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Magnetic Cannon: the Physics of the Gauss Rifle Arsène Chemin, Pauline Besserve, Aude Caussarieu, Nicolas Taberlet, Nicolas Plihon

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Magnetic Cannon: the Physics of the Gauss Rifle Arsène Chemin, Pauline Besserve, Aude Caussarieu, Nicolas Taberlet, Nicolas Plihon Magnetic Cannon: the physics of the Gauss rifle Arsène Chemin, Pauline Besserve, Aude Caussarieu, Nicolas Taberlet, Nicolas Plihon To cite this version: Arsène Chemin, Pauline Besserve, Aude Caussarieu, Nicolas Taberlet, Nicolas Plihon. Magnetic Cannon: the physics of the Gauss rifle. American Journal of Physics, American Association of Physics Teachers, 2017, 85 (7), pp.495-502. 10.1119/1.4979653. hal-02408779 HAL Id: hal-02408779 https://hal.archives-ouvertes.fr/hal-02408779 Submitted on 13 Dec 2019 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Magnetic Cannon: the physics of the Gauss rifle Ars`eneChemin,1 Pauline Besserve,1 Aude Caussarieu,1 Nicolas Taberlet,1 and Nicolas Plihon1, a) Univ Lyon, Ens de Lyon, Univ Claude Bernard, CNRS, Laboratoire de Physique, D´epartement de Physique, F-69342 Lyon, France (Dated: 13 December 2019) The magnetic cannon is a simple device that converts magnetic energy into kinetic energy: when a steel ball with low initial velocity impacts a chain made of a magnet followed by a few other steel balls, the last ball of the chain is ejected at a much larger velocity. The analysis of this spectacular device involves understanding of advanced magnetostatics, energy conversion and collision of solids. In this article, the phenomena at each step of the process are modeled to predict the final kinetic energy of the ejected ball as a function of a few parameters which can all be experimentally measured. I. INTRODUCTION Steel ball Magnet Steel balls 0 ms I A. What is a magnetic cannon? 128.7 ms The magnetic cannon, sometimes referred to as the II Gauss rifle is a simple device that accelerates a steel ball 164.7 ms through conversion of magnetic energy into kinetic en- ergy1{4. The energy conversion at work is reminiscent of III other electromagnetism-based accelerating device, such Impact Soliton 190.5 ms 5 as rail-guns . Figure1 shows a time sequence (from top IV to bottom) of a typical setup where a line of four balls 193 ms (the first one being a permanent magnet) is resting on a rail. When an additional ball approaches from the left V with a low initial velocity, it experiences an attractive 218.8 ms magnetic force from the magnet, collides with the mag- net, and the final ball on the right is ejected at high VI velocity. Note that, to highlight the various sequences in Figure1, frame-times are not equi-spaced. The video from which these frames have been extracted is provided as a supplementary material. To understand the physics FIG. 1. Time sequence (from top to bottom) showing accel- of the Gauss rifle, the process may be divided into three eration in the magnetic cannon. A line of three steel balls phases: (i) acceleration of the ferromagnetic steel ball initially stuck to the right of a strong permanent magnet, in the magnetic field created by the magnet (frames I loosely fixed on a rail, is impacted by a steel ball coming from to III in Figure1), (ii) momentum propagation into the the left hand side and accelerated in the magnetic field of the chain of steel balls which is similar to the propagation in magnet. Momentum is transferred into the chain and the last the Newton's cradle (frame IV), (iii) ejection of the final ball on the right is ejected at high velocity. Images from a ball escaping the residual magnetic attraction (frames V 2048x360 pixel film acquired at 400 frames per second. and VI). While Figure1 highlights a specific example, we will focus on the more general case described in Fig- ure2, where the initial chain is formed of n steel balls The acceleration phase is governed by the magnetic in front of the magnet and m balls behind (each ball of field created by the magnet6,7 and the magnetization of mass M having a radius R). The magnet is a strong Nd- the impacting steel ball. The determination of the mag- FeB permanent magnet which is maintained on the rail, netization of the incoming ferromagnetic steel ball differs strongly enough to prevent the chain from moving in the from classical problems in which material (dia- or para- leftward direction in the acceleration phase, but loosely magnetic balls8) or geometry (large sample of ferromag- enough to allow momentum propagation in the chain (a netic materials9{11) are different from ours. The gain of patch of putty can be observed in Figure1). Note that magnetic energy during the acceleration phase is called the dipolar axis of the magnet is spontaneously aligned U to indicate the dependence on the number