Annus MirabilisEinstein in 1905
Reading preferences

Reading preferences

Keep every explanation and worked example on the page. Nothing moves or plays by itself, and the larger instruments load only when you ask.

Line length
Type size
Contrast
Paragraph spacing

Read · Special relativity: from clock operations to electrodynamics · Results

§9 · charge and current

Clocks, rods and light first, then electricity and magnetism: follow the relativity paper through its introduction and all ten sections, including the later ones most retellings skip.

The paper's results

Pages 916, 917

The Maxwell-Hertz equations with convection current keep their form

Carried into the moving system by the transformations of § 3 and § 6, the Maxwell-Hertz equations with convection current, which § 9 states for K, keep their form, with the charge density ϱ′=β(1−vuxV2)ϱ\varrho'=\beta(1-\frac{v u_x}{V^2})\varrho and the velocity of the electricity transformed by the addition theorem of § 5; so the electrodynamic basis of Lorentz's theory of moving bodies conforms to the principle of relativity, and a body whose charge is constant in a system moving with it has a constant charge in K as well.

As printed

Transformiert man diese Gleichungen, welche im System KK gelten mögen, mit Hilfe der Transformationsgleichungen von § 3 und § 6 auf das System kk, so erhält man die Gleichungen: wobei

1V{uξϱ′+∂X′∂τ}=∂N′∂η−∂M′∂ζ,∂L′∂τ=∂Y′∂ζ−∂Z′∂η,1V{uηϱ′+∂Y′∂τ}=∂L′∂ζ−∂N′∂ξ,∂M′∂τ=∂Z′∂ξ−∂X′∂ζ,1V{uζϱ′+∂Z′∂τ}=∂M′∂ξ−∂L′∂η,∂N′∂τ=∂X′∂η−∂Y′∂ξ,\begin{aligned} \frac{1}{V} \left\{ u_\xi \varrho' + \frac{\partial X'}{\partial \tau} \right\} &= \frac{\partial N'}{\partial \eta} - \frac{\partial M'}{\partial \zeta} , &\qquad \frac{\partial L'}{\partial \tau} &= \frac{\partial Y'}{\partial \zeta} - \frac{\partial Z'}{\partial \eta} , \\ \frac{1}{V} \left\{ u_\eta \varrho' + \frac{\partial Y'}{\partial \tau} \right\} &= \frac{\partial L'}{\partial \zeta} - \frac{\partial N'}{\partial \xi} , &\qquad \frac{\partial M'}{\partial \tau} &= \frac{\partial Z'}{\partial \xi} - \frac{\partial X'}{\partial \zeta} , \\ \frac{1}{V} \left\{ u_\zeta \varrho' + \frac{\partial Z'}{\partial \tau} \right\} &= \frac{\partial M'}{\partial \xi} - \frac{\partial L'}{\partial \eta} , &\qquad \frac{\partial N'}{\partial \tau} &= \frac{\partial X'}{\partial \eta} - \frac{\partial Y'}{\partial \xi} , \end{aligned}Speed of lightCharge velocity (moving system)Charge density, moving systemElectric 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 systemCharge velocity (moving system)Position in the moving systemCharge velocity (moving system)Explain this equation
ux−v1−uxvV2=uξ,uyβ(1−uxvV2)=uη,ϱ′=∂X′∂ξ+∂Y′∂η+∂Z′∂ζ=β(1−vuxV2)ϱ.uzβ(1−uxvV2)=uζ.\begin{aligned} \frac{u_x - v}{1 - \frac{u_x v}{V^2}} &= u_\xi , \\ \frac{u_y}{\beta \left( 1 - \frac{u_x v}{V^2} \right)} &= u_\eta , &\qquad \varrho' = \frac{\partial X'}{\partial \xi} + \frac{\partial Y'}{\partial \eta} + \frac{\partial Z'}{\partial \zeta} = \beta \left( 1 - \frac{v u_x}{V^2} \right) \varrho . \\ \frac{u_z}{\beta \left( 1 - \frac{u_x v}{V^2} \right)} &= u_\zeta . \end{aligned}Charge velocity (stationary system)Speed of the moving systemSpeed of lightCharge velocity (moving system)Charge velocity (stationary system)Lorentz factorCharge velocity (moving system)Charge density, moving systemElectric field, moving systemPosition in the moving systemElectric field, moving systemPosition in the moving systemElectric field, moving systemPosition in the moving systemCharge density, stationary systemCharge velocity (stationary system)Charge velocity (moving system)Explain this equation

Da — wie aus dem Additionstheorem der Geschwindigkeiten (§ 5) folgt — der Vektor (uξ,uη,uζ)(u_\xi, u_\eta, u_\zeta) nichts anderes ist als die Geschwindigkeit der elektrischen Massen im System kk gemessen, so ist damit gezeigt, daß unter Zugrundelegung unserer kinematischen Prinzipien die elektrodynamische Grundlage der Lorentzschen Theorie der Elektrodynamik bewegter Körper dem Relativitätsprinzip entspricht.

