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We report very simple and accurate algebraic expressions for the van der Waals (vdW) potentials and the forces between two parallel and crossed carbon nanotubes. The Lennard-Jones potential for two carbon atoms and the method of the smeared out approximation suggested by L.A. Girifalco were used. It is found that interaction between parallel and crossed tubes are described by different uniform curves which depend only on dimensionless distance. The explicit functions for equilibrium vdW distances, well depths, and maximal attractive forces have been given. These results may be used as a guide for analysis of experimental data to investigate interaction between nanotubes of various natures., Comment: 4 pages, 2 figures

The analytical expressions for the van der Waals potential energy and force between two crossed carbon nanotubes are presented. The Lennard-Jones potential for two carbon atoms and the method of the smeared out approximation suggested by L.A. Girifalco were used. The exact formula is expressed in terms of rational and elliptical functions. The potential and force for carbon nanotubes were calculated. The uniform potential curves for single- and multi- wall nanotubes were plotted. The equilibrium distance, maximal attractive force, and potential energy have been estimated., Comment: 8 pages, 6 figures

We present an exact solution for the electrostatic field between a metallic hemi-ellipsoidal needle on a plate (as a cathode) and a flat anode. The basic idea is to replace the cathode by a linearly charged thread in a uniform electric field and to use a set of "image" charges to reproduce the anode. We calculate the field enhancement factor on the needle surface and ponderomotive force acting on the needle. Using the Fowler-Nordheim theory we obtain an exact analytical formula for the total current., Comment: 7 pages, 8 figures