Read · Special relativity: from clock operations to electrodynamics

§9 · charge and current

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Letters

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Einstein's letters change 12 of this paper's 40 formulas. 28 stay in today's letters, with a note saying so.

A derivation

Neutrality and current belong to a frame

How can a neutral current acquire charge density?

The 3 printed paragraphs this passage explains

§9, paragraph 1: Einstein starts from the Maxwell-Hertz equations with convection currents, electricity carried along by moving charges. With the charges bound to small rigid bodies, ions or electrons, these are the electromagnetic basis of Lorentz's electrodynamics and optics of moving bodies.
§9, paragraph 2: Transformed into the moving system, the equations keep their form, with the velocity of the charges and the density of electricity as measured there. So the electrodynamic basis of Lorentz's theory agrees with the principle of relativity.
§9, paragraph 3: A further result: if a charged body moves in any way and its charge, measured in a system moving with it, stays the same, then its charge measured from the stationary system stays constant too.

\rho' = \gamma\,\left(\rho - \frac{v\,J_x}{c^{2}}\right)

J_x' = \gamma\,\left(J_x - v\,\rho\right)

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

The primed charge density is gamma times rho minus v times the current along x over c squared; the primed current along x is gamma times the current along x minus v times rho.

The current components across the motion, $J_y$ and $J_z$ , are unchanged. For a neutral wire, $\rho = 0$ , carrying current $J_x$ , the new density is $\rho' = -\gamma vJ_x/c^2$ , which is not zero for any $v \ne 0$ . To see why, transform each kind of charge on its own: the positive ions and the moving electrons have different velocities, so their densities change by different factors.

Charge is conserved in every frame: density and current together obey the continuity equation. The total charge of one body is the same in both frames when each frame adds up the charge at one of its own instants. A fixed box of space is a different comparison, because charge flows through its walls. The charge-and-current laboratory compares the densities and the total charge of a body.

Open the foundation: Counting what crosses a boundary

Show every step here: Neutrality and current belong to a frame

Letters: in §9 the paper writes ρ for the divergence of its Gaussian electric force, which it calls 4π times the density of electricity, and $(u_x,u_y,u_z)$ for the velocity of the electricity; in k these are $\rho'$ and $(u_ξ,u_η,u_ζ)$ . This explanation writes ρ for the charge density itself and $J_x=\rho u_x$ for the current along x, in SI units, where the paper's factors of 4π and V are absorbed.

The paper extends the Maxwell–Hertz equations of §6 by the convection current: each equation for the electric force gains a term $u\rho$ . With electric masses bound to small rigid bodies, ions and electrons, these are the electromagnetic basis of Lorentz's electrodynamics and optics of moving bodies.
Transforming them with §3 and §6 gives the same equations in k, with $u_ξ=\frac{u_x-v}{1-u_xv/V^2}$ , $u_η=\frac{u_y}{\beta(1-u_xv/V^2)}$ , $u_ζ=\frac{u_z}{\beta(1-u_xv/V^2)}$ , and $\rho'=\beta\left(1-\frac{vu_x}{V^2}\right)\rho$ .
By §5, $(u_ξ,u_η,u_ζ)$ is the velocity of the electricity measured in k. So, on the paper's kinematic principles, the electrodynamic basis of Lorentz's theory agrees with the principle of relativity.
Rewrite the density rule with $J_x=\rho u_x$ : $\rho'=\gamma(\rho-vJ_x/c^2)$ . Multiply $\rho'$ by $u_ξ$ and simplify: $J'_x=\gamma(J_x-v\rho)$ . The pair is below.

\rho' = \gamma\,\left(\rho - \frac{v\,J_x}{c^{2}}\right)

J_x' = \gamma\,\left(J_x - v\,\rho\right)

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

The primed charge density is gamma times rho minus v times the current along x over c squared; the primed current along x is gamma times the current along x minus v times rho.