of balls with the axis of the steel ball chain to minimize potential n screening the magnetic field of the magnet. Part of this energy. energy is converted into kinetic energy, which will add up to the initial kinetic energy of the incoming ball Kinit, resulting in an impacting kinetic energy Kimpact. The en- ergy transfer in the chain is reminiscent of the Newton's a)Electronic mail: [email protected] cradle12,13, governed by Hertzian contact forces involv- 2 Incoming n balls Magnet m balls ball addition to the challenge that the tournament represents, it provides students with an exciting and eye-opening ex- x=0 x perience in which they learn how to design experiments K init impact with the aim of solving physics problems, and to con- U structively criticize scientific solutions. n The authors would highly recommend participation in K impact soliton propagation the IPT as a rare learning opportunity for undergraduate students. K Ejected eject ball -U C. What students can learn from this problem m-1 K final At introductory physics level, this experiment could be FIG. 2. Schematic of the sequences described in Figure1 used to foster student's motivation while working on an and associated kinetic K and (potential) magnetic U ener- open problem involving energy conservation. In a more gies. Red arrows represent conversion between magnetic en- classical laboratory work, students could reinvest their ergy and kinetic energy. knowledge on magnetism to find out whether the incom- ing steel ball is to be considered as a permanent or an induced magnet through magnetic force and magnetic ing dissipation14 in an inhomogeneous chain containing field measurements. At graduate level, the question of a magnet. Part of the impacting energy is transmitted the dependence of the final velocity on some of the pa- through the chain, resulting in a kinetic energy Keject. rameters of the system might lead to a few days exper- Finally, in the expulsion phase, the final kinetic energy imental project work in which knowledge on mechanics, of the ejected ball Kfinal is given by the energy transmit- magnetism and non linear physics can be reused. ted through the chain decreased by the energy required to escape the residual magnetic attraction on the right- hand side of the chain, Um−1. The goal of the present II. FROM MAGNETIC ENERGY TO KINETIC ENERGY article is to relate Kfinal to Kinit and the parameters of the system. Note that the energy conversion process at In this section we investigate the conversion of mag- lead in this problem is not in contradiction with the fact netic energy Un into kinetic energy in the acceleration that magnetic field do not work on charged particles (the phase, as well as the symmetric problem of the decrease Lorentz force being perpendicular to the charged par- of kinetic energy by Um−1 in the ejection phase. Direct ticles velocity). The analysis of a situation similar to measurement of the spatial dependence of the magnetic the one investigated here is discussed in details in Grif- force exerted by the magnet on the incoming (ejected) fiths’s textbook15; interested readers could also refer to ball and of the magnetic field are in agreement with a the discussion about magnetic energy provided in Jack- permanent dipole/induced dipole modeling. son's textbook16 (p. 224 and following) All effects related to the conversion of magnetic energy into kinetic energy are studied in sectionII. The transmission of kinetic en- A. Force exerted by the chain on a steel ball: a ergy through the chain is then detailed in section III. permanent/induced dipole interaction Finally, parameters influencing the global energy conver- sion of the system and the understanding of a succession Let us first focus on the direct measurement of the of Gauss rifles are discussed in sectionIV. magnetic force exerted on the steel ball. Its spatial evo- lution is measured using a weighing scale mounted as a Newton-meter, as sketched in Figure3. A steel ball is at- B. International Physicists Tournament tached to a heavy plastic block resting on the scale below the magnetic cannon chain. The steel ball attached to the The work presented here was done in preparation scale is subject to its weight in the downward direction for the International Physicists Tournament (http:// and to a magnetic force in the upward direction. The evo- iptnet.info/), a world-wide competition for undergrad- lution of the force exerted by the magnet as a function of uate students. Each national team is composed of six the distance d between the center of the ball attached to students who work throughout the academic year on a the scale and the center of the magnet, derived from the list of seventeen open questions and present their find- apparent mass, is displayed in Figure3.
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