1987 Ford F-150 ALTERNATOR RUNS LIKE MOTOR

Having electrical problem. Starter burned up and was replaced. Immediately the new solenoid also burned up on the new starter. Replaced it with the old solenoid and worked fine for 2 weeks. Then the old solenoid burned up. When I cut off the engine I noticed the alternator was running with the engine off. I disconnected the battery cable to get it to stop running. What's wrong?

Let me revise the above description. This is my brother's truck. He, like me, is a mechanical idiot. What burned up appears to be the relay, and not the solenoid, as it is mounted on the fender well and not on the starter. He did say that with the engine off and the ignition off the alternator was turning and the belt was also turning but the engine was not turning over. He replaced that relay (mounted on the right fender) with one off another 1987 Ford truck that he has and has had no problems since.