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Version vom 2009-02-14
author Klaus Thoden <kthoden@mpiwg-berlin.mpg.de>
date Thu, 02 May 2013 11:08:12 +0200
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1 <?xml version="1.0" encoding="utf-8" ?>
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4 <!--http://www.w3.org/TR/xhtml1/DTD/xhtml1-transitional.dtd-->
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5 <html>
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7 <head><title></title>
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8 <meta http-equiv="Content-Type" content="text/html; charset=iso-8859-1" />
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9 <meta name="generator" content="TeX4ht (http://www.cse.ohio-state.edu/~gurari/TeX4ht/mn.html)" />
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10 <meta name="originator" content="TeX4ht (http://www.cse.ohio-state.edu/~gurari/TeX4ht/mn.html)" />
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11 <!-- xhtml,html -->
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12 <meta name="src" content="Einst_Bemer_de_1909_02.tex" />
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13 <meta name="date" content="2005-02-25 10:07:00" />
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14 <link rel="stylesheet" type="text/css" href="Einst_Bemer_de_1909_02.css" />
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15 </head><body >
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17 <!--l. 12--><p class="noindent"><pb/></p>
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18 <div class="center" >
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20 <!--l. 13--><p class="noindent">
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21 </p><!--l. 14--><p class="noindent"><span
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22 class="cmr-12">11. </span><span
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23 class="cmbxti-10x-x-120">Bemerkungen zu unserer Arbeit: </span> <br/><span
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24 class="cmbxti-10x-x-120">,,</span><span
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25 class="cmbxti-10x-x-120">Über die</span>
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26 <span
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27 class="cmbxti-10x-x-120">elektromagnetischen Grundgleichungen </span> <br/><span
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28 class="cmbxti-10x-x-120">f</span><span
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29 class="cmbxti-10x-x-120">ür bewegte</span>
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30 <span
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31 class="cmbxti-10x-x-120">K</span><span
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32 class="cmbxti-10x-x-120">örper&#8220;; </span> <br/><span
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33 class="cmbxti-10x-x-120">von A. Einstein und J. Laub.</span></p></div>
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34 <div class="center" >
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36 <!--l. 20--><p class="noindent">
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37 </p><!--l. 21--><p class="noindent">--------</p></div>
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38 <!--l. 24--><p class="indent"> Hr. Laue war so freundlich, uns auf eine in unserer im <br/>Titel genannten Arbeit enthaltene Unrichtigkeit hinzuweisen.<sup ><span
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39 class="cmr-7">1</span></sup>) <br/>Wir sagen dort (Ann. d. Phys. 26. p. 535. 1908):
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40 </p><!--l. 29--><p class="indent"> ,,Es ist bemerkenswert, daß die Grenzbedingungen für die <br/>Vektoren G, D, H, B an der Grenze zweier Medien dieselben <br/>sind, wie für ruhende Körper. Es folgt dies direkt aus den <br/>Gleichungen (1a) bis (4a).&#8220;
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41 </p><!--l. 34--><p class="indent"> Abgesehen davon, daß für die Herleitung der Grenz-<br/>bedingungen die Gleichungen (3a) und (4a) nicht in Betracht <br/>kommen, ist diese Behauptung nur dann richtig, wenn die Be-<br/>wegungskomponente normal zur Grenzfläche verschwindet, was <br/>bei der im <span
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42 class="cmsy-10">§ </span>2 der genannten Arbeit behandelten Aufgabe
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43 <br/>tatsächlich zutrifft. Die allgemein gültigen Grenzbedingungen <br/>findet man wohl am leichtesten auf folgendem Wege, der dem <br/>von Heinrich Hertz eingeschlagenen entspricht.
