DOCSLIB.ORG

Polarity Free Magnetic Repulsion and Magnetic Bound State

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Polarity Free Magnetic Repulsion and Magnetic Bound State Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 1 September 2020 doi:10.20944/preprints202009.0001.v1 Polarity Free Magnetic Repulsion and Magnetic Bound State Hamdi Ucar Independent Researcher: Istanbul, Turkey, 34360. Email: [email protected] This is a report on a dynamic autonomous magnetic interaction which does not depend on polarities resulting in short ranged repulsion involving one or more inertial bodies and a new class of bound state based on this interaction. Both effects are new to the literature, found so far. Experimental results are generalized and reported qualitatively. Working principles of these effects are provided within classical mechanics and found consistent with observations and simulations. The effects are based on the interaction of a rigid and finite inertial body (an object having mass and moment of inertia) endowed with a magnetic moment with a cyclic inhomogeneous magnetic field which does not require to have a local minimum. Such a body having some DoF involved in driven harmonic motion by this interaction can experience a net force in the direction of the weak field regardless of its position and orientation or can find stable equilibrium with the field itself autonomously. The former is called polarity free magnetic repulsion and the latter magnetic bound state. Experiments show that a bound state can be obtained between two free bodies having magnetic dipole moment. Various schemes of trapping bodies having magnetic moments by rotating fields are realized as well as rotating bodies trapped by a static dipole field in presence of gravity. Also, a special case of bound state called bipolar bound state between free dipole bodies is investigated. KEYWORDS: magnetic bound state, magnetic levitation, polarity free repulsion, magnetic trapping 1 Introduction Things can be bound classically by force fields they possess and inertial forces balancing them. Despite some force fields have attractive and repulsive actions, it is not possible to obtain stable equilibrium through force fields alone as proven by Earnshaw’s theorem. Therefore classically, everything to be bound should have some dynamics in order to incorporate inertial forces. The common mechanism is the orbital motion and can explain bound states of celestial bodies. Orbital bound state is also possible within electrically charged bodies and used in Bohr model of the atom based on inverse square law, but stability cannot be obtained within magnetic forces where their dependence to the distance is out of range of the power figure to obtain stable orbital motion within the central force problem [1]. This precludes orbital bound state of nucleons based on magnetic forces. While classical physics fails historically to propose a binding mechanism of nucleons, nuclear forces and additional mechanisms are introduced as part of quantum physics addressing this problem. The orbital motion based on dipole-dipole interaction was later investigated by several authors including Kozorez [2], Shironosov [3]. Here is introduced a new mechanism to obtain bound state of entities having magnetic moment, using again inertial behaviors but in a different manner. It is based on magnetic forces, using attraction force to keep entities having magnetic moments together and balance this attraction with dynamically created repulsive magnetic 1 © 2020 by the author(s). Distributed under a Creative Commons CC BY license. Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 1 September 2020 doi:10.20944/preprints202009.0001.v1 action which might be classified as a force too. This repulsive force or the interaction is called polarity free magnetic repulsion (PFR) and the resulting bound state as magnetic bound state (MBS). Although it is possible to evaluate MBS as a dynamic equilibrium based on magnetic interaction directly without introducing PFR, generalization might be difficult. MBS might be considered as versatile because it allows entities having mass, moment of inertia and magnetic dipole moment to be bound without precise requirements. These entities can be compact bodies; that is, one body is not surrounded by the other. Fig. 1.1. A scaled-down MBS realization where a tiny fragment of a NdFeB magnet weighing about 5 mg is levitated at the tip of rotator assembly consisting of an irregular magnet fragment of ~3 mm in size coupled to a micro motor having dimensions ⌀4×10 mm running at 105000 RPM (11000 rad/s). Dynamically created repulsive action based on magnetism this way has a short range; that is, it diminishes about twice as fast by distance compared to magnetic forces between dipoles. This allows to obtain stable equilibrium between PFR and magnetic or other attractive forces in various schemes and configurations. While the mathematical model of the PFR is applied to specific configurations in this work, experiments show it is present regardless of magnetic orientations of magnetic