The components across the motion carry the same factor in their velocity and in the density, so $J'_y=J_y$ and $J'_z=J_z$ .
A neutral wire, ρ = 0, carrying current $J_x$ : then $\rho'=-\gamma vJ_x/c^2$ , not zero for any v other than zero.
Why, charge by charge: the positive ions are at rest with density $\rho_+$ , and the electrons move at $u$ with density $-\rho_+$ . The paper's rule gives the ions $\gamma\rho_+$ and the electrons $-\gamma\rho_+(1-vu/c^2)$ . Their sum is $\gamma\rho_+vu/c^2$ , which is $-\gamma vJ_x/c^2$ since $J_x=-\rho_+u$ .
Worked: 1 ampere through a wire of 1 square millimetre is $J_x=10^{6}$ amperes per square metre. For an observer walking along it at 1 metre per second, $\rho'$ is about $−1.1\times10^{-11}$ coulombs per cubic metre. At 0.6c it would be about $−2.5\times10^{-3}$ .
There is no single velocity to find by dividing $J_x$ by ρ: in a neutral wire that divides by zero. Transform the positive and the negative charges separately.
Charge is kept. The paper states the theorem: if a charged body moves in any way and its charge does not change as seen from a system moving with it, then its charge is also constant as seen from K. Compare the total charge of the same body, each frame adding it up at one of its own instants; a fixed box of space is a different comparison, because charge flows through its walls.

A tool for this step

Counting what crosses a boundary

If particles keep crossing the edges of a region, what decides whether it fills up or empties?

Take a short stretch of a tube. If more particles cross into it than cross out, the number inside goes up. The flux J counts the net number crossing a surface per unit area and per second, with rightward crossings positive; the concentration c counts particles per unit volume. They measure different things, and the equation below connects them.

\frac{\partial c}{\partial t} = -\frac{\partial J}{\partial x}

The rate at which the concentration changes equals minus the rate at which the flux changes along x.

The minus sign is the bookkeeping. If the rightward flux is larger at the right end of the stretch than at the left, more particles leave on the right than arrive on the left, and the concentration falls.

Worked example: Seven particles in, five out

In one second, 7 particles cross into the stretch and 5 cross out. The count inside rises by 2.
Next second, 5 cross in and 5 cross out. The count is unchanged, although 10 particles crossed.

The stretch, its two boundaries, and what crossed them

The dashed lines are the boundaries and the box between them is the stretch. Every particle that crosses is drawn separately, so the counts can be checked by counting.

More arrived than left, so the stretch holds 2 more than it did.

10 particles crossed a boundary and the count inside did not move. A steady count is not a still one.

What it shows, in words

In the first second 7 particles cross the left boundary into the stretch and 5 cross out through the right boundary. 12 particles crossed a boundary in all, and because 7 arrived against 5 that left, the stretch ends the second holding 2 more than it began with.

In the second second 5 cross in and 5 cross out. 10 particles crossed, which is nearly as many as before, and the number inside is exactly what it was. What decides whether a region fills or empties is the difference between the two crossings, never the size of either one.

Where this lesson stops

The contents change by the net crossing. A steady count does not mean the particles have stopped, only that as many leave as arrive.

Explore the equations in this step

Read each operation, check its units and assumptions, or open the mathematical step behind it. These are modern teaching equations, not the equations as printed.

Explore the equation · Modern model notation

Charge density depends on the frame

\rho' = \gamma\,\left(\rho - \frac{v\,J_x}{c^{2}}\right)

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

is times minus times over squared.

Select a term or operation. In the formula, Down enters an operation, Up returns to its parent, and Left/Right move between siblings. Escape clears selection. Tab leaves the formula.

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Each term and operation, in words

No live values

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Read the equation aloud in words

The charge density in the moving system is the Lorentz factor times the stationary charge density minus the speed of the moving system times the current density along x over the speed of light squared.

A current in one system contributes to the charge density in the other. A wire that is neutral in the stationary system but carries a current is not neutral in the moving system.

Model assumptions and every term’s meaning

The moving system moves along x at speed v relative to the stationary system, with |v| < c. Modern teaching notation: gamma is the paper's beta, c its V.
Charge density and the current along x transform together; the currents across the motion are unchanged by this boost.

Density corrected for the current: Zero density with a current gives a nonzero result. Read the prerequisite
c squared: Makes the contribution small for everyday currents and speeds. Read the prerequisite
What it asserts: Neutrality belongs to a frame. The total charge of a body is the same in both systems; this is about density. Read the prerequisite
Times gamma: The same factor as for coordinates. Read the prerequisite
A charge density from a current: v times the current density along x, over c squared, has the units of charge per cubic metre. Read the prerequisite
Speed times current: Nonzero whenever there is a current along the motion. Read the prerequisite
Current along x, stationary system: The current per square metre along the motion, in the stationary system. Read the prerequisite
Charge density, stationary system: Charge per cubic metre measured in the stationary system. Read the prerequisite
Charge density, moving system: Charge per cubic metre measured in the moving system. Read the prerequisite
Lorentz factor: One over the square root of one minus v squared over c squared. At 0.6c it is 1.25. Read the prerequisite
Speed of light: The paper's V. Read the prerequisite
Speed of the moving system: Signed along x: the sign says which way the system moves. Read the prerequisite

Every term’s units were checked when this page was built. That checks the units, not the model. This equation is written for this edition in modern notation, not transcribed from the paper.