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44 </p><!--l. 41--><p class="indent"> Ist die Grenzfläche, oder besser gesagt, die unendlich <br/>dünne Grenzübergangsschale, beliebig bewegt, so werden sich <br/>in einem momentan in ihr gelegenen ruhenden Punkt die das <br/>elektromagnetische Feld bestimmenden Größen im allgemeinen <br/>unstetig bzw. unendlich rasch mit der Zeit ändern; diese <br/>Änderungen werden aber stetig sein für einen Punkt, der sich <br/><span
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45 class="cmti-10">mit der Materie bewegt. </span>Es wird also die Anwendung des
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46 <br/>Operators
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47 </p>
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48 <center class="par-math-display" >
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49 <img
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50 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Bemer_de_1909_02/fulltext/img/Einst_Bemer_de_1909_020x.png" alt="@ @t + (d \~/ ) " class="par-math-display" /></center>
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51 <!--l. 54--><p class="nopar">
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52 </p><!--l. 58--><p class="noindent">an einem Skalar oder einem Vektor auch in der Grenzfläche <br/>----------
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53 </p><!--l. 62--><p class="indent"> 1) Hr. Laue hat uns in seinem Briefe bereits die Grenzbedingungen <br/>richtig angegeben und uns eine andere Ableitung derselben mitgeteilt. <pb/>
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54 </p><!--l. 69--><p class="indent">
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55
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56 </p><!--l. 70--><p class="noindent">nicht zu unendlich großen Werten führen. Schreiben wir nun <br/>die Gleichung (1a)<sup ><span
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57 class="cmr-7">1</span></sup>) in der Form:
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58 </p>
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59 <center class="par-math-display" >
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60 <img
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61 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Bemer_de_1909_02/fulltext/img/Einst_Bemer_de_1909_021x.png" alt=" { } 1 @D- 1 c @t +(d \~/ ) D + q = curl H + c (d \~/ ) D " class="par-math-display" /></center>
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62 <!--l. 79--><p class="nopar">
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63 </p><!--l. 83--><p class="noindent">und nehmen wir an, daß die Stromdichte <span
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64 class="cmmi-10"><img
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65 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Bemer_de_1909_02/fulltext/img/cmmi10-18.png" alt="q" class="cmmi-10x-x-18" align="middle" /> </span>auch in der Grenz-<br/>schicht endlich sei, so ist die linke Seite dieser Gleichung in <br/>der Grenzschicht endlich. Dasselbe gilt also auch für die <br/>rechte Seite der Gleichung.
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66 </p><!--l. 88--><p class="indent"> Zur leichten Interpretation dieses Resultates denken wir <br/>uns das Koordinatensystem so gelegt, daß ein bestimmtes, <br/>unendlich kleines Stück der Grenzfläche, das wir nun be-<br/>trachten wollen, der <span
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67 class="cmmi-10">Y Z</span>-Ebene parallel sei. Dann ist klar, <br/>daß die Ableitungen aller Größen nach <span
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68 class="cmmi-10">y </span>und <span
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69 class="cmmi-10">z </span>in dem be-<br/>trachteten Stück der Grenzfläche endlich bleiben. Es muß <br/>also auch der Inbegriff derjenigen Glieder der rechten Seite <br/>obiger Gleichung, die Differentiationen nach <span
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70 class="cmmi-10">x </span>enthalten, etwas <br/>Endliches liefern. Durch einfaches Entwickeln der rechten Seite <br/>und Weglassen der nach <span
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71 class="cmmi-10">y </span>und <span
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72 class="cmmi-10">z</span>
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73 differenzierten Glieder gelangt <br/>man zu dem Resultate, daß in der Grenzschicht die Ausdrücke:
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74 </p>
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75 <center class="par-math-display" >
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76 <img
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77 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Bemer_de_1909_02/fulltext/img/Einst_Bemer_de_1909_022x.png" alt="dx @Dx-, c @x @Hz-- dx @Dy-, @x c @x @Hy- + dx @Dz- @x c @x " class="par-math-display" /></center>
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78 <!--l. 112--><p class="nopar">
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79 </p><!--l. 116--><p class="noindent">endlich bleiben. Setzen wir noch voraus, daß die Geschwindig-<br/>keitskomponenten an der Grenzfläche keinen Sprung erleiden, <br/>so folgt daraus, daß die Ausdrücke:
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80 </p>
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81 <center class="par-math-display" >
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82 <img
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83 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Bemer_de_1909_02/fulltext/img/Einst_Bemer_de_1909_023x.png" alt="Dx, Hy + dxDz, c H - dxD z c y " class="par-math-display" /></center>
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84 <!--l. 128--><p class="nopar">
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85 </p><!--l. 132--><p class="noindent">auf beiden Seiten der Grenzfläche (<span
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86 class="cmmi-10">Y Z</span>-Ebene) denselben Wert <br/>
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87
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88 </p><!--l. 135--><p class="noindent">----------
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89 </p><!--l. 138--><p class="indent"> 1)1. c. <pb/>
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90 </p><!--l. 142--><p class="indent">
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91
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92 </p><!--l. 143--><p class="noindent">haben. Da D<sub ><span
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93 class="cmmi-7">x</span></sub> und die Komponenten von d stetig sind, können <br/>wir die beiden letzten Ausdrücke auch ersetzen durch:
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94 </p>
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95 <center class="par-math-display" >
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96 <img
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97 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Bemer_de_1909_02/fulltext/img/Einst_Bemer_de_1909_024x.png" alt=" 1 Hy - c(dz Dx - dx Dz), 1 Hz - c(dx Dy - dy Dx). " class="par-math-display" /></center>
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98 <!--l. 153--><p class="nopar">
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99 </p><!--l. 157--><p class="indent"> Von der speziellen Wahl der Lage der Koordinatenachsen <br/>relativ zum betrachteten Element der Grenzfläche machen wir <br/>uns frei, indem wir das Resultat in den Bezeichnungen der <br/>Vektoranalysis schreiben. Bezeichnen wir durch die Indizes <span
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100 class="cmmi-10">n</span>
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101 <br/>bzw. <span class="overline"><span
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102 class="cmmi-10">n</span></span> die Komponente des betreffenden Vektors im Sinne <br/>bzw. senkrecht zur Normale der Unstetigkeitsfläche, so folgt, daß
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103 </p>
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104 <center class="par-math-display" >
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105 <img
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106 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Bemer_de_1909_02/fulltext/img/Einst_Bemer_de_1909_025x.png" alt="{ Dn, } 1 H - c[dD] n " class="par-math-display" /></center>
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107 <!--l. 171--><p class="nopar">
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108 </p><!--l. 175--><p class="noindent">an der Grenzfläche stetig sein müssen.
