  We study static and dynamical properties of fluxons in a long annular
Josephson junction (JJ) with a current injected at one point and collected back
at a close point. Uniformly distributed dc bias current is applied too. We
demonstrate that, in the limit of the infinitely small size of the current
dipole, the critical value of the bias current density, above which static
phase distributions do not exist, that was recently found (in the Fraunhofer's
analytical form) for the annular JJ with the length much smaller than the
Josephson penetration length, is valid irrespective of the junction's length,
including infinitely long JJs. In a long annular JJ, the dipole generates free
fluxon(s) if the bias current density exceeds the critical value. For long JJs,
we also find another critical value (in an analytical form too), which is
always slightly smaller than the Fraunhofer value, except for points where both
values vanish. The static phase configuration which yields the new critical
value is based on an unstable fluxon-antifluxon bound state, therefore it will
probably not manifest itself in the usual (classical) regime. However, it
provides for a dominating instanton configuration for tunnel birth of a free
fluxon, hence it is expected to determine a quantum-birth threshold for fluxons
at ultra-low temperatures. We also consider the interaction of a free fluxon
with the complex consisting of the current dipole and antifluxon pinned by it.
A condition for suppression of the net interaction force, which makes the long
JJ nearly homogeneous for the free fluxon, is obtained in an analytical form.
The analytical results are compared with numerical simulations.
