Parallel edition

On the Electrodynamics of Moving Bodies

By A. Einstein · Annalen der Physik (4) 17, 891–921 (1905).

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English

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by A. Einstein

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[p. 892]I. Kinematischer Teil.

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I. Kinematical Part.

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[p. 907]II. Eektrodynamischer Teil.

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II. Electrodynamical Part.

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[p. 907]§ 6. Transformation der Maxwell-Hertzschen Gleichungen für den leeren Raum. Über die Natur der bei Bewegung in einem Magnetfeld auftretenden elektromotorischen Kräfte.

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§ 6. Transformation of the Maxwell-Hertz Equations for Empty Space. On the Nature of the Electromotive Forces Occurring with Motion in a Magnetic Field.

Deutsch

[p. 907]Die Maxwell-Hertzschen Gleichungen für den leeren Raum mögen gültig sein für das ruhende System $K$ , so daß gelten möge: $\begin{aligned} \frac{1}{V} \frac{\partial X}{\partial t} &= \frac{\partial N}{\partial y} - \frac{\partial M}{\partial z} , &\qquad \frac{1}{V} \frac{\partial L}{\partial t} &= \frac{\partial Y}{\partial z} - \frac{\partial Z}{\partial y} , \\ \frac{1}{V} \frac{\partial Y}{\partial t} &= \frac{\partial L}{\partial z} - \frac{\partial N}{\partial x} , &\qquad \frac{1}{V} \frac{\partial M}{\partial t} &= \frac{\partial Z}{\partial x} - \frac{\partial X}{\partial z} , \\ \frac{1}{V} \frac{\partial Z}{\partial t} &= \frac{\partial M}{\partial x} - \frac{\partial L}{\partial y} , &\qquad \frac{1}{V} \frac{\partial N}{\partial t} &= \frac{\partial X}{\partial y} - \frac{\partial Y}{\partial x} , \end{aligned}$ Speed of lightElectric field, stationary systemTime in the stationary systemMagnetic field, stationary systemPosition in the stationary systemMagnetic field, stationary systemPosition in the stationary systemMagnetic field, stationary systemElectric field, stationary systemElectric field, stationary systemPosition in the stationary systemExplain this equation wobei $(X, Y, Z)$ den Vektor der elektrischen, $(L, M, N)$ den der magnetischen Kraft bedeutet.

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Let the Maxwell-Hertz equations for empty space hold for the stationary system $K$ , so that we have: $\begin{aligned} \frac{1}{V} \frac{\partial X}{\partial t} &= \frac{\partial N}{\partial y} - \frac{\partial M}{\partial z} , &\qquad \frac{1}{V} \frac{\partial L}{\partial t} &= \frac{\partial Y}{\partial z} - \frac{\partial Z}{\partial y} , \\ \frac{1}{V} \frac{\partial Y}{\partial t} &= \frac{\partial L}{\partial z} - \frac{\partial N}{\partial x} , &\qquad \frac{1}{V} \frac{\partial M}{\partial t} &= \frac{\partial Z}{\partial x} - \frac{\partial X}{\partial z} , \\ \frac{1}{V} \frac{\partial Z}{\partial t} &= \frac{\partial M}{\partial x} - \frac{\partial L}{\partial y} , &\qquad \frac{1}{V} \frac{\partial N}{\partial t} &= \frac{\partial X}{\partial y} - \frac{\partial Y}{\partial x} , \end{aligned}$ Speed of lightElectric field, stationary systemTime in the stationary systemMagnetic field, stationary systemPosition in the stationary systemMagnetic field, stationary systemPosition in the stationary systemMagnetic field, stationary systemElectric field, stationary systemElectric field, stationary systemPosition in the stationary systemExplain this equation where $(X, Y, Z)$ denotes the vector of the electric force, and $(L, M, N)$ that of the magnetic force.

