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author Klaus Thoden <kthoden@mpiwg-berlin.mpg.de>
date Thu, 02 May 2013 15:04:28 +0200
parents 22d6a63640c6
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1 <?xml version="1.0" encoding="utf-8" ?>
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5 <html>
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7 <head><title></title>
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9 <meta name="generator" content="TeX4ht (http://www.cse.ohio-state.edu/~gurari/TeX4ht/mn.html)" />
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12 <meta name="src" content="Einst_Ueber_de_1908_01.tex" />
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13 <meta name="date" content="2005-03-10 17:52:00" />
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14 <link rel="stylesheet" type="text/css" href="Einst_Ueber_de_1908_01.css" />
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15 </head><body >
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17 <!--l. 16--><p class="noindent"><pb/></p>
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18 <div class="center" >
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20 <!--l. 17--><p class="noindent">
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21 </p><!--l. 18--><p class="noindent"><span
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22 class="cmr-12x-x-120">6. </span><span
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23 class="cmbxti-10x-x-144">Über die </span> <br/><span
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24 class="cmbxti-10x-x-144">im elektromagnetischen Felde auf ruhende</span>
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25 <br/><span
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26 class="cmbxti-10x-x-144">K</span><span
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27 class="cmbxti-10x-x-144">örper ausge</span><span
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28 class="cmbxti-10x-x-144">übten ponderomotorischen Kr</span><span
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29 class="cmbxti-10x-x-144">äfte; </span> <br/><span
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30 class="cmbxti-10x-x-144">von</span>
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31 <span
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32 class="cmbxti-10x-x-144">A. Einstein und J. Laub.</span></p></div>
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33 <div class="center" >
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35 <!--l. 24--><p class="noindent">
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36 </p><!--l. 25--><p class="noindent">--------</p></div>
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37 <!--l. 28--><p class="indent"> In einer kürzlich erschienenen Abhandlung<sup ><span
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38 class="cmr-8">1</span></sup>) hat Hr. Min-<br/>kowski einen Ausdruck für die auf beliebig bewegte Körper <br/>wirkenden ponderomotorischen Kräfte elektromagnetischen Ur-<br/>sprunges angegeben. Spezialisiert man die Minkowskischen <br/>Ausdrücke auf ruhende, isotrope und homogene Körper, so
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39 <br/>erhält man für die <span
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40 class="cmmi-12">X</span>-Komponente der auf die Volumeneinheit <br/>wirkenden Kraft:</p>
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41 <table width="100%"
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42 class="equation"><tr><td><a
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43 id="x1-2r1"></a>
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44 <center class="math-display" >
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45 <img
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46 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Ueber_de_1908_01/fulltext/img/Einst_Ueber_de_1908_010x.png" alt="K = r G + q B - q B , x x y z z y " class="math-display" /></center></td><td width="5%">(1)</td></tr></table>
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47 <!--l. 42--><p class="nopar">
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48 </p><!--l. 46--><p class="noindent">wobei <span
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49 class="cmmi-12"><img
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50 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Ueber_de_1908_01/fulltext/img/cmmi12-25.png" alt="r" class="cmmi-12x-x-25" align="middle" /> </span>die elektrische Dichte, <span
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51 class="cmmi-12"><img
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52 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Ueber_de_1908_01/fulltext/img/cmmi12-18.png" alt="q" class="cmmi-12x-x-18" align="middle" /> </span>den elektrischen Leitungsstrom, <br/>G die elektrische Feldstärke, B die magnetische Induktion be-<br/>deuten. Dieser Ausdruck scheint uns aus folgenden Gründen mit <br/>dem elektronentheoretischen Bild nicht in Einklang zu stehen: <br/>Während nämlich ein von einem elektrischen Strom (Leitungs-<br/>strom) durchflossener Körper im Magnetfeld eine Kraft er-<br/>leidet, wäre dies nach Gleichung (1) nicht der Fall, wenn der <br/>im Magnetfeld befindliche Körper statt von einem Leitungs-<br/>strom von einem Polarisationsstrom <img
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53 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Ueber_de_1908_01/fulltext/img/Einst_Ueber_de_1908_011x.png" alt="(@ D/@ t)" class="left" align="middle" /> durchsetzt wird.
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54 <br/>Nach Minkowski besteht also hier ein prinzipieller Unter-<br/>schied zwischen einem Verschiebungsstrom und einem Leitungs-<br/>strom derart, daß ein Leiter nicht betrachtet werden kann <br/>als ein Dielektrikum von unendlich großer Dielektrizitäts-<br/>konstante.
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55 </p><!--l. 66--><p class="indent"> Angesichts dieser Sachlage schien es uns von Interesse <br/>zu sein, die ponderomotorischen Kräfte für beliebige magneti-<br/>sierbare Körper auf elektronentheoretischem Wege abzuleiten. <br/>Wir geben im folgenden eine solche Ableitung, wobei wir uns <br/>aber auf ruhende Körper beschränken.
