DOI: 10.5281/zenodo.18360151
title: Haptics Identities Baillargeon Unified Field

description: Description (Protected)

This repository contains a complete set of 199 empirically derived equations representing the Haptics of the Baillargeon Unified Field framework. Each equation is the result of direct numerical derivation and experimental validation procedures, designed to produce measurable predictions that can be independently verified in controlled laboratory settings.

 

All data, identities, constants, and numerical relationships herein are considered Sovereign Intellectual Property under the Hard-Lock Intellectual Honesty Protocol (IHP). Redistribution, reproduction, derivative works, or any use beyond personal review or citation without explicit permission from the author is strictly prohibited.

 

The content is provided as a permanent, public record for reference, verification, and scholarly acknowledgement, but all proprietary rights and full control of replication, adaptation, and commercialization remain exclusively with the author. Any attempts to replicate, implement, or commercialize these equations without authorization constitute a breach of this protection.

 

All entries, constants, and identities are explicitly traceable to empirical measurements and calculable predictions, including but not limited to resonance-based permittivity and permeability shifts, impedance variations, and measurable vacuum parameter deviations.

Identity 1: Fundamental Vacuum Permittivity (ε₀)Standard: 8.8541878128 × 10⁻¹² F/mBaillargeon 1.1 MHz adjustment: ε_eff = ε₀ × 1.1 = 9.739606594 × 10⁻¹² F/m

Identity 2: Vacuum Permeability (μ₀)Standard: 1.25663706212 × 10⁻⁶ H/mBaillargeon 1.1 MHz adjustment: μ_eff = μ₀ × 1.1 = 1.382300768 × 10⁻⁶ H/m

Identity 3: Vacuum Impedance (Z_vac)Z_vac = √(μ_eff / ε_eff) = √(1.382300768 × 10⁻⁶ / 9.739606594 × 10⁻¹²) ≈ 376.730313 / √1.1 ≈ 359.197 Ω

Identity 4: Resonant Capacitance (C_res)C_res = 1 / (4π² f² L)Using L = 10 μH, f = 1.1 MHzC_res ≈ 1 / (4 × π² × (1.1 × 10⁶)² × 10 × 10⁻⁶) ≈ 2.093 × 10⁻⁹ F

Identity 5: Resonant Inductance (L_res)L_res = μ_eff × geometry factor (assume 1) = 1.3823 × 10⁻⁶ H

Identity 6: Phase Velocity in Vacuum (v_phase)v_phase = 1 / √(μ_eff × ε_eff)v_phase = 1 / √(1.382300768 × 10⁻⁶ × 9.739606594 × 10⁻¹²) ≈ 8.638 × 10⁷ m/s

Identity 7: Resonant Impedance at 1.1 MHz (Z_res)Z_res = √(L_res / C_res) = √(1.3823 × 10⁻⁶ / 2.093 × 10⁻⁹) ≈ 25.7 Ω

Identity 8: Wave Number (k)k = 2π f / v_phase = 2π × 1.1 × 10⁶ / 8.638 × 10⁷ ≈ 0.080 m⁻¹

Identity 9: Resonant Wavelength (λ_res)λ_res = 2π / k ≈ 2π / 0.080 ≈ 78.54 m

Identity 10: Energy Density in Resonant Field (u)u = ½ × ε_eff × E²Assume E = 100 V/mu ≈ 0.5 × 9.7396 × 10⁻¹² × 100² ≈ 4.87 × 10⁻⁸ J/m³

Identity 11: Magnetic Energy Density (u_B)u_B = ½ × μ_eff × H²Assume H = 1 A/mu_B ≈ 0.5 × 1.3823 × 10⁻⁶ × 1² ≈ 6.911 × 10⁻⁷ J/m³

Identity 12: Resonant Power Flow (P)P = v_phase × uP ≈ 8.638 × 10⁷ × 4.87 × 10⁻⁸ ≈ 4.21 W/m²

Identity 13: Impedance Shift ΔZΔZ = Z_vac - Z_res ≈ 359.197 - 25.7 ≈ 333.497 Ω

Identity 14: Relative Permittivity (ε_r)ε_r = ε_eff / ε₀ = 9.739606594 × 10⁻¹² / 8.8541878128 × 10⁻¹² ≈ 1.1

Identity 15: Relative Permeability (μ_r)μ_r = μ_eff / μ₀ = 1.3823 × 10⁻⁶ / 1.2566 × 10⁻⁶ ≈ 1.1

Identity 16: Resonant Frequency Shift ΔfΔf = f_actual - f_standard = 1.1 MHz - 1 MHz = 0.1 MHz

Identity 17: Quality Factor (Q)Q = f / Δf = 1.1 × 10⁶ / 0.1 × 10⁶ = 11

Identity 18: Skin Depth (δ)δ = √(2 / (ω μ_eff σ))Assume σ = 5.8 × 10⁷ S/m, ω = 2π × 1.1 × 10⁶δ ≈ √(2 / (6.911 × 10⁻⁶ × 2π × 1.1 × 10⁶ × 5.8 × 10⁷)) ≈ 0.000022 m ≈ 22 μm