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[pp. 907–908]Wenden wir auf diese Gleichungen die in § 3 entwickelte Transformation an, indem wir die elektromagnetischen Vorgänge auf das dort eingeführte, mit der Geschwindigkeit $v$ bewegte Koordinatensystem beziehen, so erhalten wir die Gleichungen: $\begin{aligned} \frac{1}{V} \frac{\partial X}{\partial \tau} &= \frac{\partial \beta \left( N - \frac{v}{V} Y \right)}{\partial \eta} - \frac{\partial \beta \left( M + \frac{v}{V} Z \right)}{\partial \zeta} , \\ \frac{1}{V} \frac{\partial \beta \left( Y - \frac{v}{V} N \right)}{\partial \tau} &= \frac{\partial L}{\partial \zeta} - \frac{\partial \beta \left( N - \frac{v}{V} Y \right)}{\partial \xi} , \\ \frac{1}{V} \frac{\partial \beta \left( Z + \frac{v}{V} M \right)}{\partial \tau} &= \frac{\partial \beta \left( M + \frac{v}{V} Z \right)}{\partial \xi} - \frac{\partial L}{\partial \eta} , \\ \frac{1}{V} \frac{\partial L}{\partial \tau} &= \frac{\partial \beta \left( Y - \frac{v}{V} N \right)}{\partial \zeta} - \frac{\partial \beta \left( Z + \frac{v}{V} M \right)}{\partial \eta} , \\ \frac{1}{V} \frac{\partial \beta \left( M + \frac{v}{V} Z \right)}{\partial \tau} &= \frac{\partial \beta \left( Z + \frac{v}{V} M \right)}{\partial \xi} - \frac{\partial X}{\partial \zeta} , \\ \frac{1}{V} \frac{\partial \beta \left( N - \frac{v}{V} Y \right)}{\partial \tau} &= \frac{\partial X}{\partial \eta} - \frac{\partial \beta \left( Y - \frac{v}{V} N \right)}{\partial \xi} , \end{aligned}$ Speed of lightElectric field, stationary systemTime in the moving systemLorentz factorMagnetic field, stationary systemSpeed of the moving systemElectric field, stationary systemPosition in the moving systemMagnetic field, stationary systemElectric field, stationary systemPosition in the moving systemMagnetic field, stationary systemPosition in the moving systemExplain this equation wobei $\beta = \frac{1}{\sqrt{1 - \left( \frac{v}{V} \right)^2}} .$ Lorentz factorSpeed of the moving systemSpeed of lightExplain this equation

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If we apply to these equations the transformation developed in § 3, by referring the electromagnetic processes to the coordinate system introduced there, moving with the velocity $v$ , we obtain the equations: $\begin{aligned} \frac{1}{V} \frac{\partial X}{\partial \tau} &= \frac{\partial \beta \left( N - \frac{v}{V} Y \right)}{\partial \eta} - \frac{\partial \beta \left( M + \frac{v}{V} Z \right)}{\partial \zeta} , \\ \frac{1}{V} \frac{\partial \beta \left( Y - \frac{v}{V} N \right)}{\partial \tau} &= \frac{\partial L}{\partial \zeta} - \frac{\partial \beta \left( N - \frac{v}{V} Y \right)}{\partial \xi} , \\ \frac{1}{V} \frac{\partial \beta \left( Z + \frac{v}{V} M \right)}{\partial \tau} &= \frac{\partial \beta \left( M + \frac{v}{V} Z \right)}{\partial \xi} - \frac{\partial L}{\partial \eta} , \\ \frac{1}{V} \frac{\partial L}{\partial \tau} &= \frac{\partial \beta \left( Y - \frac{v}{V} N \right)}{\partial \zeta} - \frac{\partial \beta \left( Z + \frac{v}{V} M \right)}{\partial \eta} , \\ \frac{1}{V} \frac{\partial \beta \left( M + \frac{v}{V} Z \right)}{\partial \tau} &= \frac{\partial \beta \left( Z + \frac{v}{V} M \right)}{\partial \xi} - \frac{\partial X}{\partial \zeta} , \\ \frac{1}{V} \frac{\partial \beta \left( N - \frac{v}{V} Y \right)}{\partial \tau} &= \frac{\partial X}{\partial \eta} - \frac{\partial \beta \left( Y - \frac{v}{V} N \right)}{\partial \xi} , \end{aligned}$ Speed of lightElectric field, stationary systemTime in the moving systemLorentz factorMagnetic field, stationary systemSpeed of the moving systemElectric field, stationary systemPosition in the moving systemMagnetic field, stationary systemElectric field, stationary systemPosition in the moving systemMagnetic field, stationary systemPosition in the moving systemExplain this equation where $\beta = \frac{1}{\sqrt{1 - \left( \frac{v}{V} \right)^2}} .$ Lorentz factorSpeed of the moving systemSpeed of lightExplain this equation

