--- { "id": "ASD-EM-MOVING-CHARGE-FIELD-EN", "path": "/en/theory/electromagnetism/moving-charge-field", "title": "Moving-Charge Fields: Geometry, Flux and Detection", "title_en": "Moving-Charge Fields: Geometry, Flux and Detection", "category": "Electromagnetism", "summary": "Examine what directed length, potential gradients and Gauss flux determine, and identify the missing conditions for a general high-speed field.", "template": "theory", "version": "1.0.0", "updated": "2026-10-24", "canonical": false, "source_tier": "research-notes", "theory_status": "OPEN", "evidence_status": "UNTESTED", "locale": "en", "module_key": "electromagnetism-research", "module_navigation": [ { "path": "/en/theory/electromagnetism", "label": "Overview", "en": "Overview" }, { "path": "/en/theory/electromagnetism/coulomb-interaction", "label": "Coulomb", "en": "Coulomb" }, { "path": "/en/theory/electromagnetism/magnetic-force", "label": "Magnetic force", "en": "Magnetic force" }, { "path": "/en/theory/electromagnetism/moving-charge-field", "label": "Moving charge", "en": "Moving charge" }, { "path": "/en/theory/electromagnetism/vector-potential", "label": "Vector potential", "en": "Vector potential" }, { "path": "/en/theory/electromagnetism/magnetic-field", "label": "Magnetic field", "en": "Magnetic field" }, { "path": "/en/theory/electromagnetism/electromagnetic-waves", "label": "EM waves", "en": "EM waves" }, { "path": "/en/tests/electromagnetism", "label": "Tests", "en": "Tests" } ], "canonical_scope": "Research task; no new canonical result", "source_chapter": "Chapter 3", "source_reference": "§II.4 (3.17)—(3.19), §VII.1 and 6, §IX, item 3", "source_version": "ASD-BOOK-ddbade997b45", "source_sha256": "dace8ff491b4e4ae8a59b2b98e2a6c9c10e689079021ce0bc117985baba68e86", "source_kind": "manifest", "source_records": [ { "id": "ASD-BOOK-ddbade997b45", "title": "各向异性空间动力学（最终合订稿）", "filename": "各向异性空间动力学.docx", "format": "docx", "size_bytes": 503564, "sha256": "ddbade997b45913248e588953b1d27ccb9c7c206c3c8d992d3aef2364a64836e" } ], "source_manifest": "final-manuscript-source-manifest.json", "source_manifest_url": "/downloads/ASD_Final_Manuscript_Source_Manifest.json", "last_reviewed": "2026-09-15", "depends_on": [ "/en/theory/electromagnetism/coulomb-interaction", "/en/theory/lve", "/en/theory/electromagnetism/magnetic-force" ], "related": [ "/en/theory/electromagnetism", "/en/theory/electromagnetism/coulomb-interaction", "/en/theory/electromagnetism/magnetic-force", "/en/theory/electromagnetism/vector-potential", "/en/theory/electromagnetism/magnetic-field", "/en/theory/electromagnetism/electromagnetic-waves", "/en/tests/electromagnetism" ], "open_problems": [ "/en/open-problems/electromagnetism/moving-charge-field", "/en/open-problems/electromagnetism/field-representation", "/en/open-problems/electromagnetism/maxwell-effective-limit", "/en/open-problems/electromagnetism/high-gamma-discriminator", "/en/open-problems/electromagnetism/longitudinal-transverse-coupling" ], "equations": [ "em-field-candidate", "em-potential-candidate", "em-flux" ], "sources": [ { "label": "Final combined manuscript — adopted scope documented on this page", "kind": "final-combined-manuscript", "url": "/downloads/ASD_Final_Manuscript_Source_Manifest.json" }, { "label": "Previous source: CH3-d9ead146f6ac · 第三章(1).pdf", "kind": "historical-source" }, { "label": "Richard Fitzpatrick · Field Due to a Moving Charge", "url": "https://farside.ph.utexas.edu/teaching/jk1/Electromagnetism/node151.html", "kind": "external-comparison" } ], "outline": [ "Co-moving charges and a stationary probe ask different questions", "Construct a radial candidate from true inverse-square intensity", "What follows from differentiating a candidate potential?", "How a flux check constrains the candidate", "Retardation and detectors complete further parts of the chain", "Candidate dependencies and source" ], "seo_title": "Moving-Charge Fields: Geometry, Flux and Detection | Anisotropic Space Dynamics", "canonical_url": "https://asd-physics.org/en/theory/electromagnetism/moving-charge-field", "translation_path": "/theory/electromagnetism/moving-charge-field", "mathematical_review": "OPEN REVIEW", "review_path": "/en/theory/electromagnetism/moving-charge-field#review-notes", "review_note": "Derivations