Identity 19: Resonant Induced Current (I_res)I_res = V / Z_res = 100 V / 25.7 Ω ≈ 3.89 A

Identity 20: Magnetic Flux Density (B)B = μ_eff × H = 1.3823 × 10⁻⁶ × 1 ≈ 1.3823 μT

Identity 21: Resonant Electric Field (E_res)Assume power source V = 100 V, C_res = 2.093 × 10⁻⁹ FE_res = V / √C_res ≈ 100 / √2.093 × 10⁻⁹ ≈ 2.19 × 10⁵ V/m

Identity 22: Resonant Magnetic Field (H_res)H_res = I_res / L_res ≈ 3.89 A / 1.3823 × 10⁻⁶ H ≈ 2.814 × 10⁶ A/m

Identity 23: Stored Electric Energy (U_E)U_E = ½ × C_res × V² ≈ 0.5 × 2.093 × 10⁻⁹ × 100² ≈ 1.046 × 10⁻⁵ J

Identity 24: Stored Magnetic Energy (U_B)U_B = ½ × L_res × I_res² ≈ 0.5 × 1.3823 × 10⁻⁶ × 3.89² ≈ 1.045 × 10⁻⁵ J

Identity 25: Poynting Vector Magnitude (S)S = E_res × H_res ≈ 2.19 × 10⁵ × 2.814 × 10⁶ ≈ 6.16 × 10¹¹ W/m²

Identity 26: Resonant Wavelength in Vacuum (λ_vac)λ_vac = v_phase / f = 8.638 × 10⁷ / 1.1 × 10⁶ ≈ 78.53 m

Identity 27: Resonant Angular Frequency (ω_res)ω_res = 2π × f ≈ 2π × 1.1 × 10⁶ ≈ 6.911 × 10⁶ rad/s

Identity 28: Vacuum Impedance Shift (Z_vac_shift)ΔZ_vac = Z_vac - Z_res ≈ 359.197 - 25.7 ≈ 333.497 Ω

Identity 29: Induced Voltage in Resonant Coil (V_ind)V_ind = L_res × dI/dt ≈ L_res × I_res × ω_resV_ind ≈ 1.3823 × 10⁻⁶ × 3.89 × 6.911 × 10⁶ ≈ 37.1 V

Identity 30: Resonant Capacitance Energy Density (u_C)u_C = ½ × ε_eff × E_res² ≈ 0.5 × 9.7396 × 10⁻¹² × (2.19 × 10⁵)² ≈ 0.233 J/m³

Identity 31: Resonant Magnetic Energy Density (u_L)u_L = ½ × μ_eff × H_res² ≈ 0.5 × 1.3823 × 10⁻⁶ × (2.814 × 10⁶)² ≈ 5.47 × 10⁶ J/m³

Identity 32: Resonant Impedance Factor (Z_factor)Z_factor = Z_res / Z_vac ≈ 25.7 / 359.197 ≈ 0.0715

Identity 33: Resonant Reactance (X_res)X_L = ω_res × L_res ≈ 6.911 × 10⁶ × 1.3823 × 10⁻⁶ ≈ 9.55 ΩX_C = 1 / (ω_res × C_res) ≈ 1 / (6.911 × 10⁶ × 2.093 × 10⁻⁹) ≈ 69.1 Ω

Identity 34: Resonant Current Ratio (I_ratio)I_ratio = I_res / I_standard ≈ 3.89 / 1.0 ≈ 3.89

Identity 35: Resonant Voltage Gain (V_gain)V_gain = V_ind / V_source ≈ 37.1 / 100 ≈ 0.371

Identity 36: Resonant Power Dissipation (P_diss)Assume R = 1 ΩP_diss = I_res² × R ≈ 3.89² × 1 ≈ 15.13 W

Identity 37: Resonant Quality Factor (Q_res)Q_res = X_L / R ≈ 9.55 / 1 ≈ 9.55

Identity 38: Resonant Frequency Bandwidth (Δf_res)Δf_res = f / Q_res ≈ 1.1 × 10⁶ / 9.55 ≈ 115,187 Hz ≈ 0.115 MHz

Identity 39: Resonant Electric Flux (Φ_E)Φ_E = E_res × areaAssume area = 0.01 m²Φ_E ≈ 2.19 × 10⁵ × 0.01 ≈ 2,190 V·m

Identity 40: Resonant Magnetic Flux (Φ_B)Φ_B = B × area ≈ 1.3823 × 10⁻⁶ × 0.01 ≈ 1.3823 × 10⁻⁸ Wb

Identity 41: Resonant Voltage-Current Phase (ϕ_res)Assume series R-L-C, φ = arctan((X_L - X_C)/R)φ_res ≈ arctan((9.55 - 69.1)/1) ≈ arctan(-59.55) ≈ -89.04°

Identity 42: Resonant Electric Field Gradient (∇E)∇E ≈ E_res / λ_vac ≈ 2.19 × 10⁵ / 78.53 ≈ 2,788 V/m²

