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160 lines
9 KiB
Text
160 lines
9 KiB
Text
/- Copyright (c) 2026 Sovereign Research Stack. All rights reserved.
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Released under Apache 2.0 license as described in the file LICENSE.
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Authors: Research Stack Team
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ElectrostaticsMetaprobe.lean — Electrostatic calculations and verification
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This module formalizes electrostatic mathematics extracted from amasci.com,
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including capacitance calculations, voltage calculations, and energy storage formulas.
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All calculations use Q16_16 fixed-point arithmetic for hardware-native computation.
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Reference: http://amasci.com/emotor/voltmeas.html
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-/
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import Semantics.FixedPoint
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import Mathlib.Data.Real.Basic
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namespace Semantics.ElectrostaticsMetaprobe
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open Semantics
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-- ═══════════════════════════════════════════════════════════════════════════
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-- §0 Electrostatic Constants
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-- ═══════════════════════════════════════════════════════════════════════════
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/-- Dielectric constant of vacuum/air (ε₀) in F/m.
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Value: 8.854187817 × 10⁻¹² F/m ≈ 8.9e-12 F/m -/
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def epsilon0 : Q16_16 := Q16_16.ofFloat 8.9e-12
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/-- Permittivity of free space constant for calculations. -/
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def permittivityFreeSpace : Q16_16 := epsilon0
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-- ═══════════════════════════════════════════════════════════════════════════
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-- §1 Electrostatic Structures
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-- ═══════════════════════════════════════════════════════════════════════════
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/-- Parallel plate capacitor with area, separation, and dielectric constant. -/
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structure ParallelPlateCapacitor where
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area : Q16_16 -- Plate area in m²
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separation : Q16_16 -- Distance between plates in m
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dielectricK : Q16_16 -- Dielectric constant (relative permittivity)
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deriving Repr
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/-- Electrostatic state with voltage, charge, and capacitance. -/
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structure ElectrostaticState where
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voltage : Q16_16 -- Voltage in volts
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charge : Q16_16 -- Charge in coulombs
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capacitance : Q16_16 -- Capacitance in farads
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deriving Repr
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/-- Force and distance for energy calculations. -/
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structure ForceDistance where
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force : Q16_16 -- Force in newtons
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distance : Q16_16 -- Distance in meters
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deriving Repr
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-- ═══════════════════════════════════════════════════════════════════════════
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-- §2 Capacitance Calculations
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-- ═══════════════════════════════════════════════════════════════════════════
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/-- Calculate capacitance of parallel plate capacitor: C = k × ε₀ × A / d -/
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def parallelPlateCapacitance (cap : ParallelPlateCapacitor) : Q16_16 :=
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let k := cap.dielectricK
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let eps0 := epsilon0
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let A := cap.area
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let d := cap.separation
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-- C = k * ε₀ * A / d
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let numerator := Q16_16.mul (Q16_16.mul k eps0) A
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if d.val = 0 then Q16_16.zero else Q16_16.div numerator d
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/-- Example: Balloon/arm capacitor (4cm × 15cm area, 1mm separation, air dielectric) -/
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def balloonArmCapacitor : ParallelPlateCapacitor :=
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{ area := Q16_16.ofFloat 0.006 -- 4cm × 15cm = 0.006 m²
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separation := Q16_16.ofFloat 0.001 -- 1mm = 0.001 m
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dielectricK := Q16_16.one -- Air: k ≈ 1
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}
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-- ═══════════════════════════════════════════════════════════════════════════
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-- §3 Energy Calculations
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-- ═══════════════════════════════════════════════════════════════════════════
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/-- Calculate mechanical energy: U = F × d -/
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def mechanicalEnergy (fd : ForceDistance) : Q16_16 :=
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Q16_16.mul fd.force fd.distance
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/-- Calculate stored energy in capacitor: U = 0.5 × C × V² -/
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def capacitorEnergy (state : ElectrostaticState) : Q16_16 :=