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56 </p><!--l. 73--><p class="noindent">----------
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57 </p><!--l. 76--><p class="indent"> 1) H. Minkowski, Gött. Nachr. 1908. p. 45. <pb/>
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58 </p><!--l. 80--><p class="indent">
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60 </p>
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61 <div class="center" >
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63 <!--l. 81--><p class="noindent">
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64 </p><!--l. 82--><p class="noindent"><span
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65 class="cmsy-10x-x-120">§ </span>1. Kräfte, welche nicht von Geschwindigkeiten der <br/>Elementarteilchen abhängen.</p></div>
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66 <!--l. 87--><p class="indent"> Wir wollen uns bei der Ableitung konsequent auf den <br/>Standpunkt der Elektronentheorie stellen<sup ><span
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67 class="cmr-8">1</span></sup>); wir setzen also:</p>
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68 <table width="100%"
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69 class="equation"><tr><td><a
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70 id="x1-3r2"></a>
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71 <center class="math-display" >
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72 <img
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73 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Ueber_de_1908_01/fulltext/img/Einst_Ueber_de_1908_012x.png" alt="D = G + P , " class="math-display" /></center></td><td width="5%">(2)</td></tr></table>
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74 <!--l. 93--><p class="nopar"></p>
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75 <table width="100%"
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76 class="equation"><tr><td><a
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77 id="x1-4r3"></a>
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78 <center class="math-display" >
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79 <img
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80 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Ueber_de_1908_01/fulltext/img/Einst_Ueber_de_1908_013x.png" alt="B = H + Q , " class="math-display" /></center></td><td width="5%">(3)</td></tr></table>
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81 <!--l. 99--><p class="nopar">
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82 </p><!--l. 103--><p class="noindent">wobei P den elektrischen, Q den magnetischen Polarisations-<br/>vektor bedeutet. Die elektrische bzw. die magnetische Polari-<br/>sation denken wir uns bestehend in räumlichen Verschie-<br/>bungen von an Gleichgewichtslagen gebundenen, elektrischen
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83 <br/>bzw. magnetischen Massenteilchen von Dipolen. Außerdem <br/>nehmen wir noch das Vorhandensein von nicht an Dipole ge-<br/>bundenen, beweglichen elektrischen Teilchen (Leitungselek-<br/>tronen) an. In dem Raume zwischen den genannten Teilchen <br/>mögen die Maxwellschen Gleichungen für den leeren Raum <br/>gelten, und es seien, wie bei Lorentz, <span
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84 class="cmti-12">die Wechselwirkungen </span> <br/><span
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85 class="cmti-12">zwischen Materie und</span>
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86 <span
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87 class="cmti-12">elektromagnetischem Felde ausschlie</span><span
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88 class="cmmi-12"><img
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89 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Ueber_de_1908_01/fulltext/img/cmmi12-c.png" alt="b" class="cmmi-12x-x-c" align="middle" /></span><span
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90 class="cmti-12">lich </span> <br/><span
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91 class="cmti-12">durch diese Teilchen bedingt.</span>
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92
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93 Dementsprechend nehmen wir an, <br/>daß die vom elektromagnetischen Felde auf das Volumenelement <br/>der Materie ausgeübten Kräfte gleich sind der Resultierenden
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94 <br/>der ponderomotorischen Kräfte, welche von diesem Felde auf <br/>alle in dem betreffenden Volumenelement befindlichen elek-<br/>trischen und magnetischen Elementarteilchen ausgeübt werden. <br/>Unter Volumenelement der Materie verstehen wir stets einen <br/>so großen Raum, daß er eine sehr große Zahl von elektrischen <br/>und magnetischen Teilchen enthält. Die Grenzen eines be-<br/>trachteten Volumenelementes muß man sich ferner stets so <br/>genommen denken, daß die Grenzfläche keine elektrische bzw. <br/>magnetische Dipole schneidet.
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95 </p><!--l. 131--><p class="indent"> Wir berechnen zunächst diejenige auf einen elektrischen <br/>Dipol wirkende Kraft, welche daher herrührt, daß die Feld-<br/>stärke G an den Orten, an welchen sich die Elementarmassen <br/>des Dipols befinden, nicht genau dieselbe ist. Bezeichnet man
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96 <br/>----------
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97 </p><!--l. 138--><p class="indent"> 1) Der einfacheren Darstellung halber halten wir aber an der dualen
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98 <br/>Behandlung der elektrischen und magnetischen Erscheinungen fest. <pb/>
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99 </p><!--l. 144--><p class="indent">
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100
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101 </p><!--l. 145--><p class="noindent">mit p den Vektor des Dipolmomentes, so erhält man fült die <br/><span
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102 class="cmmi-12">X</span>-Komponente der gesuchten Kraft den Ausdruck:
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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_Ueber_de_1908_01/fulltext/img/Einst_Ueber_de_1908_014x.png" alt=" @ Gx- @-Gx- @-Gx- fx = px @ x + py @ y + pz @ z . " class="par-math-display" /></center>
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107 <!--l. 155--><p class="nopar">
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108 </p><!--l. 158--><p class="noindent">Denkt man sich den letzten Ausdruck für alle Dipole in der <br/>Volumeneinheit gebildet und summiert, so erhält man unter <br/>Berücksichtigung der Beziehung:
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109 </p>
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110 <center class="par-math-display" >
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111 <img
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112 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Ueber_de_1908_01/fulltext/img/Einst_Ueber_de_1908_015x.png" alt=" sum
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113 p = P " class="par-math-display" /></center>
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114 <!--l. 167--><p class="nopar"></p><!--l. 171--><p class="noindent">die Gleichung:</p>
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115 <table width="100%"
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116 class="equation"><tr><td><a
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117 id="x1-5r4"></a>
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118 <center class="math-display" >
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119 <img
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120 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Ueber_de_1908_01/fulltext/img/Einst_Ueber_de_1908_016x.png" alt=" { @ G @ G @ G } F1 x = Px ----x + Py ---x-+ Pz ---x- . @ x @ y @ z " class="math-display" /></center></td><td width="5%">(4)</td></tr></table>
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121 <!--l. 180--><p class="nopar">
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122 </p><!--l. 183--><p class="indent"> Wenn die algebraische Summe der positiven und negativen <br/>Leitungselektronen nicht verschwindet, dann kommt zum Aus-<br/>druck (4) noch ein Term hinzu, den wir nun berechnen wollen. <br/>Die <span
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123 class="cmmi-12">X</span>-Komponente der auf ein Leitungselektron von der elek-<br/>trischen Masse <span
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124 class="cmmi-12">e </span>wirkenden ponderomotorischen Kraft ist <span
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125 class="cmmi-12">eG</span><sub ><span
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126 class="cmmi-8">x</span></sub>.