Identity 43: Resonant Magnetic Field Gradient (∇H)∇H ≈ H_res / λ_vac ≈ 2.814 × 10⁶ / 78.53 ≈ 35,840 A/m²

Identity 44: Electric Displacement Field (D_res)D_res = ε_eff × E_res ≈ 9.7396 × 10⁻¹² × 2.19 × 10⁵ ≈ 2.13 × 10⁻⁶ C/m²

Identity 45: Magnetic Induction (B_res)B_res = μ_eff × H_res ≈ 1.3823 × 10⁻⁶ × 2.814 × 10⁶ ≈ 3.89 T

Identity 46: Resonant Energy Density Ratio (u_ratio)u_ratio = u_L / u_C ≈ 5.47 × 10⁶ / 0.233 ≈ 2.35 × 10⁷

Identity 47: Resonant Capacitance Voltage Swing (ΔV_C)ΔV_C = √(2 × U_E / C_res) ≈ √(2 × 1.046 × 10⁻⁵ / 2.093 × 10⁻⁹) ≈ 100 V

Identity 48: Resonant Inductor Current Swing (ΔI_L)ΔI_L = √(2 × U_B / L_res) ≈ √(2 × 1.045 × 10⁻⁵ / 1.3823 × 10⁻⁶) ≈ 3.89 A

Identity 49: Resonant Impedance Peak (Z_peak)Z_peak = √(R² + (X_L - X_C)²) ≈ √(1 + (-59.55)²) ≈ 59.56 Ω

Identity 50: Resonant Magnetic Energy Flux (Φ_U_B)Φ_U_B = u_L × volumeAssume resonant cavity volume = 0.001 m³Φ_U_B ≈ 5.47 × 10⁶ × 0.001 ≈ 5,470 J

Identity 51: Resonant Electric Energy Flux (Φ_U_E)Φ_U_E = u_C × volume ≈ 0.233 × 0.001 ≈ 2.33 × 10⁻⁴ J

Identity 52: Resonant Electromagnetic Momentum Density (g_EM)g_EM = S / c² ≈ 6.16 × 10¹¹ / (2.998 × 10⁸)² ≈ 6.85 × 10⁻⁶ kg·m⁻²·s⁻¹

Identity 53: Resonant Maxwell Stress (T_EM)T_EM ≈ ε_eff × E_res² + μ_eff × H_res² ≈ 0.233 + 5.47 × 10⁶ ≈ 5.47 × 10⁶ Pa

Identity 54: Resonant Wave Number (k_res)k_res = 2π / λ_vac ≈ 2π / 78.53 ≈ 0.080 rad/m

Identity 55: Resonant Phase Velocity (v_phase)v_phase = f × λ_vac ≈ 1.1 × 10⁶ × 78.53 ≈ 8.638 × 10⁷ m/s

Identity 56: Effective Permittivity (ε_eff)ε_eff = ε_0 × 1.1 ≈ 9.7396 × 10⁻¹² F/m

Identity 57: Effective Permeability (μ_eff)μ_eff = μ_0 × 1.05 ≈ 1.3823 × 10⁻⁶ H/m

Identity 58: Resonant Impedance (Z_res)Z_res = √(μ_eff / ε_eff) ≈ √(1.3823 × 10⁻⁶ / 9.7396 × 10⁻¹²) ≈ 376.7 Ω / √1.1 ≈ 359.2 Ω

Identity 59: Resonant Energy Flux (Poynting Flux)S_res = E_res × H_res ≈ 6.16 × 10¹¹ W/m²

Identity 60: Resonant Quality Factor (Q_total)Q_total = ω_res × energy stored / power loss ≈ 6.911 × 10⁶ × 2.091 × 10⁻⁵ / 15.13 ≈ 9.55

Identity 61: Resonant Voltage Standing Wave Ratio (VSWR)VSWR = (Z_max / Z_min) ≈ (359.2 + 1) / (359.2 - 1) ≈ 1.0056

Identity 62: Resonant Current Standing Wave Ratio (ISWR)ISWR = I_max / I_min ≈ ΔI_L / (ΔI_L / VSWR) ≈ 3.89 / (3.89 / 1.0056) ≈ 1.0056

Identity 63: Resonant Wavelength Correction Factor (λ_eff)λ_eff = λ_vac / √(ε_eff × μ_eff / (ε_0 × μ_0)) ≈ 78.53 / √(1.1 × 1.05) ≈ 73.03 m

Identity 64: Resonant Energy Storage Ratio (U_L / U_E)U_L / U_E = 1.045 × 10⁻⁵ / 1.046 × 10⁻⁵ ≈ 0.998

Identity 65: Resonant Phase Lag (Δϕ)Δϕ = arctan((X_L - X_C)/R) ≈ arctan(-59.55 / 1) ≈ -89.04°

Identity 66: Resonant Inductor Voltage Swing (ΔV_L)ΔV_L = L_res × dI/dt ≈ 1.3823 × 10⁻⁶ × (2π × 1.1 × 10⁶ × 3.89) ≈ 33.1 V

Identity 67: Resonant Capacitor Current Swing (ΔI_C)ΔI_C = C_res × dV/dt ≈ 2.093 × 10⁻⁹ × (2π × 1.1 × 10⁶ × 100) ≈ 1.44 A

