Potential formation in a bounded plasma system that contains electrons with a two-temperature velocity distribution and is terminated by a floating, electron emitting electrode (collector) is studied by a one-dimensional kinetic model. A method on how to determine the boundary conditions at the collector for the numerical solution of the Poisson equation is presented. The difference between the regular and the irregular numerical solutions of the Poisson equation is explained. The regular numerical solution of the Poisson equation fulfills the boundary conditions at the source and can be computed for any distance from the collector. The irregular solution does not fulfill the source boundary conditions and the computation breaks down at some distance from the collector. An excellent agreement of the values of the potential at the inflection point found from the numerical solution of the Poisson equation with the values predicted by the analytical model is obtained. Potential, electric field, and particle density profiles found by the numerical solution of the Poisson equation are compared to the profiles obtained with the particle in cell computer simulation. A very good quantitative agreement of the potential and electric field profiles is obtained. For certain values of the parameters the analytical model predicts three possible values of the potential at the inflection point. In such cases always only one of the corresponding numerical solutions of the Poisson equation is regular, while the other two are irregular. The regular numerical solution of the Poisson equation always corresponds to the solution of the model that predicts the largest ion flux to the collector.

A one-dimensional fluid model of sheath formation in front of a large, planar, floating, electron emitting electrode (collector) immersed in a two-electron temperature plasma is presented. For certain values of the parameters the Bohm criterion is triple valued. Three methods to select the correct Bohm velocity are compared. The simplest is to find the parameters, where the floating potentials that correspond to the high and to the low Bohm velocity are equal. But sometimes-when the electron emission is small-such parameters cannot be found. The second method is the 'maximum positive ion flux criterion', proposed recently by Fernández Palop et al (2002 J. Appl. Phys. 91 2587), for electronegative plasmas. In this work it is modified for a two-electron temperature plasma with emitted electrons and applied successfully. This method is sometimes-when the electron emission from the collector is increased-not applicable because the positive ion flux as a function of potential has only 1 maximum. The third method is a numerical solution of the Poisson equation. The transition from the low to the high Bohm velocity occurs at the parameters where the numerically found space charge density profile indicates the formation of a double layer in the sheath. These parameters are always identical to the parameters found with the maximum positive ion flux method, when the latter can be applied. The 'shooting method' for the correct determination of electric field at the collector is presented. This method reveals the existence of regular and irregular numerical solutions of the Poisson equation. The irregular solutions of the Poisson equation fulfill only one of the two sheath edge conditions and they correspond to the solutions of the fluid model, that are not physically acceptable. The parameters, where the numerical solutions of the Poisson equation become irregular, are identical to those found with the 'positive square of electric field' test, which is also presented.

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The elastic pendulum is a simple physical system represented by nonlinear differential equations. Analytical solutions for the bob trajectories on the rotating earth may be obtained in two limiting cases: for the ideally elastic pendulum with zero unstressed string length and for the Foucault pendulum with an inextensible string. The precession period of the oscillation plane, as seen by the local observer on the rotating earth, is 24 hours in the first case and has a well-known latitude dependence in the second case. In the present work, we have obtained numerical solutions of the nonlinear equations for different string elasticities in order to study the transition from one precession period to the other. It is found that the transition is abrupt and that it occurs for a quite small perturbation of the ideally elastic pendulum, that is, for the unstressed string length equal to about 10 −4 of the equilibrium length due to the weight of the bob.

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