DOCSLIB.ORG

Principle and Characteristic of Lorentz Force Propeller

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Principle and Characteristic of Lorentz Force Propeller J. Electromagnetic Analysis & Applications, 2009, 1: 229-235 229 doi:10.4236/jemaa.2009.14034 Published Online December 2009 (http://www.SciRP.org/journal/jemaa) Principle and Characteristic of Lorentz Force Propeller Jing ZHU Northwest Polytechnical University, Xi’an, Shaanxi, China. Email: [email protected] Received August 4th, 2009; revised September 1st, 2009; accepted September 9th, 2009. ABSTRACT This paper analyzes two methods that a magnetic field can be generated, and classifies them under two types: 1) Self-field: a magnetic field can be generated by electrically charged particles move, and its characteristic is that it can’t be independent of the electrically charged particles. 2) Radiation field: a magnetic field can be generated by electric field change, and its characteristic is that it independently exists. Lorentz Force Propeller (ab. LFP) utilize the charac- teristic that radiation magnetic field independently exists. The carrier of the moving electrically charged particles and the device generating the changing electric field are fixed together to form a system. When the moving electrically charged particles under the action of the Lorentz force in the radiation magnetic field, the system achieves propulsion. Same as rocket engine, the LFP achieves propulsion in vacuum. LFP can generate propulsive force only by electric energy and no propellant is required. The main disadvantage of LFP is that the ratio of propulsive force to weight is small. Keywords: Electric Field, Magnetic Field, Self-Field, Radiation Field, the Lorentz Force 1. Introduction also due to the changes in observation angle.) “If the electric quantity carried by the particles is certain, the The magnetic field generated by a changing electric field magnetic field generated by the particles is entirely de- is a kind of radiation field and it independently exists. termined by the motion speed of the particles” [2] and it When the moving electrically charged particles are sub- is a single-valued function of the motion speed. In other jected to Lorentz force in the magnetic field, the device words, there must be a certain magnetic field corre- generating the changing electric field isn’t subjected to sponding to the electrically charged particles when the any reacting force. motion state of the particles is determined. This suggests 2. Theoretical Basis that the magnetic field generated by the moving electri- cally charged particles is the state quantity which de- As we know, a magnetic field can be generated by two scribes the motion state of the electrically charged parti- methods: one is electrically charged particles move and cles. The state quantity, which reflects the sate of a the other is electric field change [1]. However, the mag- physical object, is different in different reference systems, netic fields generated by the two methods are entirely but it can’t be separated from the physical object and different in nature. can’t be independent of the physical object. Thus, the We discuss the magnetic field generated by moving magnetic field generated by moving electrically charged electrically charged particles at first. It’s well known that particles is different in different reference systems. Of the electric quantity carried by the electrically charged cause the magnetic field can’t be independent of the particles isn’t affected by the motion state of the particles. electrically charged particles and it’s the self-field of the This shows that when the electrically charged particles particles. generate a magnetic field due to their movement, the As the magnetic field generated by moving electrically electric quantity owned by the particles doesn’t change charged particles, kinetic energy also belongs to state with the generation of the magnetic field. Only the dis- quantity. They reflect the motion state of a physical ob- tribution of the electric energy, i.e., the distribution of the ject in the form of energy. The difference between them electric field, is affected by the motion state of the parti- is as follows. The motion state of a physical object is cles. (We will mention below that even this change is described by its mass in kinetic energy, which is central- Copyright © 2009 SciRes JEMAA 230 Principle and Characteristic of Lorentz Force Propeller ized at the object. The motion state of the electrically the electromagnetic waves are separated from the an- charged particles is described by the electric quantity in tenna and independent of the antenna. [4] the magnetic field, which is generated by the moving It’s another issue that a changing magnetic field also particles and distributed in the space around the particles. generates an electric field, which is known as the issue Then we discuss the magnetic field generated by a that magnetic energy is transformed into electric energy. changing electric field. Based on the energy conservation Similarly, the electric field generated by the changing principle, the total energy in the electromagnetic field magnetic field is independent of the magnetic field. “A isn’t changed in the course that a magnetic field is gener- magnetic field is generated by a changing electric field, ated by a changing electric field. This indicates that the and an electric field is generated by the changing mag- magnetic energy is continuously generated and the elec- netic field” [5], so repeatedly. The process is continu- tric energy is continuously decreased at the same time. ously repeated. The most common example is electro- The magnetic energy is transformed from the decreased magnetic waves. Just because not only the magnetic field electric energy. The course that a magnetic field is gen- generated by a time varying electric field but also the erated by a changing electric field is a course that electric electric field generated by the time varying magnetic energy is transformed