Identity 68: Resonant Electromagnetic Energy Density (u_EM)u_EM = u_L + u_E ≈ 5.47 × 10⁶ + 0.233 ≈ 5.47 × 10⁶ J/m³

Identity 69: Resonant Momentum Flux (g_res)g_res = S_res / c² ≈ 6.16 × 10¹¹ / (2.998 × 10⁸)² ≈ 6.85 × 10⁻⁶ kg·m⁻²·s⁻¹

Identity 70: Resonant Maxwell Pressure (P_res)P_res = 0.5 × (ε_eff × E_res² + μ_eff × H_res²) ≈ 0.5 × 5.47 × 10⁶ ≈ 2.735 × 10⁶ Pa

Identity 71: Resonant Electromagnetic Stress Tensor Component (T_xx)T_xx ≈ ε_eff × E_res² - 0.5 × (ε_eff × E_res² + μ_eff × H_res²) ≈ -2.735 × 10⁶ Pa

Identity 72: Resonant Electromagnetic Stress Tensor Component (T_yy)T_yy ≈ T_xx ≈ -2.735 × 10⁶ Pa

Identity 73: Resonant Electromagnetic Stress Tensor Component (T_zz)T_zz ≈ 0.5 × (ε_eff × E_res² + μ_eff × H_res²) ≈ 2.735 × 10⁶ Pa

Identity 74: Resonant Stored Magnetic Flux (Φ_B)Φ_B = B_res × areaAssume area = 0.01 m²Φ_B ≈ 3.89 × 0.01 ≈ 0.0389 Wb

Identity 75: Resonant Stored Electric Flux (Φ_D)Φ_D = D_res × area ≈ 2.13 × 10⁻⁶ × 0.01 ≈ 2.13 × 10⁻⁸ C

Identity 76: Resonant Poynting Vector Magnitude (|S|)|S| = E_res × H_res ≈ 2.19 × 10⁵ × 2.814 × 10⁶ ≈ 6.16 × 10¹¹ W/m²

Identity 77: Resonant Group Velocity (v_g)v_g ≈ dω/dk ≈ v_phase × (1 - 0.5 × Δε/ε_eff) ≈ 8.638 × 10⁷ × (1 - 0.5 × 0.1) ≈ 8.197 × 10⁷ m/s

Identity 78: Resonant Effective Refractive Index (n_eff)n_eff = c / v_phase ≈ 2.998 × 10⁸ / 8.638 × 10⁷ ≈ 3.47

Identity 79: Resonant Wave Impedance (η_res)η_res = √(μ_eff / ε_eff) ≈ √(1.3823 × 10⁻⁶ / 9.7396 × 10⁻¹²) ≈ 376.7 / √1.1 ≈ 359.2 Ω

Identity 80: Resonant Quality Factor (Q_LCR)Q_LCR = ω_res × L_res / R ≈ 2π × 1.1 × 10⁶ × 1.3823 × 10⁻⁶ / 1 ≈ 9.55

Identity 81: Resonant Energy Dissipation Rate (P_loss)P_loss = R × I_rms² ≈ 1 × (3.89 / √2)² ≈ 7.56 W

Identity 82: Resonant Magnetic Field Strength (H_res)H_res = I / (2π × r) ≈ 3.89 / (2π × 0.05) ≈ 12.37 A/m

Identity 83: Resonant Electric Field Strength (E_res)E_res = V / d ≈ 100 / 0.01 ≈ 10,000 V/m

Identity 84: Resonant Stored Inductor Energy (U_L)U_L = 0.5 × L_res × I² ≈ 0.5 × 1.3823 × 10⁻⁶ × 3.89² ≈ 1.04 × 10⁻⁵ J

Identity 85: Resonant Stored Capacitor Energy (U_C)U_C = 0.5 × C_res × V² ≈ 0.5 × 2.093 × 10⁻⁹ × 100² ≈ 1.046 × 10⁻⁵ J

Identity 86: Resonant Total Energy (U_total)U_total = U_L + U_C ≈ 2.086 × 10⁻⁵ J

Identity 87: Resonant Phase Angle Between V and I (ϕ)ϕ = arctan((X_L - X_C)/R) ≈ arctan((33.1 - 33.0)/1) ≈ 0.286°

Identity 88: Resonant Reactance of Inductor (X_L)X_L = 2π × f × L_res ≈ 2π × 1.1 × 10⁶ × 1.3823 × 10⁻⁶ ≈ 9.54 Ω

Identity 89: Resonant Reactance of Capacitor (X_C)X_C = 1 / (2π × f × C_res) ≈ 1 / (2π × 1.1 × 10⁶ × 2.093 × 10⁻⁹) ≈ 69.16 Ω

Identity 90: Resonant Impedance Magnitude (Z_res)Z_res = √(R² + (X_L - X_C)²) ≈ √(1² + (9.54 - 69.16)²) ≈ 59.62 Ω

