Celso L Ladera

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.

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.

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.

The naked toy model of a jumping ring

We present a comprehensive analytical model of the well-known jumping ring—in fact an improved version of that system-–as well as the experimental results that validate the model. Particular attention is paid to the magnetic driving force, whose explicit dependences upon the phase, amplitude and frequency of the exciting current we manage to separate experimentally and plot, so that it becomes evident how the magnetic force on the ring actually arises and evolves in time. We are able to measure not only the large Foucault currents that arise in the ring, but also the magnetic field generated by the ring itself in spite of the presence of the comparable magnetic field in which the ring moves.

Simple analytical approximations to the integrals of the Bessel functions Jν: application to the transmittance of a circular aperture

Two accurate, yet simple, analytic approximations to the integral of the Bessel function J0 are presented. These first and second-order approximations are obtained by improving on the recently developed method known as two-point quasi-rational approximations. The accuracy of the first-order approximant is better than 0.05. The second-order approximant is practically indistinguishable from the true integral, even for very large values of the argument (overall accuracy is better than 0.002?05). Our approximants are, in addition, analytic and therefore replace with significant advantages both the well known power series and the asymptotic formulae of the integral. Approximants to the transmittance function of a plane wave through a circular aperture are derived, a problem which arises in diffraction theory and particle scattering. The second-order approximant to the transmittance is analytic too, and can be evaluated for small and large values of the argument, just with a hand-calculator. Its accuracy is better than 0.0011. As an extension, two first-order approximations to the integrals of the Bessel functions J?, of fractional order ?, are derived.

Holographic interferometry with the compact arrowhead holographic setup

A symmetric off-axis holographic setup, shaped as an arrowhead, which requires neither a collimator nor a beam splitter, is presented. It is applied to measure small perpendicular-to-surface displacements and deformations and the magnetostriction of a body by holographic interferometry. It offers advantages such as implicit fulfilment of several hologram recording conditions, possible use of short coherence length light sources, low-cost, and significant immunity against mechanical perturbations.

Unveiling the physics of the Thomson jumping ring

We present a new theoretical model and validating experiments that unveil the rich physics behind the flight of the conductive ring in the Thomson experiment—physics that is hard to see because of the rapid motion. The electrodynamics of the flying ring exhibits interesting features, e.g., varying mutual inductance between the ring and the electromagnet. The dependences of the ring electrodynamics upon time and position as the ring travels upward are conveniently separated and determined to obtain a comprehensive view of the ring motion. We introduce a low-cost jumping ring setup that incorporates pickup coils connected in opposition, allowing us to scrutinize the ring electrodynamics and confirm our theoretical model with good accuracy. This work is within the reach of senior students of science or engineering, and it can be implemented either as a teaching laboratory experiment or as an open-ended project.

Parametric resonances of a conductive pipe driven by an alternating magnetic field in the presence of a static magnetic field

The parametric oscillations of an oscillator driven electromagnetically are presented. The oscillator is a conductive pipe hung from a spring, and driven by the oscillating magnetic field of a surrounding coil in the presence of a static magnetic field. It is an interesting case of parametric oscillations since the pipe is neither a magnet nor a ferromagnet, and because the driving and the damping forces are functions of the magnet-to-pipe distance. We develop an analytical model of the oscillator that leads to a new kind of Mathieu equation with nonlinear terms of third order in the oscillator position, and on the product of the squared position times the first derivative of the position function. We show how the oscillations evolve as the frequency is varied up to the saturation regime and present ample experimental evidence of the parametric nature of our oscillator. Because of its peculiarities and their advanced scientific and technological applications, parametric oscillations are very important for physicists and engineers at both senior undergraduate and graduate levels. This oscillator is very easy to set up and provides an excellent opportunity to learn all facets of parametric oscillations at both levels.