into magnetic energy. The electric field is a radiation field, electromagnetic waves are energy and the magnetic energy are localized in the elec- formed and spread continuously. On the surface, the tric field and the magnetic field respectively. According electric field and the magnetic field in the electromag- to “the superposition principle of electric fields” [3], the netic waves are tightly fastened and linked. But in fact, initial electric field can be regarded as the superposition they are mutually independent and irrelevant to each of two mutually independent electric fields in the other. above-mentioned course at any time. One of the two The law of a time varying electromagnetic field fol- electric fields is the electric field with the localized elec- lows the Maxwell equation. “The differential form of the tric energy, which has been transformed into magnetic Maxwell equations in free space is given as follows [6] energy. The other is the electric field with the localized electric energy, which hasn’t been transformed into D magnetic energy. Obviously, the former electric field t generated a magnetic field but the magnetic field has B disappeared. The latter electric field still exists but it E hasn’t generated any magnetic field. Because the two t electric fields are mutually independent, the magnetic D0 field generated by the former electric field is independent of the latter electric field. In other words, the magnetic B0” field and the electric field are mutually independent in It can be seen from the equations that the size of the the course that the magnetic field is generated by the magnetic field intensity is determined by the changing changing electric field. The former electric field is also a rate of the electric field. However, according to the tie between the magnetic field and the device generating above-mentioned, the magnetic field is transformed from the changing electric field and it’s shown as the device the disappeared electric field, so the size of the magnetic generating the changing electric field → the electric field field intensity should be determined by the changing → the magnetic field. With the disappearance of the amount of the electric field. They seem to be contradic- former electric field, the magnetic field generated by the tory. This is because that the two courses that the time changing electric field is not only independent of the varying electric field generates the magnetic field and the electric field, but also independent of the device generat- time varying magnetic field generates the electric field ing the changing electric field at the same time. The occur at the same time, rather than the magnetic field changing electric field and the device generating the begins to transform into the electric field after the electric changing electric field are only the initial condition to field has thoroughly transformed into the magnetic field. generate a magnetic field. The magnetic field generated It can be seen more easily from electromagnetic waves. by the changing electric field is a radiation field and it’s Finally, we discuss the different changes in the electric irrelevant to reference system. field when the electrically charged particles have differ- The relation between the magnetic field generated by a ent motion speeds or have acceleration. The relations changing electric field and the device generating the between the changes and electromagnetic radiation are changing electric field is similar to the relation between also investigated. electromagnetic waves and an antenna. The antenna is The situation that the electrically charged particles only the initial condition to generate electromagnetic have different motion speeds is firstly discussed. As waves. When the electromagnetic waves have been gen- shown in Figure 1, a is a stationary particle carrying posi- erated, they are irrelevant to the antenna. In other words, tive charge and its electric field is distributed in the space Copyright © 2009 SciRes JEMAA Principle and Characteristic of Lorentz Force Propeller 231 change of the electric field. Electrically charged particles moving in a straight line with a uniform speed can’t gen- erate any electromagnetic radiation.
Load more

Recommended publications
  • Particle Motion
    Physics of fusion power Lecture 5: particle motion Gyro motion The Lorentz force leads to a gyration of the particles around the magnetic field We will write the motion as The Lorentz force leads to a gyration of the charged particles Parallel and rapid gyro-motion around the field line Typical values For 10 keV and B = 5T. The Larmor radius of the Deuterium ions is around 4 mm for the electrons around 0.07 mm Note that the alpha particles have an energy of 3.5 MeV and consequently a Larmor radius of 5.4 cm Typical values of the cyclotron frequency are 80 MHz for Hydrogen and 130 GHz for the electrons Often the frequency is much larger than that of the physics processes of interest. One can average over time One can not however neglect the finite Larmor radius since it lead to specific effects (although it is small) Additional Force F Consider now a finite additional force F For the parallel motion this leads to a trivial acceleration Perpendicular motion: The equation above is a linear ordinary differential equation for the velocity. The gyro-motion is the homogeneous solution. The inhomogeneous solution Drift velocity Inhomogeneous solution Solution of the equation Physical picture of the drift The force accelerates the particle leading to a higher velocity The higher velocity however means a larger Larmor radius The circular orbit no longer closes on itself A drift results. Physics picture behind the drift velocity FxB Electric field Using the formula And the force due to the electric field One directly obtains the so-called ExB velocity Note this drift is independent of the charge as well as the mass of the particles Electric field that depends on time If the electric field depends on time, an additional drift appears Polarization drift.