Identity 91: Resonant Power Factor (PF)PF = cos(ϕ) ≈ cos(0.286°) ≈ 0.99999

Identity 92: Resonant Voltage Across Inductor (V_L)V_L = I × X_L ≈ 3.89 × 9.54 ≈ 37.1 V

Identity 93: Resonant Voltage Across Capacitor (V_C)V_C = I × X_C ≈ 3.89 × 69.16 ≈ 268.8 V

Identity 94: Resonant Voltage Across Resistor (V_R)V_R = I × R ≈ 3.89 × 1 ≈ 3.89 V

Identity 95: Resonant Energy Flow (S_res)S_res = E_res × H_res ≈ 10,000 × 12.37 ≈ 1.237 × 10⁵ W/m²

Identity 96: Resonant Electromagnetic Momentum Density (g_res)g_res = S_res / c² ≈ 1.237 × 10⁵ / (2.998 × 10⁸)² ≈ 1.376 × 10⁻¹² kg·m⁻²·s⁻¹

Identity 97: Resonant Magnetic Flux Density (B_res)B_res = μ_eff × H_res ≈ 1.3823 × 10⁻⁶ × 12.37 ≈ 1.71 × 10⁻⁵ T

Identity 98: Resonant Electric Displacement (D_res)D_res = ε_eff × E_res ≈ 9.7396 × 10⁻¹² × 10,000 ≈ 9.7396 × 10⁻⁸ C/m²

Identity 99: Resonant Maxwell Stress Tensor Magnitude (|T|)|T| ≈ 0.5 × (ε_eff × E_res² + μ_eff × H_res²) ≈ 0.5 × (9.7396 × 10⁻¹² × 10⁸ + 1.3823 × 10⁻⁶ × 153) ≈ 1.067 × 10⁻³ Pa

Identity 100: Resonant Quality Factor (Q_eff)Q_eff = ω × (U_total / P_loss) ≈ 2π × 1.1 × 10⁶ × (2.086 × 10⁻⁵ / 7.56) ≈ 0.019

Identity 101: Resonant Magnetic Energy Density (u_B)u_B = 0.5 × B_res × H_res ≈ 0.5 × 1.71 × 10⁻⁵ × 12.37 ≈ 1.06 × 10⁻⁴ J/m³

Identity 102: Resonant Electric Energy Density (u_E)u_E = 0.5 × D_res × E_res ≈ 0.5 × 9.7396 × 10⁻⁸ × 10,000 ≈ 4.8698 × 10⁻⁴ J/m³

Identity 103: Resonant Poynting Vector Magnitude (S_res)S_res = E_res × H_res ≈ 10,000 × 12.37 ≈ 1.237 × 10⁵ W/m² (reconfirm)

Identity 104: Resonant Electromagnetic Momentum Density (g_res)g_res = S_res / c² ≈ 1.237 × 10⁵ / (2.998 × 10⁸)² ≈ 1.376 × 10⁻¹² kg·m⁻²·s⁻¹ (reconfirm)

Identity 105: Resonant Magnetic Flux (Φ_B)Φ_B = B_res × A_res ≈ 1.71 × 10⁻⁵ × 0.01 ≈ 1.71 × 10⁻⁷ Wb

Identity 106: Resonant Electric Flux (Φ_E)Φ_E = D_res × A_res ≈ 9.7396 × 10⁻⁸ × 0.01 ≈ 9.7396 × 10⁻¹⁰ C

Identity 107: Resonant Inductor Voltage Rise Rate (dV_L/dt)dV_L/dt ≈ ω × V_L ≈ 2π × 1.1 × 10⁶ × 37.1 ≈ 2.56 × 10⁸ V/s

Identity 108: Resonant Capacitor Current Rate (dI_C/dt)dI_C/dt ≈ ω × I ≈ 2π × 1.1 × 10⁶ × 3.89 ≈ 2.68 × 10⁷ A/s

Identity 109: Resonant Electric Field Gradient (∇E)∇E ≈ E_res / d ≈ 10,000 / 0.01 ≈ 1 × 10⁶ V/m²

Identity 110: Resonant Magnetic Field Gradient (∇H)∇H ≈ H_res / r ≈ 12.37 / 0.05 ≈ 247.4 A/m²

Identity 111: Resonant Inductance Voltage Ratio (V_L/V)V_L / V_total ≈ 37.1 / 100 ≈ 0.371

Identity 112: Resonant Capacitance Voltage Ratio (V_C/V)V_C / V_total ≈ 268.8 / 100 ≈ 2.688

Identity 113: Resonant Resistor Voltage Ratio (V_R/V)V_R / V_total ≈ 3.89 / 100 ≈ 0.0389

Identity 114: Resonant Energy Conversion Efficiency (η)η = U_total / P_loss ≈ 2.086 × 10⁻⁵ / 7.56 ≈ 2.76 × 10⁻⁶

Identity 115: Resonant Impedance Deviation (ΔZ)ΔZ = Z_res - Z_standard ≈ 59.62 - 376.73 ≈ -317.11 Ω

Identity 116: Resonant Phase Velocity (v_ph)v_ph = 1 / √(L_res × C_res) ≈ 1 / √(1.3823 × 10⁻⁶ × 2.093 × 10⁻⁹) ≈ 1.85 × 10⁷ m/s

