  In a $U(1)_{\star}$-noncommutative (NC) gauge field theory we extend the
Seiberg-Witten (SW) map to include the (gauge-invariance-violating) external
current and formulate - to the first order in the NC parameter -
gauge-covariant classical field equations. We find solutions to these equations
in the vacuum and in an external magnetic field, when the 4-current is a static
electric charge of a finite size $a$, restricted from below by the elementary
length. We impose extra boundary conditions, which we use to rule out all
singularities, $1/r$ included, from the solutions. The static charge proves to
be a magnetic dipole, with its magnetic moment being inversely proportional to
its size $a$. The external magnetic field modifies the long-range Coulomb field
and some electromagnetic form-factors. We also analyze the ambiguity in the SW
map and show that at least to the order studied here it is equivalent to the
ambiguity of adding a homogeneous solution to the current-conservation
equation.
