Results · §6 · field equations and field components

Pages 909, 910

The electric and magnetic forces in the moving system

Carried into the moving system by the equations of § 3, the Maxwell-Hertz equations for empty space keep their form when the electric and magnetic forces in that system are the printed combinations, in which the components along the motion are unchanged and those across it mix electric with magnetic, so that, as the paper concludes, the two forces have no existence independent of the state of motion of the coordinate system.

As printed

$\begin{aligned} X' &= X , &\qquad L' &= L , \\ Y' &= \beta \left( Y - \frac{v}{V} N \right) , &\qquad M' &= \beta \left( M + \frac{v}{V} Z \right) , \\ Z' &= \beta \left( Z + \frac{v}{V} M \right) , &\qquad N' &= \beta \left( N - \frac{v}{V} Y \right) . \end{aligned}$ 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
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.)
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.)
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.
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.

Display on page 909 of the German source, § 6, display 9 · Text on pages 909–910 of the German source, § 6, paragraph 9 · Text on page 910 of the German source, § 6, paragraph 10 · Text on page 910 of the German source, § 6, paragraph 7 · Text on page 910 of the German source, § 6, paragraph 8

In modern notation

E'_{x} = E_{x}

E'_{y} = \gamma\,\left(E_{y} - v\,B_{z}\right)

E'_{z} = \gamma\,\left(E_{z} + v\,B_{y}\right)

B'_{x} = B_{x}

B'_{y} = \gamma\,\left(B_{y} + \frac{v\,E_{z}}{c^{2}}\right)

B'_{z} = \gamma\,\left(B_{z} - \frac{v\,E_{y}}{c^{2}}\right)

Shown in modern letters; Einstein's are in §6 of the German source face.

Premise: The derivation assumes the Maxwell-Hertz equations for empty space, not a theory of matter, and the principle of relativity, which requires them to hold in the same form in k.
Premise: That the transformed quantities are the forces a charge at rest in k would feel is a further step. § 6 takes it by measuring force on a unit charge, and by taking, from the principle of relativity, that a charge of one unit in K is one unit in k; that the equations keep their form does not by itself prove it.
Approximation: The old manner of expression is printed with terms in the second and higher powers of $v/V$ neglected. The transformation of the forces is exact.

Decoder

$X, Y, Z$ and $L, M, N$: The components of the electric and of the magnetic force in K; primed, the same components in k. Here $L$ and $N$ are magnetic components, not the $L$ or the $N$ of the other papers.
$\beta$: $1/\sqrt{1-(v/V)^2}$ , the factor written $\gamma$ today, as in § 3.
$Y-\frac{v}{V}N$: The paper uses Gaussian units, in which electric and magnetic forces have the same dimension. In SI the same combination is $E_y-vB_z$ , and the modern layer writes the SI form.
"Alte" and "Neue Ausdrucksweise": The old and the new manner of expression. In the old, a charge moving in a field feels an electromotive force besides the electric force; in the new, the force on it is the electric force in a system in which it is at rest.

Try it

SR-08: A field that is purely electric in K, seen from k moving at 0.6 of the speed of light: what electric and magnetic forces does k measure? Choose the preset “Pure electric field at 0.6c” there.

SR-08: A field that is purely magnetic in K, seen from k moving at 10 m/s: what electric force does a charge at rest in k feel? Choose the preset “Pure magnetic field at 10 m/s” there.

Where this stops: Einstein’s Gaussian equations become SI ones by a change of units, not by renaming symbols. A common factor on all the new fields is fixed by a separate argument, going there and back and the symmetry of space; rearranging the equations does not fix it. Read the argument

Where this stops: An accelerating charge has a different momentary rest frame at each instant, not one frame for its whole path. Fields, forces and electromotive force each transform by their own rule. Read the argument

Argument: A consequence of the principle of relativity applied to the Maxwell-Hertz equations, with the forces in k defined by their action on a charge at rest there
Model: Exact within the electrodynamics of empty space
Evidence: A derivation, not evidence
History: Lorentz's 1904 theory had already transformed the free-space fields, as mathematical aids in an ether at rest; here they are the forces measured in the moving system

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§6 · field equations and field components

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