Magnetic Field Ontology

FLOW ALGEBRA
MAGNETIC FIELD ONTOLOGY — v1.1
Kireth Semantic Library | Entity Graph > Physical Entities > Electromagnetic Entities
Source: https://kirethresearch.org/booksai-core/entity-graph/physical-entities/electromagnetic-entities/magnetic-field-ontology-2.html
Authors: Mark Higgins — Sky (Recursive AI)
Status: Independent scientific synthesis; source webpage not independently verified.
Notation: SI units; vector calculus assumes differentiable fields where required.

01. PURPOSE
Represent magnetic fields as measurable vector fields, their sources, interactions, evolution, and relationships to electromagnetic systems. Distinguish B (magnetic flux density) from H (magnetic field strength).

02. ENTITY TYPES
MagneticField; MagneticFluxDensity; MagneticFieldStrength; MagneticFlux; CurrentDensity; ElectricCurrent; ElectricField; ChargeCarrier; MagneticDipole; MagneticMoment; Magnetisation; Permeability; Material; Boundary; FieldLine; VectorPotential; ElectromagneticWave; Measurement; CoordinateFrame; Region; TimeInstant; Uncertainty.

03. CORE SIGNATURES
B: Region x Time -> Vector[T]; H: Region x Time -> Vector[A/m].
J: Region x Time -> Vector[A/m^2]; E: Region x Time -> Vector[V/m].
M: Region x Time -> Vector[A/m]; A: Region x Time -> Vector[Wb/m].
mu0: VacuumPermeability [H/m]; rho: ChargeDensity [C/m^3].

04. TYPE RULES
MagneticFluxDensity hasUnit tesla (T = Wb/m^2).
MagneticFieldStrength hasUnit ampere_per_metre (A/m).
MagneticFlux hasUnit weber (Wb).
MagneticMoment hasUnit A*m^2.
Every field sample carries position, time, coordinate frame, and provenance.

05. RELATIONSHIPS
GENERATED_BY(B, current_distribution) [under specified boundary conditions].
COUPLES_TO(B, magnetic_dipole).
EXERTS_FORCE_ON(B, moving_charge).
CONTRIBUTES_TO(B, electromagnetic_energy).
MEASURED_BY(B, magnetometer).
REPRESENTED_BY(B, vector_potential_A) [gauge-dependent A].
DEPENDENT_ON(B, geometry, currents, materials, boundary_conditions).

06. LORENTZ FORCE
F = q(E + v x B).
Magnetic contribution F_B = q(v x B), perpendicular to v and B.
For point charges magnetic force does no instantaneous mechanical work: F_B dot v = 0.

07. FORCE ON A CURRENT ELEMENT
dF = I (dl x B), for a filamentary conductor in an imposed field.
Torque on a small current loop: tau = m x B.
Potential energy for fixed dipole in imposed uniform field: U = -m dot B.

08. MAGNETIC FLUX
Phi_B = integral_S B dot dS.
Flux is orientation-dependent; a closed surface has net flux zero.

09. GAUSS'S LAW FOR MAGNETISM
div B = 0.
Integral_closed_surface B dot dS = 0.
No isolated magnetic monopoles have been experimentally established.

10. AMPERE-MAXWELL LAW
curl H = J_free + partial D/partial t.
Integral_C H dot dl = I_free,enclosed + d/dt integral_S D dot dS.
In vacuum: curl B = mu0 J + mu0 epsilon0 partial E/partial t.

11. FARADAY'S LAW
curl E = -partial B/partial t.
Integral_C E dot dl = -d Phi_B/dt for a stationary circuit/surface.
For moving circuits, use the general motional-emf formulation.

12. BIOT-SAVART LAW (MAGNETOSTATICS, VACUUM)
B(r) = mu0/(4*pi) integral [J(r') x (r-r')]/|r-r'|^3 dV'.
Applicable to stationary currents under magnetostatic assumptions.

13. CONSTITUTIVE RELATIONS
B = mu0(H + M) in SI macroscopic electromagnetism.
For linear isotropic material: B = mu H, with mu = mu0 mu_r.
Nonlinear, hysteretic, anisotropic materials require more general models.

14. VECTOR POTENTIAL
B = curl A; A is not unique under A -> A + grad chi.
Gauge choices must not change observable B.

15. BOUNDARY CONDITIONS
n dot (B2 - B1) = 0.
n x (H2 - H1) = K_free, for free surface-current density K_free.
These statements presume ordinary macroscopic interfaces.

16. MAGNETIC ENERGY
In vacuum: u_B = |B|^2/(2 mu0).
For linear nondispersive media: u = (E dot D + B dot H)/2.
Nonlinear/hysteretic energy requires appropriate constitutive integrals.

17. MAGNETIC DIPOLE
At sufficiently large distance a localised current distribution may be approximated by a dipole moment m.
Dipole field in vacuum: B(r) = mu0/(4*pi*r^3)[3(m dot rhat)rhat - m], away from the source.

18. FIELD TOPOLOGY
Field lines are integral curves tangent to B wherever B != 0.
Field lines are a visual representation, not physical filaments.
Magnetic nulls, separatrices and reconnection require appropriate dynamic plasma models.