Explore the equation · Modern model notation

The current along the motion depends on the frame

J_x' = \gamma\,\left(J_x - v\,\rho\right)

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

is times minus times .

Select a term or operation. In the formula, Down enters an operation, Up returns to its parent, and Left/Right move between siblings. Escape clears selection. Tab leaves the formula.

Explain this equation

Each term and operation, in words

No live values

This equation states a relation. No instrument on this page computes its terms.

Read the equation aloud in words

Primed longitudinal current is gamma times J x minus v rho.

Charge at rest in one system is a current in the other: charge density times the speed of the system.

Model assumptions and every term’s meaning

The moving system moves along x at speed v relative to the stationary system, with |v| < c. Modern teaching notation: gamma is the paper's beta, c its V.
Charge density and the current along x transform together; the currents across the motion are unchanged by this boost.

Current corrected for the motion: Subtracting the charge that the relative motion carries along. Read the prerequisite
Charge carried by the motion: Charge density times speed is a current: charge at rest in one system flows in the other. Read the prerequisite
What it asserts: Current and charge density transform together, like position and time. Read the prerequisite
Times gamma: The same factor as for coordinates. Read the prerequisite
Current along x, stationary system: The current per square metre along the motion, in the stationary system. Read the prerequisite
Current along x, moving system: The current per square metre along the motion, in the moving system. Read the prerequisite
Charge density, stationary system: Charge per cubic metre in the stationary system. Read the prerequisite
Lorentz factor: One over the square root of one minus v squared over c squared. At 0.6c it is 1.25. Read the prerequisite
Speed of the moving system: Signed along x: the sign says which way the system moves. Read the prerequisite

Every term’s units were checked when this page was built. That checks the units, not the model. This equation is written for this edition in modern notation, not transcribed from the paper.

Modern qualifications

Einstein treats convection current, charge carried along by moving bodies, and this edition keeps his account as he gives it instead of turning it into a general formula for a single charged fluid. The four-current, which combines $\rho$ and $J$ into one four-vector, came later; it is used here only to check results.

Assumptions and limits: Neutrality and current belong to a frame

Assumed here

Charge and current are spread through a real body, and the new observer moves along x.
When total charge is compared between frames, it is the charge of the same isolated body.

What this does not establish

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.

Earlier step: Light constraints leave a scale to determine

Source context: German source · English · Interlinear gloss · Facsimile

References

A. Einstein, Über einen die Erzeugung und Verwandlung des Lichtes betreffenden heuristischen Gesichtspunkt. Annalen der Physik (4), 17, 132–148 (1905).

A. Einstein, On the motion of particles suspended in liquids at rest required by the molecular-kinetic theory of heat. Annalen der Physik (4), 17, 549–560 (1905), §§1–5. Bibliographic pointer; this preview is not a source transcription or translation.

A. Einstein, Zur Elektrodynamik bewegter Körper. Annalen der Physik (4), 17, 891–921 (1905).

A. Einstein, Does the inertia of a body depend upon its energy content? Annalen der Physik (4), 18, 639–641 (1905). External 1923 Perrett–Jeffery translation, electronically transcribed by John Walker; its notation was modernized. A reference for this explanatory preview, not this edition’s translation or its pinned facsimile.

28 foundation readings sit behind this argument.

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§9 · charge and current

Original journal scan

Neutrality and current belong to a frame

Charge density depends on the frame

The current along the motion depends on the frame

Modern qualifications

What this relies on

Assumed here

What this does not establish

References

Reading a graph

Fractions and ratios

A letter stands for a quantity

Marks that look alike and do different jobs

A sign records direction

Powers of ten and physical units

Squares and square roots

Adding and averaging

Rates of change and derivatives

Density is not probability

Dot and cross products

Charges, fields and light

Exponential change

Fields and waves

Counting what crosses a boundary

Events and distant clocks

Functions and graphs

Hyperbolic functions and rapidity

Adding continuously

Logarithms: turning products into sums

Matrices and linear maps

The push of light

Partial derivatives and held-fixed quantities

Quantities and units

Approximating a curve near a point

The units of 1905

Vectors and their components

Energy of motion and inertia

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