Moving-Charge Fields: Geometry, Flux and Detection

01Co-moving charges and a stationary probe ask different questions

The magnetic-force page gives a specified interaction for co-moving charges. A laboratory experiment holding its probe still while the source passes asks something else: what field reaches the probe, and how does it become a reading? The source–probe velocities differ. Dividing the first two-body law by probe charge does not supply the second configuration's complete high-speed field.

Let the source charge be $q$ and its constant background speed be $v$, with $\beta=v/c$, $\gamma=(1-\beta^2)^{-1/2}$. In reference coordinates, $\mathbf R$ points from the chosen source position to the field point, $R=|\mathbf R|$, and $\phi$ is its angle with source velocity; the manuscript writes $\varphi$. Source position may be simultaneous at process parameter $t$ or retarded in a causal source model. That choice must be specified together with potentials, source density and propagation kernel.

LVE supplies $R_T=K(\phi)R$, where $K(\phi)=\gamma(1-\beta\cos\phi)$. This is initially a one-dimensional directed-length input, not a complete prescription for three-dimensional components, area elements or gradients. This page retains the manuscript's candidate and open-review status, using two constitutive choices to identify the remaining work.

02Construct a radial candidate from true inverse-square intensity

First add two choices: take the direction to be radial in ordinary reference coordinates, and keep the reference sphere area element $d\mathbf S=\hat{\mathbf R}R^2d\Omega$. Here $d\Omega$ is solid angle, unrelated to the spatial curl $\boldsymbol\Omega_T$ on the magnetic-field page. Rewriting true inverse-square intensity in terms of $R$ gives:

The coefficient $1/K^2=(1-\beta^2)/(1-\beta\cos\phi)^2$ comes from squaring the length. It provides a candidate angular enhancement and suppression but does not specify induction, accompanying magnetic field, source-density transformation or stationary-probe dynamics. The added assumptions cannot be presented as LVE's unique consequence.

03What follows from differentiating a candidate potential?

Alternatively choose $\Phi_{\mathrm{cand}}=q/(4\pi\varepsilon_0RK)$ at the true distance, then differentiate in the same ordinary reference spherical coordinates. Let $\hat{\boldsymbol\phi}$ point toward increasing polar angle and $K'=dK/d\phi=\gamma\beta\sin\phi$. Radial differentiation acts on $1/R$, while angular differentiation acts on $1/K$, giving:

The radial coefficient is now $1/K$, not $1/K^2$, and the polar component is nonzero. The equipotential condition $RK(\phi)=\mathrm{const}$ belongs to this chosen potential. These calculations do not replace each other; they show that after inserting true length one must specify potential construction, gradient geometry and component interpretation. Using true-space operators also requires consistent transformations of field components, density and area elements.

Chapter 3 also checks direction reversal. Generally $K(\phi)\ne K(\pi-\phi)$, whereas a fixed three-axis stretching has an even direction factor. A symmetric ellipsoid cannot implement the complete directed geometry. At the same process parameter, imposing $R_T=ct$ gives reference wavefront radius $R=ct/K(\phi)$; $ctK(\phi)$ cannot describe that same wavefront. This corrected relation and the constant-coefficient model's limits retain manuscript equations (3.69)–(3.73).

04How a flux check constrains the candidate

For the radial candidate, integrate over the chosen ordinary reference sphere. Axial symmetry permits the azimuthal integral, followed by $z=\cos\phi$. For $\beta\ne0$:

At $\beta=0$ the integrand is 1 and gives 2 as well. Including the azimuthal integral and field normalization yields:

This verifies total charge normalization for that candidate, not unique local angular components, evolution or detector response. One integral constraint admits many distributions. In another spatial geometry, Gauss relations require a consistent set of source density, field components, area elements and differential operators; changing only the radius is insufficient.

Curl and flux test different requirements

Final (3.17) is radial with angle-dependent intensity: at nonzero $\beta$ its curl is generally nonzero, despite the correct integrated flux. The gradient candidate (3.18) is curl-free in the regular exterior region but its ordinary-sphere flux does not generally equal $q/\varepsilon_0$. A full moving field must therefore combine the source equation, induced $-\partial_t\mathbf A_\tau$ and accompanying magnetic response. These completed checks delimit the geometric inputs rather than select either candidate as the full high-speed solution.

05Retardation and detectors complete further parts of the chain

Chapter 3 supplies a causal kernel in the effective source system. There $\mathbf r'$ labels the distributed source, $R=|\mathbf r-\mathbf r'|$, and the source is evaluated at $t-R/c$. This differs from drawing an angular profile about simultaneous source position. Calculating fields from complete retarded potentials must retain $-\partial_t\mathbf A_\tau$. The effective source model still has macroscopic, slow-source and low-frequency limits; it is not prescribed as the full source equation of a high-speed closed topological charge.

Observation also passes through a detector. A mechanical probe involves its velocity, momentum and support reaction; electro-optic sampling involves crystal orientation, frequency response, optical path and calibration. An optical signal is not a direct reading of point-charge force. Finite source distributions, source–probe configurations, boundaries and temporal response must be connected within the same process before comparing measured waveforms.

This page completes checks of candidate gradients, directional consistency and flux. The general high-speed source and apparatus mapping remain in the moving-field task and high-γ open case; an angular picture does not supply a definite high-speed prediction.