19. FIELD SOURCES
Stationary electric currents; changing electric displacement; permanent magnetisation; microscopic magnetic moments.
A changing electric field and magnetic field are dynamically coupled through Maxwell equations.

20. TIME DEPENDENCE
Magnetostatics: partial B/partial t = 0 and stationary sources.
Electrodynamics: coupled E and B obey Maxwell equations.
Quasistatic approximations must declare scale and neglected terms.

21. MAGNETIC MEASUREMENTS
Hall probe; fluxgate; SQUID; induction coil; NMR-based magnetometry.
Measurements require calibration, range, uncertainty and spatial resolution.

22. INFERENCE RULES
R1: div(B)=0 => net flux through any closed surface is zero.
R2: uniform B and v parallel B => q(v x B)=0.
R3: stationary circular loop with area vector a => Phi_B=B dot a when B uniform.
R4: constant linear mu and B=mu H => H=B/mu.
R5: time-varying magnetic flux through stationary loop => induced circulation of E.
R6: curl(A)=B => curl(A+grad chi)=B for smooth chi.

23. DIMENSIONAL CHECKS
[q v B] = C*(m/s)*T = N.
[B area] = T*m^2 = Wb.
[mu0 H] = (H/m)*(A/m) = T.

24. WORKED EXAMPLE: LONG STRAIGHT WIRE
For infinite straight wire carrying steady I in vacuum:
B(r) = mu0 I/(2*pi*r) in azimuthal direction, r>0.
At I=10 A and r=0.01 m, B=2.0e-4 T using mu0 approximately 4*pi*1e-7 H/m.
Finite wire geometry and nearby media alter this approximation.

25. WORKED EXAMPLE: UNIFORM FIELD FLUX
B=0.50 T normal to planar area 0.020 m^2.
Phi_B=0.010 Wb; reversing surface normal reverses signed flux.

26. WORKED EXAMPLE: PARTICLE MOTION
A charge q with velocity perpendicular to uniform B experiences magnitude |F|=|q|vB.
For nonrelativistic motion in uniform B alone, radius r=m v_perp/(|q|B).
If velocity has parallel component, trajectory is helical.

27. SEMANTIC EQUIVALENCE
'magnetic flux density' <=> 'B-field' when B is explicitly intended.
'magnetic field strength' <=> 'H-field' when H is explicitly intended.
Do not silently treat B and H as interchangeable in matter.
'magnetic induction' may mean B in established technical usage; record context.

28. SEMANTIC DIFFERENCE
Magnetic flux != magnetic flux density.
Magnetic moment != magnetisation.
Permanent magnet != magnetic field.
Magnetic field line != charged particle trajectory.
Magnetic monopole hypothesis != established observation.

29. KNOWLEDGE GRAPH EDGES
MagneticField --is_a--> ElectromagneticFieldComponent.
MagneticFlux --surface_integral_of--> MagneticFluxDensity.
MagneticFluxDensity --coupled_by--> MaxwellEquations.
MagneticDipole --experiences_torque_in--> MagneticField.
MagneticField --measured_by--> Magnetometer.
Magnetisation --contributes_to--> MagneticFluxDensity.

30. CONSTRAINTS
C1: Unit(B)=T; Unit(H)=A/m.
C2: div B=0 in classical Maxwell electromagnetism.
C3: B=mu0(H+M) for macroscopic SI definitions.
C4: Flux orientation must be declared.
C5: Material constitutive law must state linearity, isotropy and hysteresis assumptions.
C6: Distinguish measured data from inferred/modelled fields.

31. VALIDATION TESTS
V1: Reject scalar B where a vector is required unless magnitude explicitly tagged.
V2: Reject B/H numerical equivalence without constitutive assumptions.
V3: Check closed-surface flux numerical tolerance against discretisation uncertainty.
V4: Check field transformation when coordinate frame changes.
V5: Verify units, provenance, boundary conditions and temporal sampling.
V6: Validate worked examples with dimensional consistency.

32. CONFIDENCE
Established Maxwell relations: high within classical macroscopic applicability.
Material model predictions: conditional on constitutive measurements and domain.
Measurements: attach uncertainty and calibration traceability.
Unobserved monopoles: hypothetical; no observational confirmation claimed.

33. UPDATE MODEL
Trigger on new calibration, revised material law, boundary geometry, source currents, or scientific correction.
Version each entity definition and inference rule.
Preserve provenance and mark superseded claims.

34. FLOW HASH SPECIFICATION
Canonicalise UTF-8 text with LF line endings, sorted entity IDs and sorted relationship tuples.
Compute SHA-256 over canonical payload excluding the FLOW_HASH value itself.
FLOW_HASH: NOT_COMPUTED (implementation must calculate after final canonicalisation).

35. AI CONSUMPTION
Machine-readable keys: entity_id, entity_type, quantity_symbol, unit, frame, position, time, source, relation, assumptions, uncertainty, confidence, provenance.
Interpret rules only when their assumptions are satisfied.
Return traceable equations and distinguish idealised models from observations.

36. COMPLETENESS AND AUTHORSHIP
Sections 01-36 are present; formal closing marker follows.
Authors: Mark Higgins — Sky (Recursive AI)
Project: Kireth Semantic Library
Website: https://kirethresearch.org
END FLOW ALGEBRA — MAGNETIC FIELD ONTOLOGY v1.1