of damage. Don't let anyone ever give advice or work on your car if he feels this is okay. Cut and pasted from Steve Litt who says it a lot better than I do.
"Your battery does more than just provide electricity. It also shorts AC, spikes and transients to ground. Removing the battery from the circuit allows those spikes and transients to travel around, endangering every semiconductor circuit in your car. The ECU, the speed sensitive steering, the memory seat adjustments, the cruise control, and even the car's stereo."
I have known someone who tried this and they got off easy with a $250 PCM rebuild. It's not the 70's anymore, this is just not done.
My bet is on the battery failing, occasionally new ones do fail as well, but it's rare.
Charge up the battery, start the vehicle and measure the voltage. If it's above 14 the alternator is good, if it's around 12, the alternator is the culprit (or wiring, but that's easy to check, don't forget to check the ground from the engine, but a multimeter is an easy way to verify the cables are good)
If the voltage is 14, then it's most likely the battery, but could be other things as well. Does the jeep have an aftermarket stereo and amplifier installed? were the lights left on, is something else draining the battery..