  We construct perturbative static solutions to the classical field equations
of noncommutative U(1) gauge theory for the three cases: a) space-time
noncommutativity, b) space-space noncommutativity and c) both a) and b). The
solutions tend to the Coulumb solution at spatial infinity and are valid for
intermediate values of the radial coordinate $r$. They yield a self-charge in a
sphere of radius $r$ centered about the origin which increases with decreasing
$r$ for case a), and decreases with decreasing $r$ for case b). For case a)
this may mean that the exact solution screens an infinite charge at the origin,
while for case b) it is plausible that the charge density is well behaved at
the origin, as happens in Born-Infeld electrodynamics. For both cases a) and b)
the self-energy in the intermediate region grows faster as $r$ tends to the
origin than that of the Coulumb solution. It then appears that the divergence
of the classical self-energy is more severe in the noncommutative theory than
it is in the corresponding commutative theory. We compute the lowest order
effects of these solutions on the hydrogen atom spectrum and use them to put
experimental bounds on the space-time and space-space noncommutative scales. We
find that cases a) and b) have different experimental signatures.