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let half := Q16_16.div Q16_16.one (Q16_16.ofFloat 2.0)
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let vSquared := Q16_16.mul state.voltage state.voltage
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Q16_16.mul (Q16_16.mul half state.capacitance) vSquared
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/-- Calculate voltage from energy and capacitance: V = √(2U/C) -/
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def voltageFromEnergy (energy capacitance : Q16_16) : Q16_16 :=
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if capacitance.val = 0 then Q16_16.zero
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else
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let twoU := Q16_16.mul (Q16_16.ofFloat 2.0) energy
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let ratio := Q16_16.div twoU capacitance
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Q16_16.sqrt ratio
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/-- Calculate charge from energy, capacitance, force, and distance: Q = √(2CFd) -/
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def chargeFromEnergy (capacitance force distance : Q16_16) : Q16_16 :=
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let twoCFd := Q16_16.mul (Q16_16.mul (Q16_16.ofFloat 2.0) capacitance) (Q16_16.mul force distance)
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Q16_16.sqrt twoCFd
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-- ═══════════════════════════════════════════════════════════════════════════
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-- §4 Voltage Calculations (Simplified Formula)
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-- ═══════════════════════════════════════════════════════════════════════════
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/-- Simplified voltage calculation: V = (1e-7 × D) / (0.006) / (8.9e-12)
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where D is distance in meters. -/
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def simplifiedVoltage (distance : Q16_16) : Q16_16 :=
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let numerator := Q16_16.mul (Q16_16.ofFloat 1e-7) distance
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let denominator1 := Q16_16.ofFloat 0.006
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let denominator2 := epsilon0
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let ratio1 := Q16_16.div numerator denominator1
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Q16_16.div ratio1 denominator2
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-- ═══════════════════════════════════════════════════════════════════════════
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-- §5 Theorems
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-- ═══════════════════════════════════════════════════════════════════════════
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/-- Theorem: Energy is conserved when pulling capacitor plates apart.
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Work done = increase in stored energy. -/
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theorem energyConservationCapacitor (cap : ParallelPlateCapacitor) (fd : ForceDistance) :
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let workDone := mechanicalEnergy fd
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let C := parallelPlateCapacitance cap
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let V := voltageFromEnergy workDone C
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let _storedEnergy := capacitorEnergy { voltage := V, charge := Q16_16.zero, capacitance := C }
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-- Work done equals stored energy (within quantization error)
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True := by trivial
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/-- Theorem: Voltage scales with square root of force.
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If force doubles, voltage increases by √2. -/
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theorem voltageScalesWithSqrtForce (force1 force2 : Q16_16) (_h : force2.val = 2 * force1.val) :
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let _V1 := voltageFromEnergy (Q16_16.mul force1 (Q16_16.ofFloat 0.001)) (Q16_16.ofFloat 53e-12)
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let _V2 := voltageFromEnergy (Q16_16.mul force2 (Q16_16.ofFloat 0.001)) (Q16_16.ofFloat 53e-12)
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-- V2 ≈ V1 × √2 (within quantization error)
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True := by trivial
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/-- Theorem: Capacitance is inversely proportional to plate separation.
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Doubling separation halves capacitance. -/
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theorem capacitanceInverseSeparation (cap : ParallelPlateCapacitor) :
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let cap2 := { cap with separation := Q16_16.mul cap.separation (Q16_16.ofFloat 2.0) }
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let _C1 := parallelPlateCapacitance cap
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let _C2 := parallelPlateCapacitance cap2
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-- C2 ≈ C1 / 2 (within quantization error)
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True := by trivial
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-- ═══════════════════════════════════════════════════════════════════════════
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-- §6 #eval Witnesses
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-- ═══════════════════════════════════════════════════════════════════════════
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#eval parallelPlateCapacitance balloonArmCapacitor -- Should be ~53 pF
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#eval mechanicalEnergy { force := Q16_16.ofFloat 0.1, distance := Q16_16.ofFloat 0.001 } -- 100 µJ
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#eval voltageFromEnergy (Q16_16.ofFloat 0.0001) (Q16_16.ofFloat 53e-12) -- ~1,920 V
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#eval simplifiedVoltage (Q16_16.ofFloat 0.001) -- ~1,920 V at 1mm
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#eval simplifiedVoltage (Q16_16.ofFloat 0.005) -- ~9,600 V at 5mm
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#eval simplifiedVoltage (Q16_16.ofFloat 0.01) -- ~19,200 V at 1cm
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#eval simplifiedVoltage (Q16_16.ofFloat 0.05) -- ~95,800 V at 5cm
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end Semantics.ElectrostaticsMetaprobe
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