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 Fraunhofers analytical form) for the annular JJ with the length much smaller than the Josephson penetration length, is valid irrespective of the junctions 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.
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