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Easy-to-evaluate approximate formulas are presented for the repelled-species current collected by a spherical body in a collisionless, magnetized plasma, valid over the full range of ratio of Larmor radius to sphere radius. The form is an appropriate average of the lower and upper bounds obtained by prior analytic arguments. This formulation enables rapid evaluation of the floating potential in hybrid Boltzmann-particle-in-cell (PIC) codes with a background magnetic field. The treatment is validated by comparison of hybrid PIC (ion particle) code with full PIC (electron and ion particle) code calculations. It is found that typically for no value of magnetic field is it valid to approximate the electrons as fully magnetized but the ions as unmagnetized in the absence of plasma drift.

Electron collection by a negatively charged sphere in a collisionless magnetoplasma