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109 </p><!--l. 178--><p class="indent"> In gleicher Weise schließt man aus der Gleichung (2a)<sup ><span
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110 class="cmr-7">1</span></sup>) <br/>die Stetigkeit der Komponenten:
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111 </p>
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112 <center class="par-math-display" >
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113 <img
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114 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Bemer_de_1909_02/fulltext/img/Einst_Bemer_de_1909_026x.png" alt=" Bn, { 1 } G + - [dB] -. c n " class="par-math-display" /></center>
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115 <!--l. 188--><p class="nopar">
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116 </p><!--l. 192--><p class="indent"> Bern und Würzburg, November 1908.
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117 </p><!--l. 194--><p class="noindent">----------
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118 </p><!--l. 197--><p class="indent"> 1) 1. c.
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119 </p>
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120 <div class="center" >
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121
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122 <!--l. 199--><p class="noindent">
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123 </p><!--l. 200--><p class="noindent">(Eingegangen 6. Dezember 1908.)</p></div>
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124
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125 <div class="center" >
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126
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127 <!--l. 203--><p class="noindent">
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128 </p><!--l. 204--><p class="noindent">----------</p></div>
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129 <!--l. 207--><p class="indent"> <span
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130 class="cmti-10">Nachtrag. </span>Wenn an der betrachteten Grenzfläche eine <br/>Schicht wahrer Elektrizität (<span
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131 class="cmsy-10"><img
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132 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Bemer_de_1909_02/fulltext/img/cmsy10-73.png" alt=" integral " class="10x-x-73" /></span> <span
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133 class="cmmi-10"><img
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134 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Bemer_de_1909_02/fulltext/img/cmmi10-25.png" alt="r" class="cmmi-10x-x-25" align="middle" /> d<img
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135 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Bemer_de_1909_02/fulltext/img/cmmi10-1c.png" alt="t" class="10x-x-1c" /></span>) von der Flächendichte <span
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136 class="cmmi-10"><img
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137 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Bemer_de_1909_02/fulltext/img/cmmi10-11.png" alt="j" class="cmmi-10x-x-11" align="middle" /> </span> <br/>sich befindet, so wird <span
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138 class="cmmi-10"><img
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139 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Bemer_de_1909_02/fulltext/img/cmmi10-18.png" alt="q" class="cmmi-10x-x-18" align="middle" /> </span>unendlich. Es ist dann
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140 </p>
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141 <center class="par-math-display" >
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142 <img
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143 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Bemer_de_1909_02/fulltext/img/Einst_Bemer_de_1909_027x.png" alt="curl H + 1 (d D) D - q c " class="par-math-display" /></center>
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144 <!--l. 217--><p class="nopar">
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145 </p><!--l. 221--><p class="noindent">in der Grenzschicht endlich, wobei <span
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146 class="cmmi-10"><img
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147 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Bemer_de_1909_02/fulltext/img/cmmi10-18.png" alt="q" class="cmmi-10x-x-18" align="middle" /> </span>durch (<span
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148 class="cmmi-10">d/c</span>)<span
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149 class="cmmi-10"><img
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150 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Bemer_de_1909_02/fulltext/img/cmmi10-25.png" alt="r" class="cmmi-10x-x-25" align="middle" /> </span>ersetzt <br/>werden kann. Für diesen Fall findet man ebenfalls die obigen <br/>Grenzbedigungen, mit dem Unterschiede, daß die erste der-<br/>selben durch
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151 </p>
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152 <center class="par-math-display" >
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153 <img
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154 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Bemer_de_1909_02/fulltext/img/Einst_Bemer_de_1909_028x.png" alt="Dn2 - Dn1 = j " class="par-math-display" /></center>
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155 <!--l. 230--><p class="nopar">
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156 </p><!--l. 234--><p class="noindent">zu ersetzen ist.
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157 </p>
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158 <div class="center" >
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159
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160 <!--l. 237--><p class="noindent">
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161 </p><!--l. 238--><p class="noindent">(Eingegangen 19. Januar 1909.)</p></div>
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162 <div class="center" >
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165 </p><!--l. 242--><p class="noindent">----------</p></div>
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167 </body></html>
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