use the constitutive inputs and domain stated on this page; a formal source designation is distinct from experimental validation.", "evidence_note": "Evidence is assigned per observable in the EM test matrix; no new experimental fit is claimed.", "translation_of": "/theory/electromagnetism/moving-charge-field", "translation_version": "1.0.0", "section_anchors": [ "em-probe-configurations", "em-radial-candidate", "em-potential-gradient", "em-flux-check", "em-detector-response", "em-moving-sources" ], "legacy_anchors": { "s0": "em-probe-configurations", "s1": "em-radial-candidate", "s2": "em-potential-gradient", "s3": "em-flux-check", "s4": "em-detector-response", "s5": "em-moving-sources" }, "research_details_from": 5, "final_source": { "source_id": "ASD-BOOK-ddbade997b45", "chapter": "Chapter 3", "reference": "§II.4 (3.17)—(3.19), §VII.1 and 6, §IX, item 3", "locators": [ "p1021", "p1023", "p1024", "p1026", "p1027", "p1173", "p1224", "p1230", "p1283" ], "adopted_scope": "Adopt the corresponding interaction, continuous-field, propagation and source-coupling scope of final Chapter 3.", "conditions": [ "Special two-body result: uniform exterior background, equal parallel velocities, transverse separation and virtual displacement at fixed velocity.", "Free Maxwell representation: long waves, weak perturbations, transverse constraint, constant coefficients and normalization.", "Minimal-coupling sources: macroscopic coarse graining, low-speed point sources or a low-frequency near zone; high-speed closed sources require separate construction." ], "unresolved": [ "Local results do not automatically close high-speed topological sources, full instrument mapping or global directed propagation." ], "migration_status": "reviewed", "manuscript_sha256": "ddbade997b45913248e588953b1d27ccb9c7c206c3c8d992d3aef2364a64836e", "source_manifest": "final-manuscript-source-manifest.json", "body_review": "updated" }, "previous_source": { "source_chapter": "Chapter 3 · Electromagnetism in ASD", "source_reference": "Chapter 3 / 第三章 · §§II.4, VII.1, VII.6, VIII.3; (3.17)–(3.19), (3.51)–(3.53), (3.69)–(3.73); PDF pp. 5–6, 13, 16–19", "source_version": "EM-f06476f55cae", "source_sha256": "f06476f55cae728d712f553b338c513eb182e144c2c5cc4cd5ae5ed87ed1d1c6", "source_kind": "manifest", "source_records": [ { "id": "CH3-d9ead146f6ac", "title": "第三章(1).pdf", "sha256": "d9ead146f6ac3335b8663b23cea963319306f549ea4fcd5e3ee34fb7f7ab9413", "pages": 20 } ], "source_manifest": "electromagnetism-source-manifest.json", "source_manifest_url": "/downloads/ASD_Electromagnetism_Source_Manifest.json", "sources": [ { "label": "CH3-d9ead146f6ac · 第三章(1).pdf", "kind": "supplied-manuscript" }, { "label": "Richard Fitzpatrick · Field Due to a Moving Charge", "url": "https://farside.ph.utexas.edu/teaching/jk1/Electromagnetism/node151.html", "kind": "external-comparison" } ] } } --- ## Co-moving charges and a stationary probe ask different questions The [magnetic-force page](/en/theory/electromagnetism/magnetic-force) 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. ## Construct 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. ## What 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). ## How 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 $\beta\ne0$: $$ \int_{-1}^{1}\frac{1-\beta^2}{(1-\beta z)^2}\,dz =\left[\frac{1-\beta^2}{\beta(1-\beta z)}\right]_{-1}^{1}=2. $$ 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. ## Retardation and detectors complete further parts of the chain Chapter 3 supplies a causal kernel in the [effective source system](/en/theory/electromagnetism/electromagnetic-waves#em-effective-sources). 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](/en/open-problems/electromagnetism/moving-charge-field) and [high-γ open case](/en/tests/case-studies/high-gamma-charge-field); an angular picture does not supply a definite high-speed prediction. ## Candidate dependencies and source
Based on Chapter 3, Section II.4, Sections VII.1 and VII.6, and Section VIII.3, equations (3.17)–(3.19), (3.51)–(3.53), (3.69)–(3.73). Intensity and potential candidates retain their own constitutive conditions; mathematical review remains OPEN REVIEW.