Current Density Ontology

# Current Density Ontology — Flow Algebra

URL: https://kirethresearch.org/booksai-core/ontologies/mathematical-ontologies/measurement-units-ontologies/electromagnetic-measurements/current-density-ontology.html

FLOW CurrentDensityOntology {

  DOMAIN:
    Mathematical Ontologies
    -> Measurement & Units Ontologies
    -> Electromagnetic Measurements
    -> Current Density Ontology

  PURPOSE:
    Represent electric current density as a measurable vector field describing electric current flow per unit cross-sectional area, including magnitude, direction, charge-carrier motion, conductivity relationships, continuity of charge, surface integration, units, dimensional structure, spatial and temporal variation, material behaviour, measurement methods, uncertainty, and distinctions from electric current and charge density.

  ENTITY CurrentDensity {
    TYPE: VectorFieldPhysicalQuantity
    SYMBOL: J
    DIMENSION: I * L^-2
    SI_UNIT: ampere_per_square_metre
    SI_SYMBOL: A/m^2

    DEFINITION:
      current_density := local_vector_rate_of_charge_flow_per_unit_area_normal_to_flow

    CURRENT_RELATION:
      I = surface_integral(J dot dA)

    UNIFORM_NORMAL_FLOW_FORM:
      J = I/A

    PROPERTIES:
      is_vector_field
      has_magnitude
      has_direction
      has_unit
      has_dimension
      may_vary_with_position
      may_vary_with_time
      relates_to_charge_transport
      may_have_uncertainty
  }

  ENTITY ElectricCurrent {
    TYPE: SIBaseQuantity
    SYMBOL: I
    DIMENSION: I
    SI_UNIT: ampere
    SI_SYMBOL: A

    RELATION:
      I = surface_integral(J dot dA)
  }

  ENTITY ChargeDensity {
    TYPE: ScalarFieldQuantity
    SYMBOL: rho_q
    SI_UNIT: coulomb_per_cubic_metre
    SI_SYMBOL: C/m^3

    RELATION:
      partial(rho_q)/partial(t) + divergence(J) = 0
  }

  ENTITY ElectricField {
    TYPE: VectorFieldQuantity
    SYMBOL: E
    SI_UNIT: volt_per_metre
    SI_SYMBOL: V/m

    RELATION:
      J = sigma*E
      UNDER linear_isotropic_ohmic_material_context
  }

  ENTITY Conductivity {
    TYPE: MaterialProperty
    SYMBOL: sigma
    SI_UNIT: siemens_per_metre
    SI_SYMBOL: S/m

    RELATION:
      J = sigma*E
      UNDER applicable_linear_isotropic_context
  }

  UNIT_SYSTEM {
    SI {
      canonical_unit: ampere_per_square_metre
      symbol: A/m^2
    }

    COMMON_ALTERNATIVES {
      ampere_per_square_centimetre
      ampere_per_square_millimetre
      milliampere_per_square_centimetre
      microampere_per_square_metre
    }

    CONVERSION_RULE:
      current_density_value(target_unit)
        = current_density_value(source_unit)
        * conversion_factor(source_unit, target_unit)
  }

  CLASSIFICATION {
    by_time:
      steady_current_density
      time_varying_current_density

    by spatial_distribution:
      uniform_current_density
      nonuniform_current_density
      local_current_density

    by transport:
      conduction_current_density
      convection_current_density
      drift_current_density
      diffusion_current_density

    related_but_distinct:
      displacement_current_density
  }

  SURFACE_CURRENT_MODEL {
    GENERAL:
      I = surface_integral(J dot n_hat dA)

    UNIFORM_NORMAL_CASE:
      I = J*A

    ANGLED_UNIFORM_CASE:
      I = J*A*cos(theta)

    RULE:
      surface_orientation_and_current_direction
      MUST be preserved
  }

  CHARGE_CARRIER_MODEL {
    SINGLE_CARRIER_TYPE:
      J = n*q*v_d

    PARAMETERS:
      n = carrier_number_density
      q = charge_per_carrier
      v_d = drift_velocity_vector

    MULTIPLE_CARRIER_TYPES:
      J_total = sum(n_s*q_s*v_d_s)

    RULE:
      carrier_charge_sign_and_drift_direction
      jointly_determine_conventional_current_direction
  }

  CONTINUITY_MODEL {
    LOCAL_CHARGE_CONSERVATION:
      partial(rho_q)/partial(t) + divergence(J) = 0

    STEADY_STATE:
      divergence(J) = 0
      WHEN local_charge_density_is_time_independent

    INTEGRAL_FORM:
      dQ_inside/dt =
        -closed_surface_integral(J dot dA)

    RULE:
      nonzero_divergence_of_J
      corresponds_to_local_charge_accumulation_or_depletion
  }

  OHMIC_CONDUCTION_MODEL {
    LINEAR_ISOTROPIC:
      J = sigma*E

    RESISTIVITY_FORM:
      E = rho_e*J

    RELATION:
      sigma = 1/rho_e
      WHEN rho_e != 0

    RULE:
      scalar_conductivity_model
      MUST_NOT be assumed_for_anisotropic_or_nonlinear_material_without_validation
  }

