MOTION MOUNTAIN

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212 5 electromagnetic effects Ref. 173 Challenge 205 ny Challenge 207 ny Challenge 208 e Page 38 Vol. I, page 103 Ref. 174 Challenge 206 ny For levitation methods that donot consume energy – all such methods are necessarily stationary – a well-known limitation can be found by studying Coulomb’s ‘law’ of electrostatics:nostaticarrangementofelectricfieldscanlevitate acharged object in free space or in air. The same result is valid for gravitational fields and massive objects;* in other words, we cannot produce a local minimum of potential energy in the middle of a box using electric or gravitational fields.This impossibility is calledEarnshaw’stheorem. Speaking mathematically, the solutions of the Laplace equationΔφ=0, the so-called harmonic functions, have minima or maxima only at the border, and never inside the domain of definition. (You proved this yourself on page 175 in volume I.) Earnshaw’s theorem can also be proved by noting that given a potential minimum in free space, Gauss’ theorem for a sphere around that minimum requires that a source of the field be present inside, which is in contradiction with the original assumption. We can deduce that it is also impossible to use electric fields to levitate an electrically neutral body in air: the potential energyUof such a body, with volumeVand dielectric constantε, in an environment of dielectric constantε 0 , is given by U V =−1 2 (ε−ε 0)E 2 . (83) Since the electric fieldEnever has a maximum in the absence of space charge, and since for all materialsε>ε 0 , there cannot be a minimum of potential energy in free space for a neutral body.** To sum up, using static electric or static gravitational fields it is impossible to keep an object from falling; neither quantum mechanics, which incorporates phenomena such as antimatter, nor general relativity, including phenomena such as black holes, change this basic result. For staticmagnetic fields, the argument is analogous to electrical fields: the potential energyUof a magnetizable body of volumeVand permeabilityμin a medium with permeabilityμ 0 containing no current is given by U V =−1 2 (1 μ −1 μ 0 )B 2 (84) and due to the inequalityΔB 2 ⩾0, isolated maxima of a static magnetic field are not possible, only isolated minima. Therefore, it is impossible to levitate paramagnetic (μ> μ o ) or ferromagnetic (μ≫μ 0 ) materials such as steel, including bar magnets, which are all attracted, and not repelled to magnetic field maxima. There are thus two ways to realize magnetic levitation: levitating a diamagnet or using a time-dependent field. Diamagnetic materials (μ
electromagnetic effects and challenges 213 Ref. 176 Ref. 175 Ref. 177 F I G U R E 157 Stable diamagnetic levitation of a graphite bar over rectangular permanent magnets (above) and of two graphite plates, one seen from above and another from the side (centre); below, levitation of a 4 mm diameter NdFeB permanent magnet, above a graphite plate and between two graphite plates, near a large ring magnet (not shown) (© Joachim Schlichting from Ref. 175). Indeed, diamagnetic materials, such as graphite or water, can be levitated by staticmagnetic fieldsbecause theyare attracted to magnetic fieldminima. In fact,it is possibleto levitate magnets if one uses a combination containing diamagnets. A few cases that can easily be replicated on a kitchen table are shown in Figure 157. Another well-known example of diamagnetic levitation is the levitation of superconductors. Indeed, superconductors, at least those of type I, are perfects diamagnets (μ=0).Insomecases,superconductorscaneven besuspended inmid-air,belowamagnet. Also single atoms with a magnetic momentare diamagnets; theyare routinely levitated this way and have also been photographed in this state. Single neutrons, which have a magnetic dipole moment, have been kept in magnetic bottles through magnetic Motion Mountain – The Adventure of Physics copyright © Christoph Schiller June 1990–November 2015 free pdf file available at www.motionmountain.net
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ChristophSchiller MOTION MOUNTAIN t
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Editiovicesima octava. Proprietassc
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DieMenschenstärken,die Sachenklär
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8 preface PHYSICS: Describing motio
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10 preface Your donation to the cha
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12 contents Do we see what exists?