moments of entities. Stability and other characteristics of PFR and MBS are evaluated analytically and by simulations for basic cases. These cases are extended experimentally with numerous schemes and configurations, basically by bounding two permanent magnets in terrestrial environment without a physical contact, except the ambient air. It was not possible to test the bound state of two free magnets truly (in free fall) in terrestrial environment but performed in an approximated manner. On the other hand, gravity was helpful to test the strength of bound states and some configurations allowed to obtain bound state regardless of the direction of gravity. MBS of two free bodies is also evaluated through simulations and gives confidence that it can as well be experimentally realized in space and under friction free conditions. Scalability of MBS is tested experimentally by varying mass of the trapped magnets from 0.01 to 100 g in various configurations. It should be noted that for the bound state of two free bodies, selection of moment of inertia of one body have some limitations. A known method for trapping magnetic bodies with cyclic fields [4] is based on parametric excitation and this way, the body can be kept at the local minimum of a quadrupole field. This solution is equivalent to well-known rolling marble on a rotating saddle and its dynamics can be modeled by the Mathieu equation [5]. This model is also used to trap charged particles [6]. It is also possible to trap magnetic bodies at local minimum of static or quasi-static quadrupole fields while trying to keep bodies in correct orientation in order they are pushed in the direction of the weak field. This is the main mechanism for trapping neutral atoms and particles [6, 7]. Since magnetic moments of these bodies are associated with quantum spin, they can keep their orientations parallel or anti-parallel to the field they are exposed. However, their aimed orientations can be lost or get flipped if the field becomes zero or too weak at the local minimum. This issue is addressed with different schemes. In a recently proposed scheme [8] to trap neutral particles using rotating magnetic field of an electromagnetic wave, the rotating field has a similar role of the rotating electric field in the Paul trap [6]. Solutions obtained there might be also related to [9]. 2 Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 1 September 2020 doi:10.20944/preprints202009.0001.v1 Another scheme is known as Levitron® [10, 11, 12] and a variation called Horizontal Levitron [13]. In the original Levitron, a rotating dipole body around its dipole axis is levitated by a repulsive magnetic interaction having positive stiffness balanced by gravity and negative stiffness introduced in angular degrees of freedom is balanced by the positive dynamic stiffness caused by the rotation and precession of the body. In the horizontal axis Levitron, the body is trapped between multiple dipole magnets. In this solution, precession of the top has a similar role. A detailed analysis of this solution can be found here [14]. Superconductivity also allows to trap objects or obtain bound state directly by handling magnetic field belonging to the object known as flux pinning; however, classical physics does not cover this phenomenon. Another mechanism to induce thrust using cyclic fields is ponderomotive force. With this scheme, it is possible to induce a net force in direction of the weak field on electrically charged particles regardless of their polarities by exposing them to an alternative electric field having a gradient. This is also a parametric excitation mechanism where a characteristics of the driven harmonic motion (DHM) called phase lag is present and the net force is produced while the particles oscillates through the gradient. Phase lag is the condition where the action and reaction are in opposite phases in a periodic excitation. An example of this in mechanics is a mass spring system driven through the spring. This mechanism also constitutes the principle of bass reflex speakers. While PFR and ponderomotive force have some similarities, the acted force in ponderomotive force has alternating cycle and a body needs to oscillate along the applied force direction but PFR can be obtained by a static interaction (including fields and forces) and within the local reference frame. PFR and MBS does not depend also on local minimum since they can be obtained between dipoles. In PFR, the repulsive force is basically produced by “antiparallel” orientation of a free body with an inhomogeneous magnetic field. Here the term antiparallel denotes an orientation where the angle between two vectors is greater than π/2. While such an orientation is unstable in magnetostatics, it can be stable within DHM under phase lag condition. The way to obtain PFR is exposing an inertial body embedded with a magnetic moment to an inhomogeneous rotating field or by spinning such a body which creates a rotating field and exposing it to an inhomogeneous static field. PFR might be explained in the basic case by two steps in a simpler way.