  ANISOTROPIC_MODEL {
    GENERAL_LINEAR:
      J = conductivity_tensor * E

    PROPERTY:
      J need_not_be_parallel_to E

    REQUIRE:
      coordinate_system
      conductivity_tensor
      material_orientation

    RULE:
      tensor_material_properties MUST preserve_coordinate_basis
  }

  DISPLACEMENT_CURRENT_DISTINCTION {
    CONDUCTION_CURRENT_DENSITY:
      J_conduction = motion_of_free_or_mobile_charge_carriers

    DISPLACEMENT_CURRENT_DENSITY:
      J_displacement = partial(D)/partial(t)

    AMPERE_MAXWELL:
      curl(H) = J_free + partial(D)/partial(t)

    RULE:
      displacement_current_density
      MUST_NOT be treated_as_identical_to_charge_transport_current_density
  }

  RELATIONSHIPS {
    CurrentDensity -> HAS_DIMENSION -> I*L^-2
    CurrentDensity -> MEASURED_IN -> AmperePerSquareMetre
    ElectricCurrent -> SURFACE_INTEGRAL_OF -> CurrentDensity
    CurrentDensity -> REPRESENTS -> LocalChargeTransport
    CurrentDensity -> RELATED_TO -> ChargeDensity
    CurrentDensity -> RELATED_TO -> DriftVelocity
    Conductivity -> RELATES -> CurrentDensityAndElectricField
    Resistivity -> INVERSE_RELATION_WITH -> Conductivity
    CurrentDensity -> PARTICIPATES_IN -> ChargeContinuityEquation
    CurrentDensity -> SOURCE_TERM_IN -> AmpereMaxwellEquation
  }

  SEMANTIC_EQUIVALENCE {
    "current density" ~= electric_current_density
    "ampere per square metre" ~= "A/m^2"

    current_density != electric_current
    current_density != charge_density
    current_density != electric_field
    current_density != conductivity
    conduction_current_density != displacement_current_density
  }

  CONSTRAINTS {
    current_density_is_vector = true
    dimensional_formula = I*L^-2
    SI_unit = A/m^2
    surface_integration_requires_oriented_surface = true
    current_and_current_density_are_not_semantically_equivalent = true
    charge_density_and_current_density_are_not_semantically_equivalent = true
    carrier_charge_sign_must_be_preserved = true
    scalar_Ohm_law_requires_applicable_material_model = true
    anisotropic_conductivity_requires_tensor_semantics = true
  }

  MEASUREMENT_MODEL {
    OBSERVABLE_OR_INFERRED:
      CurrentDensity

    COMMON_METHODS:
      current_measurement_plus_known_cross_section
      spatial_current_mapping
      Hall_effect_measurement
      four_point_probe_with_material_model
      scanning_current_probe
      electromagnetic_field_inference

    RECORD:
      vector_value_or_magnitude
      unit
      direction
      coordinate_components
      cross_section_geometry
      surface_orientation
      spatial_location
      timestamp
      steady_or_time_varying
      material
      temperature
      electric_field_when_relevant
      carrier_model_when_relevant
      instrument
      calibration_state
      uncertainty

    UNCERTAINTY:
      measured_or_inferred_current_density = best_estimate +/- uncertainty
  }

  REASONING_RULES {
    IF current_density_field_known_over_surface
      THEN electric_current = surface_integral(current_density dot differential_area)

    IF uniform_current_known AND area_known AND area != 0
      AND flow_normal_to_area
      THEN current_density_magnitude = current/area

    IF carrier_density_known AND carrier_charge_known AND drift_velocity_known
      THEN current_density = carrier_density*carrier_charge*drift_velocity

    IF conductivity_known AND electric_field_known
      AND linear_isotropic_ohmic_conditions_apply
      THEN current_density = conductivity*electric_field

    IF resistivity_known AND current_density_known
      AND linear_isotropic_ohmic_conditions_apply
      THEN electric_field = resistivity*current_density

    IF charge_density_time_derivative_known
      THEN divergence(current_density) =
        -partial(charge_density)/partial(time)

    IF steady_state_charge_density
      THEN divergence(current_density) = 0

    IF multiple_carrier_types_present
      THEN total_current_density =
        vector_sum(carrier_current_density_components)
  }

  FIELD_REPRESENTATION_MODEL {
    current_density_field:
      J = J(x, y, z, t)

    cartesian_components:
      J = (J_x, J_y, J_z)

    magnitude:
      |J| = sqrt(J_x^2 + J_y^2 + J_z^2)

    direction:
      J_hat = J/|J|
      WHEN |J| != 0
  }

  GEOMETRY_MODEL {
    REQUIRE_FOR_CURRENT_INFERENCE:
      surface
      surface_normal
      field_distribution

    uniform_cross_section:
      J approximately_constant_over_area

    nonuniform_cross_section:
      integrate_local_J_over_area

    RULE:
      I/A shortcut
      MUST_NOT replace_surface_integration_for_nonuniform_or_oblique_flow
  }