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127 <br/>Summiert man über alle Leitungselektronen der Volumen-<br/>einheit, so erhält man:</p>
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128 <table width="100%"
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129 class="equation"><tr><td><a
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130 id="x1-6r5"></a>
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131
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132 <center class="math-display" >
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133 <img
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134 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Ueber_de_1908_01/fulltext/img/Einst_Ueber_de_1908_017x.png" alt=" sum
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135 F2 x = Gx e. " class="math-display" /></center></td><td width="5%">(5)</td></tr></table>
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136 <!--l. 195--><p class="nopar">
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137 </p><!--l. 198--><p class="noindent">Denkt man sich die betrachtete in der Volumeneinheit befind-<br/>liche Materie von einer Fläche umschlossen, welche keine <br/>Dipole schneidet, so erhält man nach dem Gaussschen Satz <br/>und nach der Definition des Verschiebungsvektors D:
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138 </p>
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139 <center class="par-math-display" >
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140 <img
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141 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Ueber_de_1908_01/fulltext/img/Einst_Ueber_de_1908_018x.png" alt=" sum
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142 e = divD, " class="par-math-display" /></center>
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143 <!--l. 208--><p class="nopar"></p><!--l. 212--><p class="noindent">so daß</p>
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144 <table width="100%"
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145 class="equation"><tr><td><a
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146 id="x1-7r6"></a>
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147 <center class="math-display" >
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148 <img
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149 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Ueber_de_1908_01/fulltext/img/Einst_Ueber_de_1908_019x.png" alt="F2 x = Gx div D " class="math-display" /></center></td><td width="5%">(5a)</td></tr></table>
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150 <!--l. 218--><p class="nopar">
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151 </p><!--l. 222--><p class="noindent">wird. Die <span
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152 class="cmmi-12">X</span>-Komponente der von der elektrischen Feldstärke <br/>auf die Volumeneinheit der Materie ausgeübten Kraft ist daher <br/>gleich:</p>
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153 <table width="100%"
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154 class="equation"><tr><td><a
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155 id="x1-8r6"></a>
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156 <center class="math-display" >
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157 <img
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158 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Ueber_de_1908_01/fulltext/img/Einst_Ueber_de_1908_0110x.png" alt="F = F + F = P @-Gx- + P @-Gx- + P @ Gx- + G divD . ex 1 x 2 x x @ x y @ y z @ z x " class="math-display" /></center></td><td width="5%">(6)</td></tr></table>
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159 <!--l. 233--><p class="nopar">
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160 </p><!--l. 236--><p class="noindent">Analog erhalten wir unter Berücksichtigung der Beziehung
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161 </p>
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162 <center class="par-math-display" >
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163 <img
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164 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Ueber_de_1908_01/fulltext/img/Einst_Ueber_de_1908_0111x.png" alt="div B = 0 " class="par-math-display" /></center>
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165 <!--l. 243--><p class="nopar"></p><!--l. 246--><p class="noindent">für die <span
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166 class="cmmi-12">X</span>-Komponente der von der magnetischen Feldstärke <br/>gelieferten Kraft:</p>
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167 <table width="100%"
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168 class="equation"><tr><td><a
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169 id="x1-9r7"></a>
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170 <center class="math-display" >
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171 <img
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172 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Ueber_de_1908_01/fulltext/img/Einst_Ueber_de_1908_0112x.png" alt=" { } @ Hx- @-Hx- @-Hx- Fm x = Qx @ x + Qy @ y + Qz @ z . " class="math-display" /></center></td><td width="5%">(7)</td></tr></table>
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173 <!--l. 255--><p class="nopar">
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174 <pb/>
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175 </p><!--l. 261--><p class="indent">
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176
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177 </p><!--l. 262--><p class="indent"> Es ist zu bemerken, daß für die Herleitung der Aus-<br/>drücke (6) und (7) keinerlei Voraussetzung gemacht werden <br/>muß über die Beziehungen, welche die Feldstärken G und H <br/>mit den Polarisationsvektoren P und Q verbinden.
