In density mesearement mode, the potential is fixed and the current I is measured. The plasma parameters can be retrieved if a theoretical model of I is available. In principle, a model of I must be determined by expressing each component of the current using the distribution function at the surface of the object,
J is the current densities at the object surface. q is the charge of the particle. u is the speed of the particle. n is the normal to the surface. f is the distribution function. is the electric potential.
In general the determination of requires to solve the Vlasov-Poisson system of equation,
E is the electric field. B is the magnetic field. m is the mass of the particle.Fortunately, there are several possible simplifications of the problem leading to more tractable equations. For instance, it can be shown that in the case of a system with spherical symmetry and a radius much smaller than the plasma Debye-Length,
rp is the radius of the probe. ne is the electronic density. Te is the electronic temperature. is the potential of the probe.