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Will I damage my alternator by using it to charge a flat battery?
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Yes, you will cause (or at least risk) damage to the alternator if you use it to charge up a flat battery, or a battery that's deteriorated enough that it can't accept a full charge.

Here's a good description of why, from Mike Pardee:
"Bad batteries take a lot of current trying to pretend they are okay. As they
get weaker the battery voltage deteriorates, from failure to hold a charge
and sometimes from shorted cells. Either way, failing batteries (or dead
batteries) cause the alternator to put out maximum current for progressively
longer times in order to bring the voltage up to the regulator level.

"Alternators are odd things - by design, they will only put out so much
current for a given field excitation. The ratings you see on alternators are
those maximum ratings for about 14 volts on the field. The way that works is
that the voltage they produce is proportional to rpm, and the frequency at
which they operate internally is proportional to rpm. Since the stator
windings are inductors, their reactance is proportional to the frequency (in
turn proportional to rpm) so the maximum current the alternator will put
out, even into a short circuit, is limited.

"In the '60s, the alternator could put out the full rating (usually 35 amps)
indefinitely. They were big, Tim Allen devices that scarcely ran warm at
full output. Those days are long gone. Modern alternators are much smaller
and are called on to put out much more current - often 60 to 100 amps. With
a good battery, the electrical systems are carefully designed not to burn up
and still keep up with demand. With a bad battery, the alternator runs too
hot and cooks the insulation and the diodes. Eventually too much damage is
done and the alternator fails."

If you had to get a boost because your battery was unable to crank the starter, hook the battery  up to a proper trickle-charger, or buy a new one.