Load more

Recommended publications
  • Redalyc.Alternative Method to Calculate the Magnetic Field of Permanent Magnets with Azimuthal Symmetry
    Revista Mexicana de Física ISSN: 0035-001X [email protected] Sociedad Mexicana de Física A.C. México Camacho, J. M.; Sosa, V. Alternative method to calculate the magnetic field of permanent magnets with azimuthal symmetry Revista Mexicana de Física, vol. 59, núm. 1, enero-junio, 2013, pp. 8-17 Sociedad Mexicana de Física A.C. Distrito Federal, México Available in: http://www.redalyc.org/articulo.oa?id=57048158002 How to cite Complete issue Scientific Information System More information about this article Network of Scientific Journals from Latin America, the Caribbean, Spain and Portugal Journal's homepage in redalyc.org Non-profit academic project, developed under the open access initiative EDUCATION Revista Mexicana de F´ısica E 59 (2013) 8–17 JANUARY–JUNE 2013 Alternative method to calculate the magnetic field of permanent magnets with azimuthal symmetry J. M. Camacho and V. Sosaa Facultad de Ingenier´ıa, Universidad Autonoma´ de Yucatan,´ Av. Industrias no contaminantes por Periferico´ Norte, A.P. 150 Cordemex, Merida,´ Yuc., Mexico.´ aPermanent address: Departamento de F´ısica Aplicada, CINVESTAV-IPN, Unidad Merida,´ A.P. 73 Cordemex, Merida,´ Yuc., C.P. 97310, Mexico.´ Phone: (+52-999) 9429445; Fax: (+52-999) 9812917. e-mail: [email protected]. Received 6 September 2012; accepted 8 January 2013 The magnetic field of a permanent magnet is calculated analytically for different geometries. The cases of a sphere, cone, cylinder, ring and rectangular prism are studied. The calculation on the axis of symmetry is presented in every case. For magnets with cylindrical symmetry, we propose an approach based on an expansion in Legendre polynomials to obtain the field at points off the axis.
    [Show full text]
  • A Motor Driven by Permanent Magnets Alone; a Clean and Abundant Source of Electromagnetic Energy from Iron and Other Ferromagnetic Materials
    http://www.scirp.org/journal/ns Natural Science, 2017, Vol. 9, (No. 9), pp: 319-329 A Motor Driven by Permanent Magnets Alone; A Clean and Abundant Source of Electromagnetic Energy from Iron and Other Ferromagnetic Materials Kenneth Kozeka Austin Peay State University, Clarksville, USA Correspondence to: Kenneth Kozeka, Keywords: Electromagnetic Energy, Permanent Magnets, Angular Momentum, Permanent Magnet Motor Received: May 13, 2017 Accepted: September 18, 2017 Published: September 21, 2017 Copyright © 2017 by authors and Scientific Research Publishing Inc. This work is licensed under the Creative Commons Attribution International License (CC BY 4.0). http://creativecommons.org/licenses/by/4.0/ Open Access ABSTRACT Since the discovery of loadstone more than three hundred years ago, we have contemplated using the magnetic forces generated by permanent magnets to work for us. It is shown here that two permanent magnets with opposite poles facing can attract along their equatorial plane and then repel along their polar plane in sequence, without reversing polarity and without the use of another source of energy. This sequence of attract and repel between two permanent magnets is like the attract and repel sequence between an electromagnet and permanent magnet in an electric motor. The discovery described here can be used to con- struct a motor driven entirely by permanent magnets. 1. INTRODUCTION Electromagnetism is one of the four fundamental forces that act on matter and it is second in effective strength. The relationship between moving electrons and magnetic fields is exploited today in a wide va- riety of applications and technologies that includes the use of electromagnets and permanent magnets.
