Solution of the Vlasov Equations for a Plasma in an Externally Uniform Magnetic Field

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Abstract
An exact solution of the collisionless Boltzmann equation is presented that describes a slab of plasma contained in a magnetic field that is uniform external to the slab. The magnetic field dips in the plasma due to the diamagnetic currents in the plasma sheath separating the region of nearly constant density from the vacuum region. The currents are given by the distribution functions for ions and electrons and the fields obey Maxwell's equations. The solution has the qualitative features of real collisionless laboratory plasma distributions. The problem of a cylindrical plasma with axial magnetic field is also treated, and a solution presented for a limiting case. The solution to the slab problem is charge neutral. The limiting case in the cylindrical problem is also charge neutral, but the general cylindrical problem is not, and contains radial electric fields. The solutions have the merit of being substantially more simple than others that have previously been found for the plasma sheath problem.