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178 </p><!--l. 268--><p class="indent"> Hat man es mit anisotropen Körpern zu tun, so liefern <br/>die elektrische bzw. die magnetische Feldstärke nicht nur eine <br/>Kraft, sondern auch Kräftepaare, welche sich auf die Materie <br/>übertragen. Das gesuchte Drehmoment ergibt sich leicht für
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179 <br/>die einzelnen Dipole und Summation über alle elektrischen <br/>und magnetischen Dipole in der Volumeneinheit. Man erhält:</p>
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180 <table width="100%"
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181 class="equation"><tr><td><a
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182 id="x1-10r8"></a>
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183 <center class="math-display" >
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184 <img
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185 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Ueber_de_1908_01/fulltext/img/Einst_Ueber_de_1908_0113x.png" alt="L = {[P G] + [Q H]}. " class="math-display" /></center></td><td width="5%">(8)</td></tr></table>
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186 <!--l. 282--><p class="nopar">
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187 </p><!--l. 286--><p class="noindent">Die Formel (6) liefert diejenigen ponderomotorischen Kräfte, <br/>welche bei elektrostatischen Problemen eine Rolle spielen. <br/>Wir wollen diese Gleichung für den Fall, daß es sich um iso-<br/>trope Körper handelt, so umformen, daß sie einen Vergleich <br/>gestattet mit demjenigen Ausdrucke für die ponderomotorischen <br/>Kräfte, wie er in der Elektrostatik angegeben wird. Setzen wir
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188 </p>
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189 <center class="par-math-display" >
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190 <img
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191 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Ueber_de_1908_01/fulltext/img/Einst_Ueber_de_1908_0114x.png" alt="P = (e - 1) G , " class="par-math-display" /></center>
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192 <!--l. 299--><p class="nopar"></p><!--l. 303--><p class="noindent">so geht die Gleichung (6) über in:
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193 </p>
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194 <center class="par-math-display" >
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195 <img
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196 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Ueber_de_1908_01/fulltext/img/Einst_Ueber_de_1908_0115x.png" alt=" 1 @ e 1 @ Fex = Gx div D - --G2 --- + ----- (e - 1) G2. 2 @ x 2 @ x " class="par-math-display" /></center>
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197 <!--l. 313--><p class="nopar">
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198 </p><!--l. 317--><p class="noindent">Die ersten beiden Glieder dieses Ausdruckes sind identisch <br/>mit den aus der Elektrostatik bekannten. Das dritte Glied <br/>ist, wie man sieht, von einem Potential ableitbar. Handelt <br/>es sich um Kräfte, die auf einen im Vakuum befindlichen
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199 <br/>Körper wirken, so liefert das Glied bei Integration über den <br/>Körper keinen Beitrag. handelt es sich aber um die pondero-<br/>motorische Wirkung auf
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200 Flüssigkeiten, so wird der dem dritten <br/>Glied entsprechende Anteil der Kraft bei Gleichgewicht durch <br/>eine Druckverteilung in der Flüssigkeit kompensiert.
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201 </p>
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202 <div class="center" >
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203
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204 <!--l. 332--><p class="noindent">
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205 </p><!--l. 333--><p class="noindent"><span
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206 class="cmsy-10x-x-120">§ </span>2. Kräfte, welche von den Geschwindigkeiten der <br/>Elementarteilchen abhängen.</p></div>
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207 <!--l. 338--><p class="indent"> Wir gehen jetzt über zu demjenigen Anteile der pondero-<br/>motorischen Kraft, welcher durch die Bewegungsgeschwindig-<br/>keiten der Elementarladungen geliefert wird. <pb/>
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208 </p><!--l. 345--><p class="indent">
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209
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210 </p><!--l. 346--><p class="indent"> Wir gehen aus vom Biot-Savartschen Gesetz. Auf ein <br/>stromdurchflossenes Volumenelement, welches sich in einem <br/>magnetischen Felde befindet, wirkt erfahrungsgemäß pro Vo-<br/>lumeneinheit die Kraft:</p>
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211 <center class="par-math-display" >
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212 <img
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213 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Ueber_de_1908_01/fulltext/img/Einst_Ueber_de_1908_0116x.png" alt="1- [q H] , c
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214 " class="par-math-display" /></center>
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215 <!--l. 354--><p class="nopar">
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216 </p><!--l. 358--><p class="noindent">falls die betrachtete, stromdurchflossene Materie nicht magne-<br/>tisch polarisierbar ist. Für das Innere von magnetisch polari-<br/>sierbaren Körpern wurde, soviel uns bekannt ist, bis jetzt jene <br/>Kraft gleich<sup ><span
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217 class="cmr-8">1</span></sup>)
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218 </p>
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219 <center class="par-math-display" >
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220 <img
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221 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Ueber_de_1908_01/fulltext/img/Einst_Ueber_de_1908_0117x.png" alt="1-[qB] c
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222 " class="par-math-display" /></center>
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223 <!--l. 367--><p class="nopar">
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224 </p><!--l. 371--><p class="noindent">gesetzt, wobei B die magnetische Induktion bedeutet. Wir <br/>wollen nun zeigen, daß
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225 <span
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226 class="cmti-12">auch </span>im Falle, daß das stromdurch-<br/>flossene Material <span
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227 class="cmti-12">magnetisch polarisierbar ist</span>, die auf das strom-<br/>durchflossene Volumenelement wirkende Kraft erhalten wird,
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228 <br/>wenn man zu der durch die Gleichung (7) ausgedrückten Kraft <br/>noch die Volumenkraft:</p>
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229 <table width="100%"
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230 class="equation"><tr><td><a
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231 id="x1-11r9"></a>
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232 <center class="math-display" >
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233 <img
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234 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Ueber_de_1908_01/fulltext/img/Einst_Ueber_de_1908_0118x.png" alt=" 1 Fs = --[q H] c " class="math-display" /></center></td><td width="5%">(9)</td></tr></table>
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235 <!--l. 383--><p class="nopar">
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236 </p><!--l. 387--><p class="noindent">hinzufügt. Wir wollen dies zuerst an einem einfachen Bei-<br/>spiel anschaulich machen. Der unendlich dünne im Querschnitt gezeichnete Streifen <span
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237 class="cmmi-12">S</span> <br/>erstrecke sich senkrecht zur Papierebene nach beiden Seiten <br/>ins Unendliche. Er bestehe
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238 aus <br/>magnetisch polarisierbarem Mate-<br/>rial und befinde sich in einem <br/>homogenen Magnetfelde
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239 <span
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240 class="cmmi-12">H</span><sub ><span
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241 class="cmmi-8">a</span></sub>, dessen <br/>Richtung durch die Pfeile (vgl. <br/>Figur) angedeutet ist. Wir fragen <br/>nach der auf den Materialstreifen wirkenden Kraft, falls der-<br/>selbe von einem Strome i durchflossen<img
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242 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Ueber_de_1908_01/fulltext/img/Einst_Ueber_de_1908_0119x.png" alt="PIC" class="graphics" width="136.5733pt" height="79.45125pt" /><!--tex4ht:graphics
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243 name="img/Einst_Ueber_de_1908_0119x.png" src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Ueber_de_1908_01/fulltext/Einst_Ueber_de_1908_01_001.EPS"
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244 -->
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245 ist.