    [Show full text]
  • Magnetic Field of a Wire
    TUTORIAL 11: MAGNETS INSTRUCTORS MANUAL: TUTORIAL 11 Goals: 1. Understand bound currents: the two types (surface and volume), what causes them, how they compare to real currents, how to calculate them 2. Estimate the bound current in a magnet (by experiment) 3. Gain familiarity with magnetic dipoles, and the field they produce 4. Understand the atomic effects that cause different types of magnetization (electron spin vs. electron orbit) 5. See that magnetization is in the world around us (i.e. grapes) This tutorial is based on: Written by Steven Pollock, Stephanie Chasteen, Darren Tarshis and Colin Wallace, with edits by Mike Dubson, Ed Kinney and Markus Atkinson. “Push me a grape” is from Paul Doherty at the Exploratorium Materials needed: Compasses, toy magnets, meter sticks, grape demo (grapes, straw, string, stand, very strong magnet), superconductor levitation demo (liquid nitrogen, superconductor, small permanent magnet, plastic tweezers). Tutorial Summary: Students estimate the bound current of a toy magnet by comparing its field to the Earth’s. In the process, they confront magnetic dipoles and bound current. Then they apply their knowledge of magnetism to make sense of two demonstrations: repelling a grape with a magnet, and levitating a magnet over a superconductor. Special Instructions: Follow the instructions at the following links for the “Push me a Grape” activity: http://www.exploratorium.edu/snacks/diamagnetism_www/index.html and http://www.exo.net/~pauld/activities/magnetism/magnetrepelgrape.html Tutorial 11, Week 13 Instructor’s Manual © University of Colorado - Boulder Contact: [email protected] TUTORIAL 11: MAGNETS Reflections on this Tutorial This is a particularly strong tutorial.
    [Show full text]
  • FULL PAPER Comparison and Scaling Effects of Rotational Micro-Generators Using Electromagnetic and Piezoelectric Transduction
    FULL PAPER Comparison and Scaling Effects of Rotational Micro-Generators using Electromagnetic and Piezoelectric Transduction Hailing Fu*[a] and Eric M. Yeatman[b] Abstract: Rotational energy is widely distributed or easily acquirable electrical generators in kinetic wristwatches. First patented by from other energy sources (fluid flow, machine operation or human Berney in 1976 [3], this technology has been developed and motion) in many industrial and domestic scenarios. At small scales, commercialized in many Seiko kinetic watches. The harvester power generation from such rotational ambient sources can enable generates rotation from human activities using an off-center rotor, many autonomous and self-reliant sensing applications. In this paper, and the motion is then amplified by gear trains before generating three typical types of micro-generators (energy harvesters), namely electricity using a micro-EMG.[4] Kinetron is another innovative electromagnetic (EMREHs), piezoelectric resonant (PRREHs) and organization specialized in micro kinetic systems, especially in piezoelectric non-resonant rotational energy harvesters (PNRREHs) EMG design.[5] These devices are designed for applications are discussed and compared in terms of device dimensions and including wristwatches, mobile phones and pedal illumination operation frequencies. Theoretical models are established for each systems. In watch applications, the claw-pole generator MG4.0 is type to calculate maximum achievable output power as a function of the most commonly used design. A 14-pole Sm2Co17 magnet and device dimension and operating frequency. Using these theoretical 1140 windings are integrated. The typical output power is 10 mW models, scaling laws are established for each type to estimate the at 15,000 rpm (Φ4.0 × 2.2 mm).
    [Show full text]
  • F18 Magnetic and Electromagnetic Experiment V1(Li) Docx
    SF Scientific Co.,Ltd. F18 Magnetic and Electromagnetic Experiments 1 Contents 1.1 Determine earth magnetism by magnetic moment 1.2 Measure earth magnetism by tangent galvanometer 1.3 Current balance measurement 1.4 DC motor 1.5 Faraday's Law 1.6 Lenz's Law 1.7 Jumping ring 1.8 Generator 1.9 Transformer 1.10 Electromagnetic communication 2 Introduction 2.1 Determine earth magnetism by magnetic moment Put a horizontal magnetic needle on a place whose horizontal magnitude of earth magnetism is B. After the needle becomes still, needle's north magnetic pole will point to the north and south magnetic pole will point to the south. The perpendicular plane which passes through the earth's core and directs to the north and south direction determined by the magnetic needle is called magnetic meridian plane. Put a magnetic bar whose length is 2l to a place where it is far from the center of the magnetic needle P by distance d; moreover, its direction is perpendicular to the horizontal direction of earth magnetism. Then, the magnetic bar will produce a magnetic field B at P: 2�� � = ! �! − �! ! where m and -m are intensity of two poles, M=2ml is the magnetic pole of the bar. F18-1 SF Scientific Co.,Ltd. The magnetic needle is under earth magnetism's horizontal component B and the magnetic field Bm of the magnetic bar at the same time. Thus, it will tilt at an angle θ �! 2�� tanθ = = � � �! − �! ! Rewrite the above equation, we get � �! − �! !���� = � 2� The moment of inertia of the magnetic bar is I.
