Guillermo Donoso

A modified asymptotic Padé method. Application to multipole approximation for the plasma dispersion function Z

An improved asymptotic Pade method is presented. The approximations expressed as rational functions are developed both in a power series and asymptotic expansions. Identifying these developments with those for the exact function, the approximated rational functions are obtained. Three‐ and four‐pole approximations for the plasma dispersion function have been determined with this method. Our approximations (Zlm) give closer results to the exact function than all the published ones.

Magnet Fall inside a Conductive Pipe: Motion and the Role of the Pipe Wall Thickness.

Theoretical models and experimental results are presented for the retarded fall of a strong magnet inside a vertical conductive non-magnetic tube. Predictions and experimental results are in good agreement modelling the magnet as a simple magnetic dipole. The effect of varying the pipe wall thickness on the retarding magnetic drag is studied for pipes of different materials. Conductive pipes of thinner walls produce less dragging force and the retarded fall of the magnet is seen to consist of an initial transient accelerated regime followed by a stage of uniform motion. Alternative models of the magnet field are also presented that improve the agreement between theory and experiments.

Damped fall of magnets inside a conducting pipe

We consider the uniform motion of a short strong cylindrical magnet falling inside a conducting pipe and study the dependence of the magnetic braking force on the distance of the falling magnet from the pipe wall. We also consider two magnets falling together with parallel or opposite magnetic moments. We develop models for these three cases of magnetic braking and describe the experiments that validate our models. The experimental setups are inexpensive and can be readily assembled in a teaching laboratory.

A new Langmuir–Child equation including temperature effects

A generalized Langmuir–Child equation has been obtained wherein the temperature effects appear as a function of the parameter τ=T/q(Vp−VR). The space‐charge analysis for planar electrodes is done using kinetic theory and the differential equations are solved using the recently published [J. Math. Phys. 26, 705, 1186 (1985); 28, 330 (1987)] two‐point quasifractional approximants.

Grid effects on velocity analyzers of variable geometry

A complete theory of a planar electrostatic velocity analyzer is developed for the case of a discriminating grid composed of parallel cylindrical wires. Exact solutions for the potential distribution inside the analyzer and for the depletion of the discriminating potential in the grid plane are calculated by means of a Schwarz–Christoffel transformation. It is shown that the dependence of the collector current on the discriminating voltage deviates from the simple exponential law. The analysis shows the following effects: a stretching and displacement of the V axis of the characteristics and a decrease of the collector saturation current. A new expression to calculate the plasma temperature from the characteristic curve is given.

Experimental verification of the grid effects in a velocity analyzer with variable geometry

A velocity analyzer with adjustable interelectrodic distances is described here. Several experiments have been performed in a double plasma machine with this analyzer of variable geometry in order to investigate the effects of the discontinuous structure of the grids. The measured displacements of the characteristic curve for different interelectrodic distances show excellent agreement with the theoretical predictions due to the depletion of the discriminating potential in the holes of the grid, as is shown in our recent theoretical analysis using a Schwarz–Christoffel transformation. The influence of the grid structure on the temperature determined from the characteristic curve is also investigated and the correlation experiment theory is qualitatively correct.

Magnetically coupled magnet–spring oscillators

A system of two magnets hung from two vertical springs and oscillating in the hollows of a pair of coils connected in series is a new, interesting and useful example of coupled oscillators. The electromagnetically coupled oscillations of these oscillators are experimentally and theoretically studied. Its coupling is electromagnetic instead of mechanical, and easily adjustable by the experimenter. The coupling of this new coupled oscillator system is determined by the currents that the magnets induce in two coils connected in series, one to each magnet. It is an interesting case of mechanical oscillators with field-driven coupling, instead of mechanical coupling. Moreover, it is both a coupled and a damped oscillating system that lends itself to a detailed study and presentation of many properties and phenomena of such a system of oscillators. A set of experiments that validates the theoretical model of the oscillators is presented and discussed.

Space‐charge effects in a velocity analyzer of variable geometry

A velocity analyzer of adjustable interelectrode distance has been used to study the space‐charge effects on the characteristic curve. The effects become apparent for distances larger than about 10λD. The theoretical analysis considering the trapped ions in the vicinity of the entrance grid leads to a better correlation with the experiments than that omitting them.

Nonlinear dynamics of a magnetically driven Duffing-type spring-magnet oscillator in the static magnetic field of a coil

We study the nonlinear oscillations of a forced and weakly dissipative spring–magnet system moving in the magnetic fields of two fixed coaxial, hollow induction coils. As the first coil is excited with a dc current, both a linear and a cubic magnet-position dependent force appear on the magnet–spring system. The second coil, located below the first, excited with an ac current, provides the oscillating magnetic driving force on the system. From the magnet–coil interactions, we obtain, analytically, the nonlinear motion equation of the system, found to be a forced and damped cubic Duffing oscillator moving in a quartic potential. The relative strengths of the coefficients of the motion equation can be easily set by varying the coils’ dc and ac currents. We demonstrate, theoretically and experimentally, the nonlinear behaviour of this oscillator, including its oscillation modes and nonlinear resonances, the fold-over effect, the hysteresis and amplitude jumps, and its chaotic behaviour. It is an oscillating system suitable for teaching an advanced experiment in nonlinear dynamics both at senior undergraduate and graduate levels.

Anharmonic Oscillations of a Spring-Magnet System inside a Magnetic Coil.

We consider the nonlinear oscillations of a simple spring–magnet system that oscillates in the magnetic field of an inductive coil excited with a dc current. Using the relations for the interaction of a coil and a magnet we obtain the motion equation of the system. The relative strengths of the terms of this equation can be adjusted easily by varying the coil excitation current. Both the elastic constant and the shape of the potential energy function of the system can therefore be modified by varying that current. It is shown and demonstrated that this system is a case of anharmonic oscillations in a double-well potential. This nonlinear oscillator can be easily assembled with commonly available laboratory components, and monitored with a digital oscilloscope. Its simplicity is to be compared with the setups of many other nonlinear oscillators recently described. This oscillator is ideal for an advanced undergraduate laboratory experiment or for project work.