On the Dipole Moment of Quantized Vortices in the Presence of Flows

The polarization charge $$\rho _\mathrm {pol}$$ of an inhomogeneous superfluid system is expressed as a function of the order parameter $$\Phi (\mathbf{r}_1 ,\mathbf{r}_2 )$$ . It is shown that if the order parameter changes on macroscopic distances, the polarization charge $$\rho _\mathrm{pol} $$ is proportional to $$A\nabla ^2n$$ , and the polarization $$\mathbf{P}$$ is proportional to $$A\nabla n$$ , where n is the density of the system. For noninteracting atoms, the proportionality coefficient A is independent of density, and in the presence of interaction A is proportional to n. The change of the Bose gas density is found in the presence of a flow $$\mathbf{w}=\mathbf{v}_n -\mathbf{v}_s $$ passing the vortex. It is found that a vortex in a superfluid film creates an electric potential above the film. This potential has the form of a potential of a dipole, allowing to assign a dipole moment to the vortex. The dipole moment is a sum of two terms, the first one is proportional to the relative flow velocity $$\mathbf{w}$$ and the second one is proportional to $$\left[ {{\mathbf {\kappa }} \times \mathbf{w}} \right] $$ , where $$\mathbf{\kappa }$$ is the vortex circulation.