Proximity effect in a thin two-layer tubular conductor caused by a parallel filament

In this paper, we determine in a closed form the current distribution in a thin two-layer tubular conductor with a sinusoidal current in the presence of a filament with an equal but opposite current. The method of treating the problem consists in solving an integral equation for the current density in one layer which enables to find the current density in the other layer by imposing equality of the tangential components of the electric fields at the layer interface. An important factor, the AC to DC resistance ratio for the conductor is also found in a closed form. Some numerical examples, for current distribution and resistance ratio, versus frequency or geometry parameters, are presented.