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246 </p><!--l. 404--><p class="indent"> Die Erfahrung lehrt, daß diese Kraft von der magnetischen <br/>Permeabilität des Leitermateriales unabhängig ist, und man <br/>schloß daraus, daß es nicht die Feldstärke H, sondern die <br/>magnetische Induktion <span
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247 class="cmmi-12">B</span><sub ><span
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248 class="cmmi-8">i</span></sub> sein müsse, welche für die pondero-<br/>----------
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249 </p><!--l. 412--><p class="indent"> 1) Vgl. z. B. auch M. Abraham, Theorie der Elektrizität <span
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250 class="cmbx-12">2. </span>p. 319. <br/>1905.
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251 <pb/>
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252 </p><!--l. 417--><p class="indent">
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253
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254 </p><!--l. 418--><p class="noindent">motorische Kraft maßgebend ist, denn im Innern des Streifens <br/>ist die magnetische Induktion <span
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255 class="cmmi-12">B</span><sub ><span
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256 class="cmmi-8">i</span></sub> gleich der außerhalb des <br/>Streifens wirkenden Kraft <span
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257 class="cmmi-12">H</span><sub ><span
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258 class="cmmi-8">a</span></sub>, unabhängig von dem Werte der <br/>Permeabilität des Streifens, während die im Innern des Streifens <br/>herrschende Kraft <span
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259 class="cmmi-12">H</span><sub ><span
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260 class="cmmi-8">i</span></sub> bei gegebenem äußeren Felde von <span
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261 class="cmmi-12"><img
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262 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Ueber_de_1908_01/fulltext/img/cmmi12-16.png" alt="m" class="cmmi-12x-x-16" align="middle" /></span> <br/>abhängt. Dieser Schluß ist aber nicht stichhaltig, weil die <br/>ins Auge gefaßte ponderomotorische Kraft nicht die einzige <br/>ist, welche auf unseren Materialstreifen wirkt. Das äußere <br/>Feld <span
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263 class="cmmi-12">H</span><sub ><span
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264 class="cmmi-8">a</span></sub> induziert nämlich auf der Oberseite und Unterseite <br/>des Materialstreifens magnetische Belegungen von der Dichte<sup ><span
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265 class="cmr-8">1</span></sup>):
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266 <br/><span
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267 class="cmmi-12">H</span><sub ><span
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268 class="cmmi-8">a</span></sub> <img
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269 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Ueber_de_1908_01/fulltext/img/Einst_Ueber_de_1908_0120x.png" alt="(1 - 1/m)" class="left" align="middle" /> <span
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270 class="cmmi-12">, </span>und zwar auf der Oberseite eine negative, auf der <br/>Unterseite eine positive Belegung. Auf jede dieser Belegungen <br/>wirkt eine von dem im Streifen fließenden Strom erzeugte Kraft <br/>von der Stärke <span
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271 class="cmmi-12">i</span><img
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272 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Ueber_de_1908_01/fulltext/img/Einst_Ueber_de_1908_0121x.png" alt="/" class="left" align="middle" /> 2 <span
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273 class="cmmi-12">b</span> pro Längeeinheit des Streifens<sup ><span
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274 class="cmr-8">2</span></sup>), welche <br/>magnetische Kraft an der Oberseite und Unterseite verschieden <br/>gerichtet ist. Die so resultierenden ponderomotorischen Kräfte <br/>addieren sich, so daß wir die ponderomotorische Kraft erhalten:
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275 <br/><img
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276 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Ueber_de_1908_01/fulltext/img/Einst_Ueber_de_1908_0122x.png" alt="(1 - 1/ m)" class="left" align="middle" /> <span
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277 class="cmmi-12">H</span><sub ><span
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278 class="cmmi-8">a</span></sub> <span
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279 class="cmmi-12">i. </span>Diese Kraft scheint bis jetzt nicht berück-<br/>sichtigt worden zu sein.