    [Show full text]
  • Analytical Models for the Design of High-Precision Permanent Magnet Actuators
    Technical Article (Semi-) analytical models for the design of high-precision permanent magnet actuators Abstract — This paper discusses (semi-) analytical modeling techniques which are applied in the analysis and design of high- precision actuators which contain large multi-dimensional magnet arrays. Such high-precision actuators are being designed for application in the lithographic industry. The methods presented in this paper offer not only a significant reduction of the calculation time compared to three-dimensional finite element simulations, but also allow the calculation of force distributions inside the permanent magnet structures. The application of these models for the design of magnetically levitated planar actuators and for gravity compensation is discussed in detail. I. INTRODUCTION Fig. 1. Magnetically levitated moving-magnet planar actuator. Actuators which are applied in high-precision equipment have to combine both a high force density and good predictability of this force. The high force density is needed to achieve sufficient acceleration and speed levels, and to gain an adequate throughput of products, whereas a very good predictability of the force is required to obtain the necessary accuracy. Examples of such actuators are electromagnetic vibration isolators (gravity compensators) and magnetically levitated planar actuators. Usually, these actuators have three-dimensional designs and lack periodicity or symmetries due to the integrated active magnetic bearings. The use of three-dimensional finite element methods is unpractical for the design of these actuators because electromagnetic quantities have to be calculated with a high spatial resolution and many design Fig. 2. Top view of the planar actuator. For the shown position of the magnet array, the dark coils are active.
    [Show full text]
  • Electric Force and Field
    Magnetism Introduction Just as static electricity was known in ancient times, so was magnetism. In fact, the term “magnetism” comes from the rocks that were found in the ancient Greek province of Magnesia 4000 years ago; Lucretius, a Roman poet and philosopher, wrote about magnets more than 2000 years ago. Magnetic materials were considered even more special than electric materials since they came out of the ground with their force of attraction, while amber had to be rubbed to develop its static electric force. It was also noted very early in human history that if magnetic rock was shaped into the form of a needled and floated on a surface of water, it always pointed in the same direction. We now call that direction “magnetic north.” This was used by Chinese mariners more than 4000 years ago to navigate. Because this direction was the same regardless of your location and which direction you were traveling, it was always possible to know which way you were sailing. There is an important similarity between electricity and magnetism. Electric charge can be either positive or negative; like charges repel and unlike charges attract. Magnets have two poles, called “north” and “south”; like poles repel and unlike poles attract. But there is a key difference as well; while it is possible to separate positive and negative electric charges (that’s what happens when you rub a plastic rod with cloth), it is impossible to separate north and south poles. In fact, a north pole can only exist in the presence of an exactly equal south pole: “there are no magnetic monopoles.” If a magnet is cut in half, the result is not the separation of the north and south poles, it’s the creation of more equal and opposite poles.