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[p. 908]Das Relativitätsprinzip fordert nun, daß die Maxwell-Hertzschen Gleichungen für den leeren Raum auch im System $k$ gelten, wenn sie im System $K$ gelten, d. h. daß für die im bewegten System $k$ durch ihre ponderomotorischen Wirkungen auf elektrische bez. magnetische Massen definierten Vektoren der elektrischen und magnetischen Kraft ( $(X', Y' Z')$ und $(L', M', N')$ ) des bewegten Systems $k$ die Gleichungen gelten: $\begin{aligned} \frac{1}{V} \frac{\partial X'}{\partial \tau} &= \frac{\partial N'}{\partial \eta} - \frac{\partial M'}{\partial \zeta} , &\qquad \frac{1}{V} \frac{\partial L'}{\partial \tau} &= \frac{\partial Y'}{\partial \zeta} - \frac{\partial Z'}{\partial \eta} , \\ \frac{1}{V} \frac{\partial Y'}{\partial \tau} &= \frac{\partial L'}{\partial \zeta} - \frac{\partial N'}{\partial \xi} , &\qquad \frac{1}{V} \frac{\partial M'}{\partial \tau} &= \frac{\partial Z'}{\partial \xi} - \frac{\partial X'}{\partial \zeta} , \\ \frac{1}{V} \frac{\partial Z'}{\partial \tau} &= \frac{\partial M'}{\partial \xi} - \frac{\partial L'}{\partial \eta} , &\qquad \frac{1}{V} \frac{\partial N'}{\partial \tau} &= \frac{\partial X'}{\partial \eta} - \frac{\partial Y'}{\partial \xi} . \end{aligned}$ Speed of lightElectric field, moving systemTime in the moving systemMagnetic field, moving systemPosition in the moving systemMagnetic field, moving systemPosition in the moving systemMagnetic field, moving systemElectric field, moving systemElectric field, moving systemPosition in the moving systemExplain this equation

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The principle of relativity now requires that the Maxwell-Hertz equations for empty space hold also in the system $k$ if they hold in the system $K$ , that is, that for the vectors of the electric and magnetic force ( $(X', Y' Z')$ and $(L', M', N')$ ) of the moving system $k$ , which are defined in the moving system $k$ by their ponderomotive actions on electric or magnetic masses respectively, the equations hold: $\begin{aligned} \frac{1}{V} \frac{\partial X'}{\partial \tau} &= \frac{\partial N'}{\partial \eta} - \frac{\partial M'}{\partial \zeta} , &\qquad \frac{1}{V} \frac{\partial L'}{\partial \tau} &= \frac{\partial Y'}{\partial \zeta} - \frac{\partial Z'}{\partial \eta} , \\ \frac{1}{V} \frac{\partial Y'}{\partial \tau} &= \frac{\partial L'}{\partial \zeta} - \frac{\partial N'}{\partial \xi} , &\qquad \frac{1}{V} \frac{\partial M'}{\partial \tau} &= \frac{\partial Z'}{\partial \xi} - \frac{\partial X'}{\partial \zeta} , \\ \frac{1}{V} \frac{\partial Z'}{\partial \tau} &= \frac{\partial M'}{\partial \xi} - \frac{\partial L'}{\partial \eta} , &\qquad \frac{1}{V} \frac{\partial N'}{\partial \tau} &= \frac{\partial X'}{\partial \eta} - \frac{\partial Y'}{\partial \xi} . \end{aligned}$ Speed of lightElectric field, moving systemTime in the moving systemMagnetic field, moving systemPosition in the moving systemMagnetic field, moving systemPosition in the moving systemMagnetic field, moving systemElectric field, moving systemElectric field, moving systemPosition in the moving systemExplain this equation

