If the solar wind is capable of driving magnetospheric convection, then solar-wind flow past any spinning, magnetized planet with a conducting ionosphere must cause the magnetic field lines in the outer part of its magnetospheric tail to be twisted into a helix. Such a magnetic field configuration requires magnetically field-aligned (Birkeland) currents in the tail that flow in and near the magnetopause and close by driving Pedersen currents through the planetary ionosphere. The strength of the Birkeland currents (and, by current continuity, the Pedersen currents) is, to first order, independent of the angle between the planetary-spin vector and the solar-wind velocity vector. Rather, the total current is a funciton of the magnetic moment of the planet, the radius of the tail, the angular velocity of planetary spin, the conductivity of the ionosphere, and the solar wind speed. For Jupiter, Saturn, ionosphere, and the solar wind speed. For Jupiter, Saturn, Uranus, and perhaps Neptune, the power these currents deliver to the ionosphere is significant with regard to magnetospheric dynamics such as the production of aurora and the generation of low-frequency radio emissions. For Mercury, Venus, Earth, Mars, and probably Pluto, these currents are relatively small, although observable effects may be marginally detectable for the case of the Earth's magnetosphere.