Exact current to a spherical electrode in a collisionless, large-Debye-length magnetoplasma

Exact calculations of the steady-state current drawn from a collisionless, Maxwellian plasma in a uniform magnetic field by a spherical, perfectly absorbing electrode are presented for a range of dimensionless electrode potentials and magnetic-field strengths. These calculations are valid in the limit of large Debye length. The results are compared with the theory of Rubinstein and Laframboise, which gives upper and lower bounds for both the attracted-species and the repelled-species current. It is found that as the electrode potential increases from space potential with magnetic-field strength fixed, the electron (i.e., attracted-species) current decreases, but not as quickly as the adiabatic-limit (effectively lower-bound) current. The ion current also diverges immediately from the adiabatic-limit current. As the electrode potential increases further, the electron current rises and moves monotonically toward the canonical upper bound, which is the warm-plasma generalization of the well-known Parker and Murphy upper bound. It is unclear whether the current approaches the upper bound asymptotically as the electrode potential becomes large, or instead a constant proportion of the upper bound which varies with magnetic-field strength. The dependence on magnetic-field strength is more complicated. As expected for small fixed electrode potentials, the attracted-species current approaches the adiabatic-limit current monotonically as the magnetic-field strength increases. However, for large electrode potentials this pattern reverses: the current approaches the canonical upper bound monotonically as the magnetic-field strength increases. These patterns are expected to persist when the Debye length is finite. Interpretation of these results leads to an inference that for large electrode potentials, the effect of decreasing the Debye length may be to reduce the current, as in the nonmagnetic case.