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Light, Charges and Brains Inourques
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16 1 electricity and fields F I G U
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18 1 electricityandfields F I G U R
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20 1 electricity and fields nylon r
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24 1 electricity and fields TA B L
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28 1 electricity and fields Challen
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30 1 electricity and fields If elec
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34 1 electricityandfields F I G U R
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42 1 electricity and fields Oersted
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44 1 electricity and fields Vol. IV
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48 1 electricity and fields Ref. 24
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50 1 electricity and fields Page 21
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52 1 electricity and fields Ref. 29
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64 1 electricity and fields battery
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Vol. IV, page 231 Chapter 2 THE DES
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76 2 the description of electromagn
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Chapter 3 WHAT IS LIGHT? Ref. 57 Pa
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96 3 what is light? Electric field
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98 3 what is light? spark transmitt
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100 3 what is light? Ref. 59 Page 1
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102 3 what is light? primary infrar
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104 3 what is light? Page 84 gases
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106 3 what is light? Fre - Waveleng
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108 3 what is light? Challenge 110
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110 3 what is light? incoming light
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112 3 what is light? Ref. 70 Challe
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114 3 what is light? F I G U R E 69
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116 3 what is light? light Ref. 72
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118 3 what is light? light Challeng
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120 3 what is light? momentumLis gi
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122 3 what is light? Ref. 80 Challe
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126 3 what is light? Ref. 87 Page 9
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130 3 what is light? v ph v So F I
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132 3 what is light? The forerunner
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134 3 what is light? vocabulary. On
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136 3 what is light? source beam sp
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138 3 what is light? B B Beam in A
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Chapter 4 IMAGES AND THE EYE - OPTI
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142 4 images and the eye - optics 2
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144 4 images and the eye - optics C
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146 4 images and the eye - optics R
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148 4 images and the eye - optics T
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160 4 images and the eye - optics
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262 8 thought and language F I G U
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264 8 thought and language Challeng
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266 8 thought and language Ref. 241
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268 8 thought and language Vol. IV,
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270 8 thought and language . . 0 1
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272 8 thought and language Ref. 251
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274 8 thought and language ⊳ Ever
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276 8 thought and language F I G U
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Chapter 10 CLASSICAL PHYSICS IN A N
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322 10 classical physics in a nutsh
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328 a units, measurements and const
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342 challenge hints and solutions n
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348 challenge hints and solutions v
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350 challenge hints and solutions b
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354 challenge hints and solutions o
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358 challenge hints and solutions C
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360 challenge hints and solutions V
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362 bibliography 10 Thiseffect hasf
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364 bibliography 37 A discussionof
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366 bibliography M.B. van der Mark
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368 bibliography Vol. I, page 96 19
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370 bibliography S.W. McCuskey, F.C
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372 bibliography 119 A complete lis
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374 bibliography F I G U R E 178 Th
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376 bibliography in unserer Zeit33,
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378 bibliography private communicat
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380 bibliography success of structu
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382 bibliography Vol. IV, page 135
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384 bibliography and 304. 260 For a
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386 bibliography Springer, 1996. wh
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CREDITS Acknowledgements Many peopl
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NAME INDEX A Abbott A Abbott,T.A.37
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396 name index D Democritus Democri
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398 name index H Holmes Holmes, C.D
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400 name index M Mohr Mohr, P.J. 38
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402 name index S Sheldon Sheldon, E
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404 name index Z Zybin Motion Mount
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406 subject index A units 328 Aplys
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408 subject index C classical class
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410 subject index E Earth Earth age
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412 subject index F finite finite 2
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414 subject index I infrared photog
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416 subject index L Lorentz Lorentz
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418 subject index M motion motion a
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420 subject index P physiologists p
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422 subject index S Scarabeus Scara
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424 subject index T table d’Unit
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426 subject index W Wien’s Wien
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