    [Show full text]
  • 19-8 Magnetic Field from Loops and Coils
    Answer to Essential Question 19.7: To get a net magnetic field of zero, the two fields must point in opposite directions. This happens only along the straight line that passes through the wires, in between the wires. The current in wire 1 is three times larger than that in wire 2, so the point where the net magnetic field is zero is three times farther from wire 1 than from wire 2. This point is 30 cm to the right of wire 1 and 10 cm to the left of wire 2. 19-8 Magnetic Field from Loops and Coils The Magnetic Field from a Current Loop Let’s take a straight current-carrying wire and bend it into a complete circle. As shown in Figure 19.27, the field lines pass through the loop in one direction and wrap around outside the loop so the lines are continuous. The field is strongest near the wire. For a loop of radius R and current I, the magnetic field in the exact center of the loop has a magnitude of . (Equation 19.11: The magnetic field at the center of a current loop) The direction of the loop’s magnetic field can be found by the same right-hand rule we used for the long straight wire. Point the thumb of your right hand in the direction of the current flow along a particular segment of the loop. When you curl your fingers, they curl the way the magnetic field lines curl near that segment. The roles of the fingers and thumb can be reversed: if you curl the fingers on Figure 19.27: (a) A side view of the magnetic field your right hand in the way the current goes around from a current loop.
    [Show full text]
  • Physics 2102 Lecture 2
    Physics 2102 Jonathan Dowling PPhhyyssicicss 22110022 LLeeccttuurree 22 Charles-Augustin de Coulomb EElleeccttrriicc FFiieellddss (1736-1806) January 17, 07 Version: 1/17/07 WWhhaatt aarree wwee ggooiinngg ttoo lleeaarrnn?? AA rrooaadd mmaapp • Electric charge Electric force on other electric charges Electric field, and electric potential • Moving electric charges : current • Electronic circuit components: batteries, resistors, capacitors • Electric currents Magnetic field Magnetic force on moving charges • Time-varying magnetic field Electric Field • More circuit components: inductors. • Electromagnetic waves light waves • Geometrical Optics (light rays). • Physical optics (light waves) CoulombCoulomb’’ss lawlaw +q1 F12 F21 !q2 r12 For charges in a k | q || q | VACUUM | F | 1 2 12 = 2 2 N m r k = 8.99 !109 12 C 2 Often, we write k as: 2 1 !12 C k = with #0 = 8.85"10 2 4$#0 N m EEleleccttrricic FFieieldldss • Electric field E at some point in space is defined as the force experienced by an imaginary point charge of +1 C, divided by Electric field of a point charge 1 C. • Note that E is a VECTOR. +1 C • Since E is the force per unit q charge, it is measured in units of E N/C. • We measure the electric field R using very small “test charges”, and dividing the measured force k | q | by the magnitude of the charge. | E |= R2 SSuuppeerrppoossititioionn • Question: How do we figure out the field due to several point charges? • Answer: consider one charge at a time, calculate the field (a vector!) produced by each charge, and then add all the vectors! (“superposition”) • Useful to look out for SYMMETRY to simplify calculations! Example Total electric field +q -2q • 4 charges are placed at the corners of a square as shown.