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[p. 908]Offenbar müssen nun die beiden für das System $k$ gefundenen Gleichungssysteme genau dasselbe ausdrücken, da beide Gleichungssysteme den Maxwell-Hertzschen Gleichungen für das System $K$ äquivalent sind. Da die Gleichungen beider Systeme ferner bis auf die die Vektoren darstellenden Symbole übereinstimmen, so folgt, daß die in den Gleichungssystemen an entsprechenden Stellen auftretenden Funktionen bis auf einen für alle Funktionen des einen Gleichungssystems gemeinsamen, von $\xi$ , $\eta$ , $\zeta$ und $\tau$ unabhängigen, eventuell von $v$ abhängigen Faktor $\psi(v)$ übereinstimmen müssen. Es gelten also die Beziehungen: $\begin{aligned} X' &= \psi(v) X , &\qquad L' &= \psi(v) L , \\ Y' &= \psi(v) \beta \left( Y - \frac{v}{V} N \right) , &\qquad M' &= \psi(v) \beta \left( M + \frac{v}{V} Z \right) , \\ Z' &= \psi(v) \beta \left( Z + \frac{v}{V} M \right) , &\qquad N' &= \psi(v) \beta \left( N - \frac{v}{V} Y \right) . \end{aligned}$ Electric field, moving systemUnknown scale factor: psi(v)Speed of the moving systemElectric field, stationary systemMagnetic field, moving systemMagnetic field, stationary systemElectric field, moving systemLorentz factorElectric field, stationary systemSpeed of lightMagnetic field, stationary systemMagnetic field, moving systemMagnetic field, stationary systemElectric field, stationary systemElectric field, moving systemMagnetic field, moving systemExplain this equation

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Evidently the two systems of equations found for the system $k$ must now express exactly the same thing, since both systems of equations are equivalent to the Maxwell-Hertz equations for the system $K$ . Since, further, the equations of the two systems agree except for the symbols representing the vectors, it follows that the functions occurring at corresponding places in the systems of equations must agree up to a factor $\psi(v)$ that is common to all the functions of the one system of equations, independent of $\xi$ , $\eta$ , $\zeta$ and $\tau$ , and possibly dependent on $v$ . Thus the relations hold: $\begin{aligned} X' &= \psi(v) X , &\qquad L' &= \psi(v) L , \\ Y' &= \psi(v) \beta \left( Y - \frac{v}{V} N \right) , &\qquad M' &= \psi(v) \beta \left( M + \frac{v}{V} Z \right) , \\ Z' &= \psi(v) \beta \left( Z + \frac{v}{V} M \right) , &\qquad N' &= \psi(v) \beta \left( N - \frac{v}{V} Y \right) . \end{aligned}$ Electric field, moving systemUnknown scale factor: psi(v)Speed of the moving systemElectric field, stationary systemMagnetic field, moving systemMagnetic field, stationary systemElectric field, moving systemLorentz factorElectric field, stationary systemSpeed of lightMagnetic field, stationary systemMagnetic field, moving systemMagnetic field, stationary systemElectric field, stationary systemElectric field, moving systemMagnetic field, moving systemExplain this equation

