--- { "id": "ASD-EM-MAGNETIC-FORCE-EN", "path": "/en/theory/electromagnetism/magnetic-force", "title": "Why Does Magnetic Interaction Arise?", "title_en": "Why Does Magnetic Interaction Arise?", "category": "Electromagnetism", "summary": "Derive the magnetic correction from true transverse distance, with explicit all-speed assumptions, low-speed limits and definitions of measurable force.", "template": "theory", "version": "1.0.0", "updated": "2026-10-24", "canonical": true, "source_tier": "canonical", "theory_status": "DERIVED", "evidence_status": "UNTESTED", "locale": "en", "module_key": "electromagnetism", "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": "Chapter 3 · status assigned per claim", "source_chapter": "Chapter 3", "source_reference": "§I.1—4 (3.1)—(3.10)", "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/dynamics" ], "related": [ "/en/theory/electromagnetism", "/en/theory/electromagnetism/coulomb-interaction", "/en/theory/electromagnetism/moving-charge-field", "/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": [ "magnetic-distance", "magnetic-force", "em-magnetic-vector", "magnetic-term", "em-si-normalization", "em-virtual-work", "em-standard-transverse", "em-low-speed" ], "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": "BIPM · The International System of Units, 9th edition (2026 update)", "url": "https://www.bipm.org/en/publications/si-brochure", "kind": "external-reference" }, { "label": "Richard Fitzpatrick · Field Due to a Moving Charge", "url": "https://farside.ph.utexas.edu/teaching/jk1/Electromagnetism/node151.html", "kind": "external-comparison" } ], "outline": [ "Keep motion and the static baseline in one comparison", "How transverse distance enters the inverse-square interaction", "Why this result needs no low-speed truncation", "True force and coordinate generalized force in the same work", "A same-configuration comparison retains quantity definitions", "Dependencies and source" ], "seo_title": "Why Does Magnetic Interaction Arise? | Anisotropic Space Dynamics", "canonical_url": "https://asd-physics.org/en/theory/electromagnetism/magnetic-force", "translation_path": "/theory/electromagnetism/magnetic-force", "mathematical_review": "CONDITIONAL", "review_path": "/en/theory/electromagnetism/magnetic-force#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/magnetic-force", "translation_version": "1.0.0", "section_anchors": [ "em-configuration", "em-distance-correction", "em-speed-domain", "em-force-mapping", "em-comparison", "em-force-sources" ], "legacy_anchors": { "s0": "em-configuration", "s1": "em-distance-correction", "s2": "em-speed-domain", "s3": "em-force-mapping", "s4": "em-comparison", "s5": "em-force-sources" }, "research_details_from": 5, "final_source": { "source_id": "ASD-BOOK-ddbade997b45", "chapter": "Chapter 3", "reference": "§I.1—4 (3.1)—(3.10)", "locators": [ "p0961", "p0964", "p0968", "p0970", "p0973", "p0975", "p0984", "p0990", "p0993", "p1281", "p1282" ], "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 · §I.1–4; (3.1)–(3.10), PDF pp. 2–4", "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": "BIPM · The International System of Units, 9th edition (2026 update)", "url": "https://www.bipm.org/en/publications/si-brochure", "kind": "external-reference" }, { "label": "Richard Fitzpatrick · Field Due to a Moving Charge", "url": "https://farside.ph.utexas.edu/teaching/jk1/Electromagnetism/node151.html", "kind": "external-comparison" } ] } } --- ## Keep motion and the static baseline in one comparison The magnetic question begins with a specified change: hold charges and reference transverse separation fixed while giving both charges a common motion. Charges $q_1,q_2$ move with the same constant parallel velocity $\mathbf v$, their separation perpendicular to it, relative to a local homogeneous spatial background. Let $r>0$, let $\hat{\mathbf r}$ point from $q_1$ to $q_2$, and write $\beta=v/c$, $\gamma=(1-\beta^2)^{-1/2}$. The [Coulomb page](/en/theory/electromagnetism/coulomb-interaction) fixes the exterior inverse-square law, source strengths and direction. Finite size, external gradients and boundary corrections remain neglected. $\mathbf F_0$ is the static baseline at the same charges and reference separation; $\mathbf F_T$ is the interaction paired with true physical displacement. In the magnitude formulas below, $F_0,F_T$ are nonnegative. ## How transverse distance enters the inverse-square interaction LVE gives $K(\phi)=\gamma(1-\beta\cos\phi)$. At transverse angle $\phi=\pi/2$, $K=\gamma$, hence: Now use Chapter 3's constitutive input: $r_T$ is the actual Coulomb interaction