Es möge noch kurz bemerkt werden, daß aus den entwickelten Gleichungen leicht der folgende wichtige Satz gefolgert werden kann: Bewegt sich ein elektrisch geladener Körper beliebig im Raume und ändert sich hierbei seine Ladung nicht, von einem mit dem Körper bewegten Koordinatensystem aus betrachtet, so bleibt seine Ladung auch — von dem „ruhenden“ System KK aus betrachtet — konstant.

Text on page 916 of the German source, § 9, paragraph 2 · Display on page 916 of the German source, § 9, display 3 · Display on page 916 of the German source, § 9, display 4 · Text on page 917 of the German source, § 9, paragraph 2 · Text on page 917 of the German source, § 9, paragraph 3

In modern notation

ρ′=γ (ρ−v Jxc2)\rho' = \gamma\,\left(\rho - \frac{v\,J_x}{c^{2}}\right)
Jx′=γ (Jx−v ρ)J_x' = \gamma\,\left(J_x - v\,\rho\right)

Shown in modern letters; Einstein's are in §9 of the German source face.

  • Premise: The electric masses are bound unchangeably to small rigid bodies (ions, electrons); on that supposition the equations are, as § 9 says, the basis of Lorentz's electrodynamics and optics of moving bodies.
  • An inference beyond the argument: What § 9 shows is that the transformed equations have the same form, with charge density and velocity transformed as stated, and it draws from them the constancy of a body's charge. It does not write a four-current or an invariant, and it does not discuss the continuity equation.
  • A comparison, not a premise: Joining charge and current density into one four-vector, with (cρ)2−∣J∣2(c\rho)^2-|J|^2 unchanged, is the later language of Poincaré and Minkowski. That a wire carrying a current appears charged to a moving observer is a modern consequence the paper does not discuss.

Decoder

ϱ\varrho, ϱ′\varrho'
4π4\pi times the density of electricity, as § 9 defines it, in K and in k.
ux,uy,uzu_x, u_y, u_z and uξ,uη,uζu_\xi, u_\eta, u_\zeta
The velocity of the electricity in K and in k. § 9 notes that the second is the first transformed by the addition theorem of § 5, so it is the velocity measured in k.
β\beta
1/1−(v/V)21/\sqrt{1-(v/V)^2}, the factor written γ\gamma today, as in § 3.
uϱu\varrho
The convection current: electricity carried along by moving charges, in the equations beside the rate of change of the electric force. In SI the charge density transforms as ρ′=γ(ρ−vJx/c2)\rho'=\gamma(\rho-vJ_x/c^2), the modern layer's form.
∂L′∂τ\frac{\partial L'}{\partial\tau}, ∂M′∂τ\frac{\partial M'}{\partial\tau}, ∂N′∂τ\frac{\partial N'}{\partial\tau}, as printed
The three magnetic equations of k are printed without the factor 1/V1/V before the time derivative, which the same equations in K carry, as do those of § 6. They contain no convection term and transform as in § 6, so the factor belongs there. Every face keeps the printed reading.

Try it

SR-12: Charge moving at half the speed of light, seen from a system moving at 0.6 of it: what charge density and current does that system measure? Choose the preset “Convection current at 0.5c boosted by 0.6c” there.

SR-12: A uniformly charged sphere seen from a system moving at 0.6 of the speed of light: does its charge density change, and does its total charge? Choose the preset “Uniformly charged sphere at 0.6c” there.

Where this stops: Charge density is not total charge: a density can change between frames while the body’s charge stays fixed. In a neutral wire the positive and negative charges move at different speeds, so there is no single velocity of the charge. Read the argument

Common wrong turns

Argument
A consequence of the transformations of §§ 3, 5 and 6 applied to the Maxwell-Hertz equations with convection current
Model
Exact within Lorentz's electrodynamics of charges bound to small rigid bodies
Evidence
A derivation, not evidence
History
The equations are the basis of Lorentz's theory; that they conform to the principle of relativity under these kinematics is shown in this paper, and the four-current is the later language of Poincaré and Minkowski

§9 · charge and current

Neutrality and current belong to a frame

Charge density and the current along the motion mix when the frame changes, so a wire that is neutral while it carries a current has a charge density in a frame moving along it. There is no single charge velocity to find by dividing current by charge density: in a neutral wire that ratio divides by zero.

Read the argument