    [Show full text]
  • Electricity and Magnetism
    Electricity and Magnetism • Review for Quiz #3 –Exp EB – Force in B-Field • on free charge • on current – Sources of B-Field •Biot-Savart • Ampere’s Law –Exp MF – Magnetic Induction • Faradays Law •Lenz’Rule Apr 5 2002 Electrical Breakdown • Need lot’s of free charges • But electrons stuck in potential well of nucleus U • Need energy ∆U to jump out of well • How to provide this energy? e- ∆U Apr 5 2002 Impact Ionization e- Ukin > ∆U E • Define Vion = ∆U/q Ionization potential •One e- in, two e- out E • Avalanche? Apr 5 2002 Impact Ionization (2) (1) - λmfp e λmfp : Mean Free Path E • To get avalanche we need: ∆Ukin between collisions (1) and (2) > Vion * e • Acceleration in uniform Field ∆Ukin = V2 –V1 = e E d12 • Avalanche condition then E > Vion /λmfp Apr 5 2002 Impact Ionization How big is Mean Free Path? (i) If Density n is big -> λmfp small (ii) If size σ of molecules is big -> λmfp small λmfp = 1/(n σ ) Avalanche if E > Vion /λmfp = Vion n σ Apr 5 2002 Magnetism • Observed New Force between –two Magnets – Magnet and Iron – Magnet and wire carrying current – Wire carrying current and Magnet – Two wires carrying currents • Currents (moving charges) can be subject to and source of Magnetic Force Apr 5 2002 Magnetic Force • Force between Magnets • Unlike Poles attract F F N S N S • Like Poles repel F F N S SN Apr 5 2002 Magnetic Force • Magnets also attract non-magnets! F Iron nail Apr 5 2002 Current and Magnet Wire, I = 0 Wire, I > 0 N N S X S Apr 5 2002 Magnet and Current F Wire I F • Force on wire if I != 0 • Direction of Force depends
    [Show full text]
  • Basic Properties of Magnets ・・・PDF［2.0MB］
    Basic Properties of Magnets In order to achieve an optimal design according to the use, we will provide a guide on the basic properties of the magnets required and the fundamental equations for designing the magnetic circuit for your consideration. Please make use of them as reference material for a more efficient design. 1. Magnetic Hysteresis Loop (BH Curve) As shown in Figure 2, the magnetic flux density in the magnet increases as the current in the coil increases gradually to magnetize the magnet until it finally reaches saturation at a certain point. (Point a in the figure) ＋B a Remanence （Br） Magnetic flux density b Demagnetization curve －H ＋H 0 f Coercivity c Magnetic field strength （oersted） （HcB） e d －B Magnetic hysteresis loop 〈Figure 1〉 Subsequently, when the strength of the magnetic field is reduced by lowering the current from this saturated state, the magnetic flux density decreases from point a to point b without returning to 0. The magnetic flux density remains at point b even if the strength of the magnetic field becomes 0. This value is known as the remanence Br.Next, if the direction of the current is reversed and the magnetic field is increased in the opposite direction, the magnetic flux density decreases gradually from point b until it finally reaches 0 at point c. The strength of the magnetic field at this point is known as the coercivity or coercive force HcB. If this reverse magnetic field is increased further, the magnet will be magnetized in the opposite direction until it reaches saturation at point d.
    [Show full text]
  • Electricity and Magnetism
    Electricity and Magnetism •Reminder – RC circuits – Electric Breakdown Experiment •Today – Magnetism Apr 1 2002 RC Circuits Variable time constant τ = RC R C VC Vin Vin t Apr 1 2002 In-Class Demo Vin t VC t I t Apr 1 2002 In-Class Demo Vin t VC I t t Apr 1 2002 In-Class Demo •Changes in R or C change τ •Large τ smoothes out signals • Sharp edges/rapid changes get removed – high frequencies are suppressed • RC circuits are low-pass filters Apr 1 2002 Experiment EB • Electrical Breakdown – You have seen many examples • Lightning! • Sparks (e.g. Faraday Cage Demo!) • Fluorescent tubes – Study in more detail – Reminder: Ionization Apr 1 2002 Ionization • Electrons and nucleus bound together • Electrons stuck in potential U well of nucleus • Need energy ∆U to jump out of well • How to provide this energy? e- ∆U Apr 1 2002 Impact Ionization e- Ukin > ∆U E • Define Vion = ∆U/q • Ionization potential •One e- in, two e- out E • Avalanche? Apr 1 2002 Impact Ionization (2) (1) - λmfp e λmfp : Mean Free Path E • To get avalanche we need: ∆Ukin between collisions (1) and (2) > Vion * e • Acceleration in uniform Field ∆Ukin = V2 –V1 = e E d12 • Avalanche condition then E > Vion /λmfp Apr 1 2002 Impact Ionization How big is Mean Free Path? (i) If Density n is big -> λmfp small (ii) If size σ of molecules is big -> λmfp small Effective cross-section λmfp = 1/(n σ ) Apr 1 2002 Impact Ionization Avalanche condition E > Vion /λmfp = Vion n σ Experiment EB: Relate E, Vion ,σ Example: Air n ~ 6x1023/22.4 10-3 m3 = 3 x 1025 m-3 σ ~ π r2 ~ 3 x (10-10m)2 = 3