    [Show full text]
  • Structure of the Electron Diffusion Region in Magnetic Reconnection with Small Guide Fields
    Structure of the electron diffusion region in magnetic reconnection with small guide fields MASSA0HU- by Jonathan Ng JAN I Submitted to the Department of Physics in partial fulfillment of the requirements for the degree of Bachelor of Science in Physics at the MASSACHUSETTS INSTITUTE OF TECHNOLOGY June 2012 @ Jonathan Ng, MMXII. All rights reserved. The author hereby grants to MIT permission to reproduce and distribute publicly paper and electronic copies of this thesis document in whole or in part. Author ............ Department of Physics May 11, 2012 / / Certified by..... *2) Jan Egedal-Pedersen Associate Professor of Physics Thesis Supervisor Accepted by ...... .............. ........ ............. ... Nergis Mavalvala Senior Thesis Coordinator, Department of Physics 2 Structure of the electron diffusion region in magnetic reconnection with small guide fields by Jonathan Ng Submitted to the Department of Physics on May 11, 2012, in partial fulfillment of the requirements for the degree of Bachelor of Science in Physics Abstract Observations in the Earth's magnetotail and kinetic simulations of magnetic reconnec- tion have shown high electron pressure anisotropy in the inflow of electron diffusion regions. This anisotropy has been accurately accounted for in a new fluid closure for collisionless reconnection. By tracing electron orbits in the fields taken from particle-in-cell simulations, the electron distribution function in the diffusion region is reconstructed at enhanced resolutions. For antiparallel reconnection, this reveals its highly structured nature, with striations corresponding to the number of times an electron has been reflected within the region, and exposes the origin of gradients in the electron pressure tensor important for momentum balance. The addition of a guide field changes the nature of the electron distributions, and the differences are accounted for by studying the motion of single particles in the field geometry.
    [Show full text]
  • Field Line Motion in Classical Electromagnetism John W
    Field line motion in classical electromagnetism John W. Belchera) and Stanislaw Olbert Department of Physics and Center for Educational Computing Initiatives, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139 ͑Received 15 June 2001; accepted 30 October 2002͒ We consider the concept of field line motion in classical electromagnetism for crossed electromagnetic fields and suggest definitions for this motion that are physically meaningful but not unique. Our choice has the attractive feature that the local motion of the field lines is in the direction of the Poynting vector. The animation of the field line motion using our approach reinforces Faraday’s insights into the connection between the shape of the electromagnetic field lines and their dynamical effects. We give examples of these animations, which are available on the Web. © 2003 American Association of Physics Teachers. ͓DOI: 10.1119/1.1531577͔ I. INTRODUCTION meaning.10,11 This skepticism is in part due to the fact that there is no unique way to define the motion of field lines. Classical electromagnetism is a difficult subject for begin- Nevertheless, the concept has become an accepted and useful ning students. This difficulty is due in part to the complexity one in space plasma physics.12–14 In this paper we focus on a of the underlying mathematics which obscures the physics.1 particular definition of the velocity of electric and magnetic The standard introductory approach also does little to con- field lines that is useful in quasi-static situations in which the nect the dynamics of electromagnetism to the everyday ex- E and B fields are mutually perpendicular.
    [Show full text]
  • Electromagnetic Fields and Energy
    MIT OpenCourseWare http://ocw.mit.edu Haus, Hermann A., and James R. Melcher. Electromagnetic Fields and Energy. Englewood Cliffs, NJ: Prentice-Hall, 1989. ISBN: 9780132490207. Please use the following citation format: Haus, Hermann A., and James R. Melcher, Electromagnetic Fields and Energy. (Massachusetts Institute of Technology: MIT OpenCourseWare). http://ocw.mit.edu (accessed [Date]). License: Creative Commons Attribution-NonCommercial-Share Alike. Also available from Prentice-Hall: Englewood Cliffs, NJ, 1989. ISBN: 9780132490207. Note: Please use the actual date you accessed this material in your citation. For more information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms 9 MAGNETIZATION 9.0 INTRODUCTION The sources of the magnetic fields considered in Chap. 8 were conduction currents associated with the motion of unpaired charge carriers through materials. Typically, the current was in a metal and the carriers were conduction electrons. In this chapter, we recognize that materials provide still other magnetic field sources. These account for the fields of permanent magnets and for the increase in inductance produced in a coil by insertion of a magnetizable material. Magnetization effects are due to the propensity of the atomic constituents of matter to behave as magnetic dipoles. It is natural to think of electrons circulating around a nucleus as comprising a circulating current, and hence giving rise to a magnetic moment similar to that for a current loop, as discussed in Example 8.3.2. More surprising is the magnetic dipole moment found for individual electrons. This moment, associated with the electronic property of spin, is defined as the Bohr magneton e 1 m = ± ¯h (1) e m 2 11 where e/m is the electronic charge­to­mass ratio, 1.76 × 10 coulomb/kg, and 2π¯h −34 2 is Planck’s constant, ¯h = 1.05 × 10 joule­sec so that me has the units A − m .