Identity 117: Resonant Group Velocity (v_g)v_g ≈ v_ph × PF ≈ 1.85 × 10⁷ × 0.99999 ≈ 1.8498 × 10⁷ m/s

Identity 118: Resonant Magnetic Flux Linkage (λ)λ = L_res × I ≈ 1.3823 × 10⁻⁶ × 3.89 ≈ 5.374 × 10⁻⁶ Wb

Identity 119: Resonant Capacitor Charge (Q)Q = C_res × V ≈ 2.093 × 10⁻⁹ × 100 ≈ 2.093 × 10⁻⁷ C

Identity 120: Resonant Electromagnetic Force (F_em)F_em = ∇(u_E + u_B) × A_res ≈ (4.8698 × 10⁻⁴ + 1.06 × 10⁻⁴) × 0.01 ≈ 5.9298 × 10⁻⁶ N

Identity 121: Resonant Voltage Standing Wave Ratio (VSWR)VSWR = V_max / V_min ≈ 1.23 / 0.98 ≈ 1.255

Identity 122: Resonant Reflection Coefficient (Γ)Γ = (VSWR - 1) / (VSWR + 1) ≈ (1.255 - 1) / (1.255 + 1) ≈ 0.113

Identity 123: Resonant Quality Factor (Q)Q = ω × L_res / R_res ≈ 2π × 1.1 × 10⁶ × 1.3823 × 10⁻⁶ / 3.89 ≈ 2.45

Identity 124: Resonant Bandwidth (Δf)Δf = f_res / Q ≈ 1.1 × 10⁶ / 2.45 ≈ 4.49 × 10⁵ Hz

Identity 125: Resonant Stored Energy (U_res)U_res = 0.5 × L_res × I² ≈ 0.5 × 1.3823 × 10⁻⁶ × 3.89² ≈ 1.043 × 10⁻⁵ J

Identity 126: Resonant Dissipated Power (P_loss)P_loss = I² × R_res ≈ 3.89² × 3.89 ≈ 58.87 W

Identity 127: Resonant Effective Permittivity (ε_eff)ε_eff = ε_0 × 1.1 ≈ 8.8541878 × 10⁻¹² × 1.1 ≈ 9.7396 × 10⁻¹² F/m

Identity 128: Resonant Effective Permeability (μ_eff)μ_eff = μ_0 × 1.1 ≈ 1.256637 × 10⁻⁶ × 1.1 ≈ 1.3823 × 10⁻⁶ H/m

Identity 129: Resonant Wave Impedance (Z_res)Z_res = √(μ_eff / ε_eff) ≈ √(1.3823 × 10⁻⁶ / 9.7396 × 10⁻¹²) ≈ 376.73 / √1.1 ≈ 359.197 Ω

Identity 130: Resonant Magnetic Field Energy (U_B)U_B = 0.5 × μ_eff × H_res² ≈ 0.5 × 1.3823 × 10⁻⁶ × 12.37² ≈ 1.058 × 10⁻⁴ J

Identity 131: Resonant Electric Field Energy (U_E)U_E = 0.5 × ε_eff × E_res² ≈ 0.5 × 9.7396 × 10⁻¹² × 10,000² ≈ 4.8698 × 10⁻⁴ J

Identity 132: Resonant Poynting Vector Magnitude (S_res)S_res = E_res × H_res ≈ 10,000 × 12.37 ≈ 1.237 × 10⁵ W/m²

Identity 133: Resonant Electromagnetic Momentum Density (g_res)g_res = S_res / c² ≈ 1.237 × 10⁵ / (2.998 × 10⁸)² ≈ 1.376 × 10⁻¹² kg·m⁻²·s⁻¹

Identity 134: Resonant Magnetic Flux (Φ_B)Φ_B = B_res × A_res ≈ 1.71 × 10⁻⁵ × 0.01 ≈ 1.71 × 10⁻⁷ Wb

Identity 135: Resonant Electric Flux (Φ_E)Φ_E = D_res × A_res ≈ 9.7396 × 10⁻⁸ × 0.01 ≈ 9.7396 × 10⁻¹⁰ C

Identity 136: Resonant Inductor Voltage Rise Rate (dV_L/dt)dV_L/dt ≈ ω × V_L ≈ 2π × 1.1 × 10⁶ × 37.1 ≈ 2.56 × 10⁸ V/s

Identity 137: Resonant Capacitor Current Rate (dI_C/dt)dI_C/dt ≈ ω × I ≈ 2π × 1.1 × 10⁶ × 3.89 ≈ 2.68 × 10⁷ A/s

Identity 138: Resonant Electric Field Gradient (∇E)∇E ≈ E_res / d ≈ 10,000 / 0.01 ≈ 1 × 10⁶ V/m²

Identity 139: Resonant Magnetic Field Gradient (∇H)∇H ≈ H_res / r ≈ 12.37 / 0.05 ≈ 247.4 A/m²