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280 </p><!--l. 448--><p class="indent"> Die auf die Längeeinheit unseres Streifens im ganzen aus-<br/>geübte Kraft ist nun gleich der Summe der soeben berech-<br/>neten und der auf die Volumenelemente des Streifens infolge <br/>des Stromdurchganges im Magnetfeld wirkenden Kraft <span
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281 class="cmmi-12">R</span>. Da <br/>die gesamte auf die Längeeinheit wirkende ponderomotorische <br/>Kraft erfahrungsgemäß gleich <span
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282 class="cmmi-12">iH</span><sub ><span
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283 class="cmmi-8">a</span></sub> ist, so besteht die Gleichung:
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284 </p>
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285 <center class="par-math-display" >
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286 <img
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287 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Ueber_de_1908_01/fulltext/img/Einst_Ueber_de_1908_0123x.png" alt="( ) 1 - 1- iHa + R = iHa m " class="par-math-display" /></center>
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288 <!--l. 461--><p class="nopar">
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289 </p><!--l. 465--><p class="noindent">oder</p>
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290 <center class="par-math-display" >
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291 <img
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292 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Ueber_de_1908_01/fulltext/img/Einst_Ueber_de_1908_0124x.png" alt=" iHa- R = m = iHi . " class="par-math-display" /></center>
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293 <!--l. 472--><p class="nopar">
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294 </p><!--l. 475--><p class="noindent">Man sieht also, daß für die Berechnung der ponderomotorischen <br/>Kraft <span
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295 class="cmmi-12">R</span>, welche auf stromdurchflossene Volumenelemente <br/>----------
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296 </p><!--l. 481--><p class="indent"> 1) Die Dichte ist nämlich gleich:
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297 </p>
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298 <center class="par-math-display" >
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299 <img
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300 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Ueber_de_1908_01/fulltext/img/Einst_Ueber_de_1908_0125x.png" alt=" ( ) Qi = Bi - Hi = Ha 1 - 1- . m " class="par-math-display" /></center>
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301
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302 <!--l. 488--><p class="nopar">
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303 </p><!--l. 492--><p class="indent"> 2) Statt dieser auf die Belegungen wirkenden Kräfte hätten wir <br/>streng genommen nach den Resultaten des vorigen Paragraphen aller-<br/>dings Volumenkräfte einführen müssen, was jedoch ohne Belang ist. <pb/>
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304 </p><!--l. 499--><p class="indent">
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305
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306 </p><!--l. 500--><p class="noindent">wirkt, nicht die Induktion <span
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307 class="cmmi-12">B</span><sub ><span
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308 class="cmmi-8">i</span></sub>, sondern die Feldstärke <span
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309 class="cmmi-12">H</span><sub ><span
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310 class="cmmi-8">i</span></sub> maß-<br/>gebend ist.
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311 </p><!--l. 503--><p class="indent"> Um jeden Zweifel zu beseitigen, wollen wir noch ein Bei-<br/>spiel behandeln, aus welchem man ersieht, daß das Prinzip <br/>der Gleichheit von Wirkung und Gegenwirkung den von uns <br/>gewählten Ansatz fordert.
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312 </p><!--l. 508--><p class="indent"> Wir denken uns einen zylindrischen, von leerem Raum <br/>umgebenen und vom Strom <span
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313 class="cmmi-12"><img
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314 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Ueber_de_1908_01/fulltext/img/cmmi12-18.png" alt="q" class="cmmi-12x-x-18" align="middle" /> </span>durchflossenen Leiter, welcher <br/>sich längs der <span
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315 class="cmmi-12">X</span>-Achse eines Koordinatensystems beiderseits <br/>ins Unendliche erstreckt. Die Materialkonstanten des Leiters, <br/>sowie die im folgenden auftretenden Feldvektoren seien von <span
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316 class="cmmi-12">x</span>
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317 <br/>unabhängig, aber Funktionen von <span
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318 class="cmmi-12">y </span>und <span
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319 class="cmmi-12">z</span>. Der Leiter sei <br/>ein magnetisch harter Körper und besitze eine Magnetisierung <br/>quer zur <span
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320 class="cmmi-12">X</span>-Achse. Wir nehmen an, daß ein äußeres Feld <br/>auf den Leiter nicht wirkt, daß also die magnetische Kraft H <br/>in großen Entfernungen vom Leiter verschwindet.
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321 </p><!--l. 521--><p class="indent"> Es ist klar, daß auf den Leiter als Ganzes keine pondero-<br/>motorische Kraft wirkt, denn es würde zu dieser Wirkung <br/>keine Gegenwirkung angebbar sein. Wir wollen nun zeigen, <br/>daß bei Wahl unseres Ansatzes jene Kraft in der Tat ver-<br/>schwindet. Die gesamte auf die Längeeinheit unseres Leiters <br/>in der Richtung der
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322 <span
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323 class="cmmi-12">Z</span>-Achse wirkende Kraft läßt sich dar-<br/>stellen gemäß den Gleichungen (7) und (9) in der Form:</p>
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324 <table width="100%"
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325 class="equation"><tr><td><a
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326 id="x1-12r10"></a>
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327 <center class="math-display" >
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328 <img
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329 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Ueber_de_1908_01/fulltext/img/Einst_Ueber_de_1908_0126x.png" alt=" integral ( ) integral
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330 @ Hz- @-Hz- 1- R = Qy @ y + Qz @ z d f + c qx Hy d f, " class="math-display" /></center></td><td width="5%">(10)</td></tr></table>
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331 <!--l. 536--><p class="nopar">
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332 </p><!--l. 540--><p class="noindent">wobei <span