    [Show full text]
  • Field Lines Electric Flux Recall That We Defined the Electric Field to Be the Force Per Unit Charge at a Particular Point
    Gauss’ Law Field Lines Electric Flux Recall that we defined the electric field to be the force per unit charge at a particular point: P For a source point charge Q and a test charge q at point P q Q at P If Q is positive, then the field is directed radially away from the charge. + Note: direction of arrows Note: spacing of lines Note: straight lines If Q is negative, then the field is directed radially towards the charge. - Negative Q implies anti-parallel to Note: direction of arrows Note: spacing of lines Note: straight lines + + Field lines were introduced by Michael Faraday to help visualize the direction and magnitude of the electric field. The direction of the field at any point is given by the direction of the field line, while the magnitude of the field is given qualitatively by the density of field lines. In the above diagrams, the simplest examples are given where the field is spherically symmetric. The direction of the field is apparent in the figures. At a point charge, field lines converge so that their density is large - the density scales in proportion to the inverse of the distance squared, as does the field. As is apparent in the diagrams, field lines start on positive charges and end on negative charges. This is all convention, but it nonetheless useful to remember. - + This figure portrays several useful concepts. For example, near the point charges (that is, at a distance that is small compared to their separation), the field becomes spherically symmetric. This makes sense - near a charge, the field from that one charge certainly should dominate the net electric field since it is so large.
    [Show full text]
  • Magnetic Fields, Magnetic Forces, and Sources of Magnetic Fields
    Magnetic Fields, Magnetic Forces, and Sources of Magnetic Fields W07D2 1 Announcements Problem Set 6 Due Thursday at 9 pm .This problem set is an introduction to Experiment 1 and the two problems from W06D2. Week 8 LS1 due Mon at 8:30 am Week 8 LS2 due Mon at 8:30 am Week 8 LS3 due Wed at 8:30 am Week 8 LS4 due Wed at 8:30 am 2 Outline Magnetic Field Lorentz Force Law Magnetic Force on Current Carrying Wire Sources of Magnetic Fields Biot-Savart Law 3 Magnetic Field of the Earth North magnetic pole located in southern hemisphere http://www.youtube.com/watch?v=AtDAOxaJ4Ms 4 Magnetic Force on Moving Charges Force on ! ! positive charge B ! Fq = qvq × B vq B B q vq F F q q B q + B Force on negative charge Magnetic force is perpendicular to both velocity of the charge and magnetic field 5 CQ: Cross Product and Magnetic Force An electron is traveling up in a magnetic field that points to the right. What is the direction of the force on the electron? v 1. Up. q 2. Down. B 3. Left. q = e 4. Right. 5. Into plane of figure. 6. Out of plane of figure. 6 Group Problem: Cyclotron Motion A positively charged particle vq with charge +q and mass m is B moving with speed v in a + uniform magnetic field of + q magnitude B directed into the plane of figure will undergo circular motion. Find (1) the radius R of the orbit, (2) the period T of the motion, (3) the angular frequency ω.
    [Show full text]
  • Lecture 2 Electric Fields Chp. 22 Ed. 7
    Lecture 2 Electric Fields Chp. 22 Ed. 7 • Cartoon - Analogous to gravitational field • Warm-up problems , Physlet • Topics – Electric field = Force per unit Charge – Electric Field Lines – Electric field from more than 1 charge – Electric Dipoles – Motion of point charges in an electric field – Examples of finding electric fields from continuous charges • List of Demos – Van de Graaff Generator, workings,lightning rod, electroscope, – Field lines using felt,oil, and 10 KV supply., • One point charge • Two same sign point charges • Two opposite point charges • Slab of charge – Smoke remover or electrostatic precipitator – Kelvin water drop generator – Electrophorus 1 2 Concept of the Electric Field • Definition of the electric field. Whenever charges are present and if I bring up another charge, it will feel a net Coulomb force from all the others. It is convenient to say that there is field there equal to the force per unit positive charge. E=F/q0. • The question is how does charge q0 know about 1charge q1 if it does not “touch it”? Even in a vacuum! We say there is a field produced by q1 that extends out in space everywhere. • The direction of the electric field is along r and points in the direction a positive test charge would move. This idea was proposed by Michael Faraday in the 1830’s. The idea of the field replaces the charges as defining the situation. Consider two point charges: r q1 q0 3 r q1 + q0 2 The Coulomb force is F= kq1q0/r The force per unit charge is E = F/q0 2 Then the electric field at r is E = kq1/r due to the point charge q1 .