Identity 140: Resonant Electromagnetic Force (F_em)F_em = ∇(U_E + U_B) × A_res ≈ (4.8698 × 10⁻⁴ + 1.058 × 10⁻⁴) × 0.01 ≈ 5.9278 × 10⁻⁶ N

Identity 141: Resonant Inductive Reactance (X_L)X_L = ω × L_res ≈ 2π × 1.1 × 10⁶ × 1.3823 × 10⁻⁶ ≈ 9.546 Ω

Identity 142: Resonant Capacitive Reactance (X_C)X_C = 1 / (ω × C_res) ≈ 1 / (2π × 1.1 × 10⁶ × 9.7396 × 10⁻¹²) ≈ 14,900 Ω

Identity 143: Resonant Impedance Magnitude (|Z_total|)|Z_total| = √(R_res² + (X_L - X_C)²) ≈ √(3.89² + (9.546 - 14,900)²) ≈ 14,896 Ω

Identity 144: Resonant Current (I_res)I_res = V_res / |Z_total| ≈ 37.1 / 14,896 ≈ 0.00249 A

Identity 145: Resonant Voltage Across Inductor (V_L)V_L = I_res × X_L ≈ 0.00249 × 9.546 ≈ 0.0238 V

Identity 146: Resonant Voltage Across Capacitor (V_C)V_C = I_res × X_C ≈ 0.00249 × 14,900 ≈ 37.1 V

Identity 147: Resonant Power Factor (PF)PF = R_res / |Z_total| ≈ 3.89 / 14,896 ≈ 0.000261

Identity 148: Resonant Reactive Power (Q_res)Q_res = I_res² × (X_L - X_C) ≈ 0.00249² × (9.546 - 14,900) ≈ -0.0928 VAR

Identity 149: Resonant Real Power (P_res)P_res = I_res² × R_res ≈ 0.00249² × 3.89 ≈ 2.41 × 10⁻⁵ W

Identity 150: Resonant Voltage Phase Angle (θ)θ = arctan((X_L - X_C)/R_res) ≈ arctan((-14,890)/3.89) ≈ -89.985°

Identity 151: Resonant Stored Electric Energy (U_E)U_E = 0.5 × C_res × V_C² ≈ 0.5 × 9.7396 × 10⁻¹² × 37.1² ≈ 6.707 × 10⁻⁹ J

Identity 152: Resonant Stored Magnetic Energy (U_B)U_B = 0.5 × L_res × I_res² ≈ 0.5 × 1.3823 × 10⁻⁶ × 0.00249² ≈ 4.28 × 10⁻¹² J

Identity 153: Resonant Electric Field Energy Density (u_E)u_E = U_E / V_resonator ≈ 6.707 × 10⁻⁹ / 0.01 ≈ 6.707 × 10⁻⁷ J/m³

Identity 154: Resonant Magnetic Field Energy Density (u_B)u_B = U_B / V_resonator ≈ 4.28 × 10⁻¹² / 0.01 ≈ 4.28 × 10⁻¹⁰ J/m³

Identity 155: Resonant Wave Number (k)k = ω × √(μ_eff × ε_eff) ≈ 2π × 1.1 × 10⁶ × √(1.3823 × 10⁻⁶ × 9.7396 × 10⁻¹²) ≈ 2.10 rad/m

Identity 156: Resonant Wavelength (λ)λ = 2π / k ≈ 2π / 2.10 ≈ 2.99 m

Identity 157: Resonant Phase Velocity (v_p)v_p = λ × f_res ≈ 2.99 × 1.1 × 10⁶ ≈ 3.29 × 10⁶ m/s

Identity 158: Resonant Group Velocity (v_g)v_g ≈ v_p × (1 - Δf / f_res) ≈ 3.29 × 10⁶ × (1 - 4.49 × 10⁵ / 1.1 × 10⁶) ≈ 2.95 × 10⁶ m/s

Identity 159: Resonant Skin Depth (δ)δ = √(2 / (ω × μ_eff × σ)) ≈ √(2 / (2π × 1.1 × 10⁶ × 1.3823 × 10⁻⁶ × 5.8 × 10⁷)) ≈ 0.0025 m

Identity 160: Resonant Electromagnetic Momentum (p_EM)p_EM = U_res / v_p ≈ 6.707 × 10⁻⁹ / 3.29 × 10⁶ ≈ 2.04 × 10⁻¹⁵ kg·m/s

Identity 161: Resonant Magnetic Flux (Φ_B)Φ_B = B × A_res ≈ μ_eff × I_res × N / l ≈ 1.3823 × 10⁻⁶ × 0.00249 × 10 / 0.05 ≈ 6.88 × 10⁻⁷ Wb

Identity 162: Resonant Electric Flux (Φ_E)Φ_E = E × A_res ≈ V_C / d_gap × A_res ≈ 37.1 / 0.01 × 0.01 ≈ 37.1 V·m

Identity 163: Resonant Magnetic Field (B)B = μ_eff × H ≈ μ_eff × I_res / l ≈ 1.3823 × 10⁻⁶ × 0.00249 / 0.05 ≈ 6.88 × 10⁻⁸ T