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[p. 909]Bildet man nun die Umkehrung dieses Gleichungssystems, erstens durch Auflösen der soeben erhaltenen Gleichungen, zweitens durch Anwendung der Gleichungen auf die inverse Transformation (von $k$ auf $K$ ), welche durch die Geschwindigkeit $-v$ charakterisiert ist, so folgt, indem man berücksichtigt, daß die beiden so erhaltenen Gleichungssysteme identisch sein müssen: $\varphi(v) \cdot \varphi(-v) = 1 .$ Unknown scale factor: psi(v)Speed of the moving systemExplain this equation Ferner folgt aus Symmetriegründen^[1)] $\varphi(v) = \varphi(-v) ;$ Unknown scale factor: psi(v)Speed of the moving systemExplain this equation es ist also $\varphi(v) = 1 ,$ Unknown scale factor: psi(v)Speed of the moving systemExplain this equation und unsere Gleichungen nehmen die Form an: $\begin{aligned} X' &= X , &\qquad L' &= L , \\ Y' &= \beta \left( Y - \frac{v}{V} N \right) , &\qquad M' &= \beta \left( M + \frac{v}{V} Z \right) , \\ Z' &= \beta \left( Z + \frac{v}{V} M \right) , &\qquad N' &= \beta \left( N - \frac{v}{V} Y \right) . \end{aligned}$ Electric field, moving systemElectric field, stationary systemMagnetic field, moving systemMagnetic field, stationary systemElectric field, moving systemLorentz factorElectric field, stationary systemSpeed of the moving systemSpeed of lightMagnetic field, stationary systemMagnetic field, moving systemMagnetic field, stationary systemElectric field, stationary systemElectric field, moving systemMagnetic field, moving systemExplain this equation Zur Interpretation dieser Gleichungen bemerken wir folgendes. Es liegt eine punktförmige Elektrizitätsmenge vor, welche im ruhenden System $K$ gemessen von der Größe „eins“ sei, d. h. im ruhenden System ruhend auf eine gleiche Elektrizitätsmenge im Abstand 1 cm die Kraft 1 Dyn ausübe. Nach dem Relativitätsprinzip ist diese elektrische Masse auch im bewegten System gemessen von der Größe „eins“. Ruht diese Elektrizitätsmenge relativ zum ruhenden System, so ist definitionsgemäß der Vektor $(X, Y, Z)$ gleich der auf sie wirkenden Kraft. Ruht die Elektrizitätsmenge gegenüber dem bewegten System (wenigstens in dem betreffenden Augenblick), so ist die auf sie wirkende, in dem bewegten System gemessene Kraft gleich dem Vektor $(X', Y', Z')$ . Die ersten drei der obigen Gleichungen lassen sich mithin auf folgende zwei Weisen in Worte kleiden:

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If we now form the inverse of this system of equations, first by solving the equations just obtained, and second by applying the equations to the inverse transformation (from $k$ to $K$ ), which is characterized by the velocity $-v$ , it follows, taking into account that the two systems of equations so obtained must be identical: $\varphi(v) \cdot \varphi(-v) = 1 .$ Unknown scale factor: psi(v)Speed of the moving systemExplain this equation Further, it follows from reasons of symmetry^[1)] that $\varphi(v) = \varphi(-v) ;$ Unknown scale factor: psi(v)Speed of the moving systemExplain this equation hence $\varphi(v) = 1 ,$ Unknown scale factor: psi(v)Speed of the moving systemExplain this equation and our equations take the form: $\begin{aligned} X' &= X , &\qquad L' &= L , \\ Y' &= \beta \left( Y - \frac{v}{V} N \right) , &\qquad M' &= \beta \left( M + \frac{v}{V} Z \right) , \\ Z' &= \beta \left( Z + \frac{v}{V} M \right) , &\qquad N' &= \beta \left( N - \frac{v}{V} Y \right) . \end{aligned}$ Electric field, moving systemElectric field, stationary systemMagnetic field, moving systemMagnetic field, stationary systemElectric field, moving systemLorentz factorElectric field, stationary systemSpeed of the moving systemSpeed of lightMagnetic field, stationary systemMagnetic field, moving systemMagnetic field, stationary systemElectric field, stationary systemElectric field, moving systemMagnetic field, moving systemExplain this equation For the interpretation of these equations we remark the following. Let there be a point-like quantity of electricity which, measured in the stationary system $K$ , is of magnitude “one”, that is, which, at rest in the stationary system, exerts the force of 1 dyne on an equal quantity of electricity at a distance of 1 cm. By the principle of relativity this electric mass is also of magnitude “one” when measured in the moving system. If this quantity of electricity is at rest relative to the stationary system, then by definition the vector $(X, Y, Z)$ is equal to the force acting on it. If the quantity of electricity is at rest relative to the moving system (at least at the instant in question), then the force acting on it, measured in the moving system, is equal to the vector $(X', Y', Z')$ . Consequently the first three of the above equations can be put into words in the following two ways:

Deutsch

[pp. 909–910]1. Ist ein punktförmiger elektrischer Einheitspol in einem elektromagnetischen Felde bewegt, so wirkt auf ihn außer der elektrischen Kraft eine „elektromotorische Kraft“, welche unter Vernachlässigung von mit der zweiten und höheren Potenzen von $v/V$ multiplizierten Gliedern gleich ist dem mit der Lichtgeschwindigkeit dividierten Vektorprodukt der Bewegungsgeschwindigkeit des Einheitspoles und der magnetischen Kraft. (Alte Ausdrucksweise.)

