Electric currents in the solar corona and the kink instability of magnetic flux rope

We found the dispersion equation for magnetohydrodynamic kink oscillations of a force-free magnetic flux rope with uncompensated longitudinal electric current under the conditions of the solar corona using the energy method and the thin magnetic flux tube approximation. It is shown that the eigenfunctions, along with the eigenvectors, impose additional restrictions on the stability conditions of the kink instability of a flux rope. This allows us to obtain not only the necessary, but also a sufficient condition for stability. The observed weak twist of coronal loops with a small (< 2–3) number of the turns of magnetic field lines around the axis indicates the dominance of unshielded magnetic flux rope in the corona of the Sun, in which the longitudinal electric currents do not exceed $10^{11}$–$10^{12}$ A.