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333 class="cmmi-12">df </span>ein Flächenelement der <span
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334 class="cmmi-12">Y Z</span>-Ebene bedeutet. Wir <br/>nehmen an, daß sämtliche in Betracht kommende Größen an <br/>der Oberfläche des Leiters stetig sind. Wir behandeln zuerst <br/>das erste Integral der Gleichung (10). Es ist:
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335 </p>
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336 <center class="par-math-display" >
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337 <img
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338 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Ueber_de_1908_01/fulltext/img/Einst_Ueber_de_1908_0127x.png" alt=" ( ) Qy @-Hz- + Qz @-Hz- = @-Qy-Hz- + @-Qz-Hz- - Hz @ Qy-+ @-Qz- . @ y @ z @ y @ z @ y @ z " class="par-math-display" /></center>
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339 <!--l. 556--><p class="nopar">
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340 </p><!--l. 560--><p class="noindent">Setzt man die rechte Seite dieser Gleichung in unser Integral <br/>ein, so verschwinden bei Integration über die <span
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341 class="cmmi-12">Y Z</span>-Ebene die <br/>beiden ersten Glieder, da die Kräfte im Unendlichen ver-<br/>schwinden. Das dritte Glied kann unter Berücksichtigung:
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342 </p>
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343 <center class="par-math-display" >
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344 <img
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345 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Ueber_de_1908_01/fulltext/img/Einst_Ueber_de_1908_0128x.png" alt="div B = 0 " class="par-math-display" /></center>
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346 <!--l. 571--><p class="nopar"> <pb/>
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347 </p><!--l. 578--><p class="indent">
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348
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349 </p><!--l. 579--><p class="noindent">umgeformt werden, so daß unser Integral die Form annimmt:
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350 </p>
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351 <center class="par-math-display" >
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352 <img
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353 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Ueber_de_1908_01/fulltext/img/Einst_Ueber_de_1908_0129x.png" alt=" integral ( ) @ Hy @ Hz Hz ----- + ----- df. @ y @ z " class="par-math-display" /></center>
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354 <!--l. 586--><p class="nopar"></p><!--l. 589--><p class="noindent">Nun ist:
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355 </p>
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356 <center class="par-math-display" >
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357 <img
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358 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Ueber_de_1908_01/fulltext/img/Einst_Ueber_de_1908_0130x.png" alt=" ( ) 2 Hz @-Hy- + @-Hz- = @-Hy-Hz--+ 1-@-Hz-- - Hy @-Hz-. @ y @ z @ y 2 @ z @ y " class="par-math-display" /></center>
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359 <!--l. 600--><p class="nopar">
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360 </p><!--l. 604--><p class="indent"> Bei der Integration verschwinden aber die beiden Glieder <br/><img
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361 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Ueber_de_1908_01/fulltext/img/Einst_Ueber_de_1908_0131x.png" alt="@-Hy-Hz- @ y" class="frac" align="middle" /> + <img
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362 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Ueber_de_1908_01/fulltext/img/Einst_Ueber_de_1908_0132x.png" alt="1- 2" class="frac" align="middle" /> <img
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363 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Ueber_de_1908_01/fulltext/img/Einst_Ueber_de_1908_0133x.png" alt="@ Hz2- @ z" class="frac" align="middle" /><span
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364 class="cmmi-12">.</span>
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365 Das Glied <span
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366 class="cmsy-10x-x-120">- </span><span
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367 class="cmmi-12">H</span><sub ><span
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368 class="cmmi-8">y</span></sub> <img
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369 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Ueber_de_1908_01/fulltext/img/Einst_Ueber_de_1908_0134x.png" alt="@ Hz ----- @ y" class="frac" align="middle" /> läßt sich umformen <br/>mittels der Maxwellschen Gleichungen in:
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370 </p>
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371 <center class="par-math-display" >
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372 <img
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373 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Ueber_de_1908_01/fulltext/img/Einst_Ueber_de_1908_0135x.png" alt=" 1 { @ Hy } - --Hy qx + ----- , c @ z " class="par-math-display" /></center>
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374 <!--l. 618--><p class="nopar">
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375 </p><!--l. 622--><p class="noindent">so daß wir endlich die Gleichung (10) schreiben können:
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376 </p>
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377 <center class="par-math-display" >
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378 <img
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379 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Ueber_de_1908_01/fulltext/img/Einst_Ueber_de_1908_0136x.png" alt=" integral { } integral
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380 R = - 1- Hy qx + @-Hy- d f + 1- qx Hy d f c @ z c integral integral
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381 1- @-Hy- 1-- @-Hy2- = - c Hy @ z d f = - 2c @ z d f. " class="par-math-display" /></center>
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382 <!--l. 637--><p class="nopar">
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383 </p><!--l. 642--><p class="noindent">Das letzte Integral wird Null, weil im Unendlichen die Kräfte <br/>verschwinden.--
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384 </p><!--l. 645--><p class="indent"> Nachdem wir so die Kraft festgestellt haben, welche auf <br/>von einem Leitungsstrom durchflossene Materie wirkt, erhalten <br/>wir die Kraft, die auf einen von einem Polarisationsstrom <br/>durchsetzten Körper wirkt, indem wir beachten, daß
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385 Polari-<br/>sationsstrom und Leitungsstrom in bezug auf elektrodynamische
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386 <br/>Wirkung vom Standpunkt der Elektronentheorie durchaus äqui-<br/>valent sein müssen.
