Special relativity: from clock operations to electrodynamics: §9 · charge and current

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.

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.

Return to how a neutral wire carries charge density in another frame. (included in this file)

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.

1. 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.
2. 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 $\rho'=\beta\left(1-\frac{vu_x}{V^2}\right)\rho$.
3. 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.
4. 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.

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.

1. The components across the motion carry the same factor in their velocity and in the density, so $J'_y=J_y$ and $J'_z=J_z$.
2. A neutral wire, ρ = 0, carrying current $J_x$: then $\rho'=-\gamma vJ_x/c^2$, not zero for any v other than zero.
3. 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 $J_x=-\rho_+u$.
4. 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 $−2.5\times10^{-3}$.
5. 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.
6. 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.

Return to how a neutral wire carries charge density in another frame. (included in this file)

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.