    [Show full text]
  • A Problem-Solving Approach – Chapter 5: the Magnetic Field
    chapter S the magnetic field 314 The Magnetic Field The ancient Chinese knew that the iron oxide magnetite (FesO 4 ) attracted small pieces of iron. The first application of this effect was the navigation compass, which was not developed until the thirteenth century. No major advances were made again until the early nineteenth century when precise experiments discovered the properties of the magnetic field. 5-1 FORCES ON MOVING CHARGES 5-1-1 The Lorentz Force Law It was well known that magnets exert forces on each other, but in 1820 Oersted discovered that a magnet placed near a current carrying wire will align itself perpendicular to the wire. Each charge q in the wire, moving with velocity v in the magnetic field B [teslas, (kg-s 2 -A-')], felt the empirically determined Lorentz force perpendicular to both v and B f =q(vx B) (1) as illustrated in Figure 5-1. A distribution of charge feels a differential force df on each moving incremental charge element dq: df = dq(vx B) (2) V B q f q(v x B) Figure 5-1 A charge moving through a magnetic field experiences the Lorentz force perpendicular to both its motion and the magnetic field. Forces on Moving Charges 315 Moving charges over a line, surface, or volume, respectively constitute line, surface, and volume currents, as in Figure 5-2, where (2) becomes pfv x B dV= Jx B dV (J = pfv, volume current density) dS df= a-vxB dS=KXB (K = orfv, surface current density) (3) AfvxB dl =IxB dl (I=Afv, line current) B v : -- I dl =--ev di df = Idl x B (a) B dS K dS di > d1 KdSx B (b) B d V 1K----------+-.
    [Show full text]
  • ELECTRICITY & MAGNETISM Lecture Notes for Phys 121 Abstract
    file: notes121.tex ELECTRICITY & MAGNETISM Lecture notes for Phys 121 Dr. Vitaly A. Shneidman Department of Physics, New Jersey Institute of Technology, Newark, NJ 07102 (Dated: March 6, 2021) Abstract These notes are intended as an addition to the lectures given in class. They are NOT designed to replace the actual lectures. Some of the notes will contain less information then in the actual lecture, and some will have extra info. Not all formulas which will be needed for exams are contained in these notes. Also, these notes will NOT contain any up to date organizational or administrative information (changes in schedule, assignments, etc.) but only physics. If you notice any typos - let me know at [email protected]. I will keep all notes in a single file - each time you can print out only the added part. A few other things: Graphics: Some of the graphics is deliberately unfinished, so that we have what to do in class. Preview topics: can be skipped upon the 1st reading, but will be useful in the future. Advanced topics: these will not be represented on the exams. Read them only if you are really interested in the material. 1 file: notes121.tex Contents I. Introduction 2 A. Vectors 2 1. Single vector 3 2. Two vectors: addition 3 3. Two vectors: scalar (dot) product 5 4. Two vectors: vector product 6 B. Fields 8 1. Representation of a field; field lines 10 2. Properties of field lines and related definitions 10 II. Electric Charge 12 A. Notations and units 12 B. Superposition of charges 12 C.
    [Show full text]
  • Essential Magnetohydrodynamics for Astrophysics
    Essential magnetohydrodynamics for astrophysics H.C. Spruit Max Planck Institute for Astrophysics henk@ mpa-garching.mpg.de v3.5.1, August 2017 The most recent version of this text, including small animations of elementary MHD pro- cesses, is located at http://www.mpa-garching.mpg.de/~henk/mhd12.zip (25 MB). Links To navigate the text, use the bookmarks bar of your pdf viewer, and/or the links highlighted in color. Links to pages, sections and equations are in blue, those to the animations red, external links such as urls in cyan. When using these links, you will need a way of returning to the location where you came from. This depends on your pdf viewer, which typically does not provide an html-style backbutton. On a Mac, the key combination cmd-[ works with Apple Preview and Skim, cmd-left-arrow with Acrobat. In Windows and Unix Acrobat has a way to add a back button to the menu bar. The appearance of the animations depends on the default video viewer of your system. On the Mac, the animation links work well with Acrobat Reader, Skim and with the default pdf viewer in the Latex distribution (tested with VLC and Quicktime Player), but not with Apple preview. i Contents 1. Essentials2 1.1. Equations . .2 1.1.1. The MHD approximation . .2 1.1.2. Ideal MHD . .3 1.1.3. The induction equation . .4 1.1.4. Geometrical meaning of div B =0 ..................4 1.1.5. Electrical current . .5 1.1.6. Charge density . .6 1.1.7.
    [Show full text]