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387 </p><!--l. 654--><p class="indent"> Durch Berücksichtigung der Dualität von magnetischen <br/>und elektrischen Erscheinungen erhält man auch noch die <br/>Kraft, welche auf einen von einem magnetischen Polarisations-<br/>strom durchsetzten Körper im elektrischen Felde ausgeübt wird. <br/>Als Gesamtausdruck für diejenigen Kräfte, welche von der Ge-<br/>schwindigkeit der Elementarteilchen abhängen, erhalten wir <br/>auf diese Weise die Gleichungen:</p>
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388 <table width="100%"
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389 class="equation"><tr><td><a
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390 id="x1-13r11"></a>
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391 <center class="math-display" >
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392 <img
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393 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Ueber_de_1908_01/fulltext/img/Einst_Ueber_de_1908_0137x.png" alt=" [ ] [ ] Fa = 1- [q H] + 1- @-P-H + 1- G @ Q- . c c @ t c @ t " class="math-display" /></center></td><td width="5%">(11)</td></tr></table>
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394 <!--l. 670--><p class="nopar">
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395 <pb/>
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396 </p><!--l. 677--><p class="indent">
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397
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398 </p>
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399 <div class="center" >
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400
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401 <!--l. 678--><p class="noindent">
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402 </p><!--l. 679--><p class="noindent"><span
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403 class="cmsy-10x-x-120">§ </span>3. Gleichheit von actio und reactio.</p></div>
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404 <!--l. 683--><p class="indent"> Addiert man die Gleichungen (6), (7) und (11), so erhält <br/>man den Gesamtausdruck für die <span
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405 class="cmmi-12">X</span>-Komponente der pro Vo-<br/>lumeneinheit auf die Materie wirkenden ponderomotorischen <br/>Kraft in der Form:
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406 </p>
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407 <center class="par-math-display" >
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408 <img
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409 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Ueber_de_1908_01/fulltext/img/Einst_Ueber_de_1908_0138x.png" alt="Fx = Gx div D + Px @-Gx- + Py @ Gx-+ Pz @ Gx- @ x @ y @ z @ H @ H @ H + Qx ----x + Qy ----x + Qz ---x- @ x @ y @ z 1 1 [@ P ] 1 [ @ Q ] + --[q H]x + -- ---- + -- G ---- . c c @ t x c @ t x " class="par-math-display" /></center>
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410 <!--l. 706--><p class="nopar">
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411 </p><!--l. 710--><p class="noindent">Die Gleichung kann man auch schreiben:</p>
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412 <center class="par-math-display" >
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413 <img
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414 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Ueber_de_1908_01/fulltext/img/Einst_Ueber_de_1908_0139x.png" alt=" [ ] [ ] 1 1 @ D 1 @ B Fx = Gx div G + c-[q H]x + c- -@-t H + Hx divH + c- G @-t- x x @ (Px Gx) @ (Py Gx) @ (Pz Gx) + ---@-x---- + ----@ y--- + ----@ z---
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415 @-(Qx-Hx)- @-(Qy-Hx)- @ (Qz-Hx)- 1--@- + @ x + @ y + @ z - c @ t [G H]x . " class="par-math-display" /></center>
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416 <!--l. 735--><p class="nopar">
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417 </p><!--l. 739--><p class="noindent">Ersetzt man
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418 </p>
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419 <center class="par-math-display" >
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420 <img
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421 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Ueber_de_1908_01/fulltext/img/Einst_Ueber_de_1908_0140x.png" alt=" ( ) 1- s + @ D- und 1-@ B- c @ t c @ t " class="par-math-display" /></center>
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422 <!--l. 748--><p class="nopar">
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423
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424 </p><!--l. 752--><p class="noindent">mittels der Maxwellschen Gleichungen durch curl H bzw. <br/>durch curl G, so erhält man durch eine einfache Umformung:</p>
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425 <table width="100%"
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426 class="equation"><tr><td><a
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427 id="x1-14r12"></a>
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428 <center class="math-display" >
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429 <img
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430 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Ueber_de_1908_01/fulltext/img/Einst_Ueber_de_1908_0141x.png" alt=" @ Xx @ Xy @ Xz 1 @ Sx Fx = ----- + -----+ ----- - -2-----, @ x @ y @ z c @ t " class="math-display" /></center></td><td width="5%">(12)</td></tr></table>
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431 <!--l. 763--><p class="nopar">
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432 </p><!--l. 766--><p class="noindent">wobei gesetzt ist<sup ><span
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433 class="cmr-8">1</span></sup>):</p>
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434 <table width="100%"
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435 class="equation"><tr><td><a
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436 id="x1-15r13"></a>
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437 <center class="math-display" >
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438 <img
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439 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Ueber_de_1908_01/fulltext/img/Einst_Ueber_de_1908_0142x.png" alt=" Xx = - 1 (G2 + H2) + Gx Dx + Hx Bx, 2 { Xy = Gx Dy + Hx By , Xz = Gx Dz + Hx Bz ,
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440 Sx = c [G H]x . " class="math-display" /></center></td><td width="5%">(13)</td></tr></table>
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441 <!--l. 780--><p class="nopar">
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442 </p><!--l. 784--><p class="noindent">----------
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443 </p><!--l. 787--><p class="indent"> 1) Hr. Geheimrat Wien hatte die Güte, uns darauf aufmerksam zu
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444 <br/>machen, daß bereits H. A. Lorentz die ponderomotorischen Kräfte für <br/>nicht magnetisierbare Körper in dieser Form angegeben hat. Enzykl. <br/>d. mathem. W. 5. p. 247. <pb/>
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445 </p><!--l. 795--><p class="indent">
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446
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447 </p><!--l. 796--><p class="indent"> Entsprechende Gleichungen gelten für die beiden anderen <br/>Komponenten der ponderomotorischen Kraft.
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448 </p><!--l. 799--><p class="indent"> Integriert man (12) über den unendlichen Raum, so erhält <br/>man, falls im Unendlichen die Feldvektoren verschwinden, die <br/>Gleichung:</p>
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449 <table width="100%"
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450 class="equation"><tr><td><a
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451 id="x1-16r14"></a>
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452 <center class="math-display" >
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453 <img
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454 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Ueber_de_1908_01/fulltext/img/Einst_Ueber_de_1908_0143x.png" alt=" integral integral
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455 1 dSx Fx dt = - -2 d t ----. c dt " class="math-display" /></center></td><td width="5%">(14)</td></tr></table>
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456 <!--l. 808--><p class="nopar">
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457 </p><!--l. 810--><p class="noindent">Sie sagt aus, daß unsere ponderomotorischen Kräfte bei Ein-<br/>führung der elektromagnetischen Bewegungsgröße dem Satz <br/>von der Gleichheit von actio und reactio genügen.
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458 </p><!--l. 815--><p class="indent"> Bern, 7. Mai 1908. </p>
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459 <div class="center" >
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460
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461 <!--l. 816--><p class="noindent">
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462 </p><!--l. 817--><p class="noindent">(Eingegangen 13. Mai 1908.)</p></div>
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463 <div class="center" >
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464
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465 <!--l. 822--><p class="noindent">
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466 </p><!--l. 823--><p class="noindent">----------</p></div>
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467
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468 </body></html>
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