distance. Inserting it into $F_T=|q_1q_2|/(4\pi\varepsilon_0r_T^2)$ changes the denominator to $4\pi\varepsilon_0\gamma^2r^2$. Factoring out the static baseline and using $\gamma^{-2}=1-\beta^2$ yields: The moving interaction still lies along the charge separation, but its strength changes. Subtract the same baseline using signed vectors: This is how ASD organizes magnetic interaction in the specified configuration. $\Delta\mathbf F_m$ is the correction relative to the static baseline, while $\mathbf F_T$ is the total force. Adding the correction again to an interaction already calculated at $r_T$ would double-count the scale effect. For like charges $\mathbf F_0$ points outward and the correction inward, reducing repulsion. For unlike charges the baseline points inward and the correction outward, reducing the attraction magnitude. Define the positive reduction separately: $F_m^{\mathrm{red}}$ is a magnitude difference, not the directed vector $\Delta\mathbf F_m$. Using SI electromagnetic normalization $\varepsilon_0\mu_0=c^{-2}$ gives the equivalent correction: $\mu_0$ is the vacuum permeability. This rewrites normalization without introducing another basic magnetic force law. The product relation does not determine the microscopic origin of either constant. ## Why this result needs no low-speed truncation
The length relation uses the complete $\gamma$, followed by an exact identity. The correction is exactly proportional to $\beta^2$; calling it second order does not mean that higher terms in a longer series were discarded. It applies for the stated constitutive law and common constant-velocity, parallel, transverse configuration throughout $0\leq v The figure compares normalized like-charge interactions; lengths and arrows are schematic. Its velocity control tests the algebra, not experimental data. The later slow-source near-zone potential approximation does not retroactively make this configuration-specific result a low-speed truncation. ## True force and coordinate generalized force in the same work
An experimental comparison must specify which displacement is paired with force. At fixed common velocity, $r_T=\gamma r$ implies $dr_T=\gamma\,dr$. With interaction energy $U_T=q_1q_2/(4\pi\varepsilon_0r_T)$, virtual work in the two variables is: In this virtual-work relation $F_T,F_0,F_r$ may be signed radial components along $\hat{\mathbf r}$; for like charges they also equal the above magnitudes. $F_T=-\partial U_T/\partial r_T$ pairs with true displacement; $F_r=-\partial U_T(\gamma r)/\partial r$ pairs with reference displacement. The chain rule gives $F_r=\gamma F_T$ rather than another physical interaction. Chapter 2 relates force to mechanical momentum change and work–energy. Physical displacement, mechanical momentum and process parameter must be paired within the same representation; reference generalized force pairs with its coordinate's conjugate momentum. Trajectory curvature, bunch momentum exchange, support reaction or electro-optic signals each require an apparatus mapping. An instrument reading cannot simply be declared to be $F_T$ or $F_r$. Acceleration also requires the velocity dependence of scale and accompanying-field energy exchange. ## A same-configuration comparison retains quantity definitions Introduce standard Maxwell–Lorentz electrodynamics separately as the comparison theory. For like charges with the same common velocity and transverse separation, its laboratory electric force, positive magnetic reduction and total transverse force are: These relations were not premises of the ASD derivation. ASD's true force is $F_0/\gamma^2$ and its coordinate generalized force is $F_0/\gamma$. The latter matches the standard laboratory total numerically, but pairs with different variables. A low-speed expansion shows: Matching the leading magnetic reduction is not matching the complete force at that order: the standard electric-field correction also contributes. ASD's subsequent quasistatic potential and minimal coupling reproduce the magnetic leading term without completing a general source coupling consistent with the true total force. Detectable differences require both models to calculate the same apparatus, reading and measurement process. The [experimental matrix](/en/tests/electromagnetism) and [high-speed discriminator task](/en/open-problems/electromagnetism/high-gamma-discriminator) retain this requirement. ## Dependencies and source
Based on Chapter 3, Section I, equations (3.1)–(3.10). The algebra closes under the stated geometry and constitutive inputs; a general two-source problem and instrument mapping are not supplied automatically.