  We examine the angular momentum of an electric charge e placed at rest
outside a dilaton black hole with magnetic charge Q. The electromagnetic
angular momentum which is stored in the electromagnetic field outside the black
hole shows several common features regardless of the dilaton coupling strength,
though the dilaton black holes are drastically different in their spacetime
structure depending on it. First, the electromagnetic angular momentum depends
on the separation distance between the two objects and changes monotonically
from eQ to 0 as the charge goes down from infinity to the horizon, if
rotational effects of the black hole are discarded. Next, as the black hole
approaches extremality, however, the electromagnetic angular momentum tends to
be independent of the distance between the two objects. It is then precisely
$eQ$ as in the electric charge and monopole system in flat spacetime. We
discuss why these effects are exhibited and argue that the above features are
to hold in widely generic settings including black hole solutions in theories
with more complicated field contents, by addressing the no hair theorem for
black holes and the phenomenon of field expulsion exhibited by extremal black
holes.