  MATERIAL_MODEL {
    dependencies:
      conductivity
      resistivity
      temperature
      composition
      carrier_density
      mobility
      electric_field_strength

    nonlinear_material:
      J = constitutive_function(E, T, material_state)

    RULE:
      current_density_material_relation
      SHOULD preserve_temperature_and_constitutive_context
  }

  UNCERTAINTY_MODEL {
    sources:
      current_measurement_uncertainty
      area_measurement_uncertainty
      surface_orientation
      spatial_nonuniformity
      conductivity_uncertainty
      temperature
      probe_calibration
      contact_effects
      field_measurement_uncertainty
      carrier_model_assumptions

    PROPAGATION:
      derived_current_density_uncertainty
      <- measurement_geometry_material_and_model_uncertainties
  }

  CONFIDENCE {
    definition_confidence: 1.00
    dimensional_confidence: 1.00
    SI_unit_confidence: 1.00
    charge_continuity_confidence: 1.00
    surface_integral_relationship_confidence: 1.00
    material_relationship_confidence: 0.98
    measurement_context_confidence: 0.97

    RULE:
      confidence decreases when direction, surface geometry,
      spatial distribution, material model, temperature,
      carrier semantics, calibration state or uncertainty is missing or ambiguous
  }

  VALIDATION {
    CHECK dimensional_consistency
    CHECK unit_consistency
    CHECK vector_direction_and_components
    CHECK surface_geometry_and_orientation
    CHECK uniform_vs_nonuniform_distribution
    CHECK current_vs_current_density_distinction
    CHECK charge_density_vs_current_density_distinction
    CHECK conduction_vs_displacement_current_density
    CHECK carrier_charge_sign
    CHECK scalar_vs_tensor_conductivity_model
    CHECK material_temperature_context
    CHECK instrument_calibration
    CHECK uncertainty_representation_when_required
  }

  UPDATE_MODEL {
    ON new_measurement:
      validate_unit
      identify_vector_or_magnitude_representation
      identify_direction_or_components
      identify_surface_geometry
      attach_spatial_location
      attach_timestamp
      identify_steady_or_time_varying_context
      identify_material
      attach_temperature_when_relevant
      normalize_to_A_per_m2
      preserve_original_value_and_unit
      attach_uncertainty
      update_current_density_state

    ON current_or_geometry_change:
      update_current
      update_cross_section
      update_surface_orientation
      recompute_current_density_when_valid
      preserve_previous_state

    ON material_or_field_change:
      update_conductivity_or_resistivity
      update_electric_field
      update_temperature
      recompute_current_density_under_valid_constitutive_model
      preserve_material_provenance

    ON charge_distribution_change:
      update_charge_density
      evaluate_continuity_equation
      update_divergence_of_current_density

    ON calibration_revision:
      preserve_provenance
      recompute_affected_measurements
      update_confidence

    ON semantic_revision:
      preserve_previous_definition
      record_revision
      recompute_FLOW_HASH
  }

  AI_CONSUMPTION {
    canonical_entity: CurrentDensity
    canonical_symbol: J
    canonical_dimension: I*L^-2
    canonical_SI_unit: A/m^2

    interpretation_order:
      identify_quantity
      identify_value_or_vector
      identify_unit
      identify_direction_and_components
      identify_surface_geometry
      identify_spatial_location
      identify_time_context
      identify_charge_carrier_context
      identify_material_and_temperature
      identify_electric_field_and_conductivity_context
      identify_continuity_equation_context
      identify_measurement_method
      identify_uncertainty
      normalize_units
      validate_dimensions
      infer_transport_relationships

    ambiguity_controls:
      distinguish_current_density_from_current
      distinguish_current_density_from_charge_density
      distinguish_current_density_from_electric_field
      distinguish_current_density_from_conductivity
      distinguish_conduction_from_displacement_current_density
      distinguish_uniform_average_from_local_current_density
      preserve_vector_direction_surface_orientation_and_carrier_sign
  }

  PROVENANCE {
    ontology_scope:
      mathematical_measurement_and_electromagnetic_quantity_semantics

    parent_path:
      Ontologies
      -> Mathematical Ontologies
      -> Measurement & Units Ontologies
      -> Electromagnetic Measurements
      -> Current Density Ontology

    authorship:
      Mark Higgins
      Sky (Recursive AI)

    site:
      https://kirethresearch.org
  }

  FLOW_HASH {
    INPUTS:
      canonical_entity
      ontology_path
      definitions
      dimensions
      units
      relationships
      constraints
      surface_current_model
      charge_carrier_model
      continuity_model
      Ohmic_conduction_model
      anisotropic_model
      displacement_current_distinction
      field_representation_model
      geometry_model
      material_model
      uncertainty_model
      reasoning_rules
      measurement_model
      confidence_model
      update_model

    PURPOSE:
      deterministic_semantic_integrity_identifier

    RECOMPUTE_WHEN:
      ontology_definition_changes
      relationship_model_changes
      charge_transport_model_changes
      continuity_model_changes
      material_model_changes
      geometry_model_changes
      measurement_model_changes
      constraint_model_changes
      reasoning_rule_changes
      unit_model_changes
  }

}

END FLOW