    [Show full text]
  • Dynamical Interactions Between Two Uniformly Magnetized Spheres
    European Journal of Physics Eur. J. Phys. 38 (2017) 015205 (25pp) doi:10.1088/0143-0807/38/1/015205 Dynamical interactions between two uniformly magnetized spheres Boyd F Edwards1,3 and John M Edwards2 1 Department of Physics, Utah State University, Logan, UT 84322, USA 2 Department of Informatics and Computer Science, Idaho State University, Pocatello, ID 83209, USA E-mail: [email protected] Received 27 May 2016, revised 6 September 2016 Accepted for publication 26 September 2016 Published 8 November 2016 Abstract Studies of the two-dimensional motion of a magnet sphere in the presence of a second, fixed sphere provide a convenient venue for exploring magnet–magnet interactions, inertia, friction, and rich nonlinear dynamical behavior. These studies exploit the equivalence of these magnetic interactions to the interac- tions between two equivalent point dipoles. We show that magnet–magnet friction plays a role when magnet spheres are in contact, table friction plays a role at large sphere separations, and eddy currents are always negligible. Web- based simulation and visualization software, called MagPhyx, is provided for education, exploration, and discovery. Keywords: force between uniformly magnetized spheres, torque between uniformly magnetized spheres, equivalence between force between magnet spheres and point dipoles, simulation of magnet interactions, magnet dynamics simulation (Some figures may appear in colour only in the online journal) 1. Introduction Small neodymium magnet spheres are used both in and out of the classroom to teach prin- ciples of mathematics, physics, chemistry, biology, and engineering [1, 2]. They offer engaging hands-on exposure to principles of magnetism and are particularly useful in studying lattice structures, where they offer greater versatility than standard ball-and-stick models because they can connect at a continuous range of angles.
    [Show full text]
  • ALTERNATIVE METHOD to CALCULATE the MAGNETIC FIELD of PERMANENT MAGNETS with AZIMUTHAL SYMMETRY 9 1 ~M · Nˆ Φ (~X) = , 2.1
    EDUCATION Revista Mexicana de F´ısica E 59 (2013) 8–17 JANUARY–JUNE 2013 Alternative method to calculate the magnetic field of permanent magnets with azimuthal symmetry J. M. Camacho and V. Sosaa Facultad de Ingenier´ıa, Universidad Autonoma´ de Yucatan,´ Av. Industrias no contaminantes por Periferico´ Norte, A.P. 150 Cordemex, Merida,´ Yuc., Mexico.´ aPermanent address: Departamento de F´ısica Aplicada, CINVESTAV-IPN, Unidad Merida,´ A.P. 73 Cordemex, Merida,´ Yuc., C.P. 97310, Mexico.´ Phone: (+52-999) 9429445; Fax: (+52-999) 9812917. e-mail: [email protected]. Received 6 September 2012; accepted 8 January 2013 The magnetic field of a permanent magnet is calculated analytically for different geometries. The cases of a sphere, cone, cylinder, ring and rectangular prism are studied. The calculation on the axis of symmetry is presented in every case. For magnets with cylindrical symmetry, we propose an approach based on an expansion in Legendre polynomials to obtain the field at points off the axis. The case of a cylinder magnet was analyzed with this method by calculating the force between two magnets of this shape. Experimental results are presented too, showing a nice agreement with theory. Keywords: Permanent magnet; calculation; analytical; symmetry. PACS: 75.50 Ww; 41.20 Gz. 1. Introduction develop an alternative method of calculation and find ana- lytical expressions for these fields. This will allow estima- tion of many important variables in certain applications, such Since ancient times, magnetism has captured the interest of as the force between magnets. We will see that the results human beings. The feeling of an unseen force (but no less obtained theoretically and experimentally are consistent with invisible than the force of gravity) acting with great inten- each other.
    [Show full text]