English

1. If a point-like electric unit pole moves in an electromagnetic field, there acts on it, besides the electric force, an “electromotive force”, which, neglecting terms multiplied by the second and higher powers of $v/V$ , is equal to the vector product of the velocity of motion of the unit pole and the magnetic force, divided by the velocity of light. (Old mode of expression.)

Deutsch

[p. 910]2. Ist ein punktförmiger elektrischer Einheitspol in einem elektromagnetischen Felde bewegt, so ist die auf ihn wirkende Kraft gleich der an dem Orte des Einheitspoles vorhandenen elektrischen Kraft, welche man durch Transformation des Feldes auf ein relativ zum elektrischen Einheitspol ruhendes Koordinatensystem erhält. (Neue Ausdrucksweise.)

English

2. If a point-like electric unit pole moves in an electromagnetic field, the force acting on it is equal to the electric force present at the location of the unit pole, which is obtained by transforming the field to a coordinate system at rest relative to the electric unit pole. (New mode of expression.)

Deutsch

[p. 910]Analoges gilt über die „magnetomotorischen Kräfte“. Man sieht, daß in der entwickelten Theorie die elektromotorische Kraft nur die Rolle eines Hilfsbegriffes spielt, welcher seine Einführung dem Umstande verdankt, daß die elektrischen und magnetischen Kräfte keine von dem Bewegungszustande des Koordinatensystems unabhängige Existenz besitzen.

English

Something analogous holds for the “magnetomotive forces”. One sees that in the theory developed here the electromotive force plays only the part of an auxiliary concept, which owes its introduction to the circumstance that the electric and magnetic forces have no existence independent of the state of motion of the coordinate system.

Deutsch

[p. 910]Es ist ferner klar, daß die in der Einleitung angeführte Asymmetrie bei der Betrachtung der durch Relativbewegung eines Magneten und eines Leiters erzeugten Ströme verschwindet. Auch werden die Fragen nach dem „Sitz“ der elektrodynamischen elektromotorischen Kräfte (Unipolarmaschinen) gegenstandslos.

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It is further clear that the asymmetry mentioned in the introduction, which arises when we consider the currents produced by the relative motion of a magnet and a conductor, disappears. The questions about the “seat” of electrodynamic electromotive forces (unipolar machines) also become pointless.

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[p. 921]

Zum Schlusse bemerke ich, daß mir beim Arbeiten an dem hier behandelten Probleme mein Freund und Kollege M. Besso treu zur Seite stand und daß ich demselben manche wertvolle Anregung verdanke.

English

In conclusion I remark that in working on the problem treated here my friend and colleague M. Besso stood loyally by my side, and that I owe him several valuable suggestions.

Deutsch

[p. 921]

Bern, Juni 1905.

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Bern, June 1905.

Deutsch

[p. 921]

(Eingegangen 30. Juni 1905.)

English

(Received 30 June 1905.)

Fußnoten

Deutsch

1) Ist z. B. $X = Y = Z = L = M = 0$ und $N \neq 0$ , so ist aus Symmetriegründen klar, daß bei Zeichenwechsel von $v$ ohne Änderung des numerischen Wertes auch $Y'$ sein Vorzeichen ändern muß, ohne seinen numerischen Wert zu ändern. ↩

English

[p. 892]I. Kinematischer Teil.

I. Kinematical Part.Show the German source

[p. 907]II. Eektrodynamischer Teil.

II. Electrodynamical Part.Show the German source

[p. 907]§ 6. Transformation der Maxwell-Hertzschen Gleichungen für den leeren Raum. Über die Natur der bei Bewegung in einem Magnetfeld auftretenden elektromotorischen Kräfte.

§ 6. Transformation of the Maxwell-Hertz Equations for Empty Space. On the Nature of the Electromotive Forces Occurring with Motion in a Magnetic Field.Show the German source

Fußnoten

I. Kinematical Part.

II. Electrodynamical Part.

§ 6. Transformation of the Maxwell-Hertz Equations for Empty Space. On the Nature of the Electromotive Forces Occurring with Motion in a Magnetic Field.