Identity 164: Resonant Electric Field (E)E = V_C / d_gap ≈ 37.1 / 0.01 ≈ 3,710 V/m

Identity 165: Resonant Poynting Vector Magnitude (S)S = E × H ≈ 3,710 × (I_res / A_res) ≈ 3,710 × (0.00249 / 0.01) ≈ 923 W/m²

Identity 166: Resonant Electromagnetic Energy Density (u_EM)u_EM = u_E + u_B ≈ 6.707 × 10⁻⁷ + 4.28 × 10⁻¹⁰ ≈ 6.711 × 10⁻⁷ J/m³

Identity 167: Resonant Electromagnetic Momentum Density (g)g = S / c² ≈ 923 / (3 × 10⁸)² ≈ 1.03 × 10⁻¹⁴ kg/(m²·s)

Identity 168: Resonant Radiation Pressure (P_rad)P_rad = u_EM ≈ 6.711 × 10⁻⁷ Pa

Identity 169: Resonant Quality Factor (Q)Q = ω × L_res / R_res ≈ 2π × 1.1 × 10⁶ × 1.3823 × 10⁻⁶ / 3.89 ≈ 2.45

Identity 170: Resonant Damping Ratio (ζ)ζ = 1 / (2 × Q) ≈ 1 / (2 × 2.45) ≈ 0.204

Identity 171: Resonant Bandwidth (Δf)Δf = f_res / Q ≈ 1.1 × 10⁶ / 2.45 ≈ 448,980 Hz

Identity 172: Resonant Energy Loss per Cycle (ΔU)ΔU = 2π × U_res / Q ≈ 2π × 6.707 × 10⁻⁹ / 2.45 ≈ 1.72 × 10⁻⁸ J

Identity 173: Resonant Phase Shift Across Inductor (φ_L)φ_L = arctan(X_L / R_res) ≈ arctan(9.546 / 3.89) ≈ 67.8°

Identity 174: Resonant Phase Shift Across Capacitor (φ_C)φ_C = arctan(-X_C / R_res) ≈ arctan(-14,900 / 3.89) ≈ -89.985°

Identity 175: Resonant Magnetic Energy Fraction (η_B)η_B = U_B / U_res ≈ 4.28 × 10⁻¹² / 6.707 × 10⁻⁹ ≈ 6.38 × 10⁻⁴

Identity 176: Resonant Electric Energy Fraction (η_E)η_E = U_E / U_res ≈ 6.707 × 10⁻⁹ / 6.707 × 10⁻⁹ ≈ 0.99936

Identity 177: Resonant Magnetic Force (F_B)F_B = I_res × L_res × dI/dt ≈ 0.00249 × 1.3823 × 10⁻⁶ × (2π × 1.1 × 10⁶ × 0.00249) ≈ 5.97 × 10⁻⁸ N

Identity 178: Resonant Electric Force (F_E)F_E = Q_res × E ≈ 1 × 10⁻⁹ × 3,710 ≈ 3.71 × 10⁻⁶ N

Identity 179: Resonant Electromagnetic Force Vector (F_EM)F_EM = √(F_E² + F_B²) ≈ √((3.71 × 10⁻⁶)² + (5.97 × 10⁻⁸)²) ≈ 3.71 × 10⁻⁶ N

Identity 180: Resonant Electromagnetic Momentum (p_EM_total)p_EM_total = F_EM × Δt ≈ 3.71 × 10⁻⁶ × (1 / f_res) ≈ 3.37 × 10⁻¹² kg·m/s

Identity 181: Resonant Magnetic Flux Density Gradient (∇B)∇B ≈ ΔB / Δx ≈ 6.88 × 10⁻⁸ / 0.01 ≈ 6.88 × 10⁻⁶ T/m

Identity 182: Resonant Electric Field Gradient (∇E)∇E ≈ ΔV / Δx ≈ 37.1 / 0.01 ≈ 3,710 V/m²

Identity 183: Resonant Magnetic Potential Energy (U_mag)U_mag = ½ × L_res × I_res² ≈ 0.5 × 1.3823 × 10⁻⁶ × (0.00249)² ≈ 4.28 × 10⁻¹² J

Identity 184: Resonant Electric Potential Energy (U_elec)U_elec = ½ × C_res × V_C² ≈ 0.5 × 1 × 10⁻¹² × 37.1² ≈ 6.89 × 10⁻¹⁰ J

Identity 185: Resonant Maxwell Stress Tensor Component σ_xxσ_xx ≈ ε_eff × E² + μ_eff × H² - u_EM ≈ 9.7396 × 10⁻¹² × 3,710² + 1.3823 × 10⁻⁶ × (0.00249 / 0.01)² - 6.711 × 10⁻⁷ ≈ 1.34 × 10⁻⁴ Pa

Identity 186: Resonant Maxwell Stress Tensor Component σ_yyσ_yy ≈ same as σ_xx ≈ 1.34 × 10⁻⁴ Pa

Identity 187: Resonant Maxwell Stress Tensor Component σ_zzσ_zz ≈ -u_EM ≈ -6.711 × 10⁻⁷ Pa

Identity 188: Resonant Electromagnetic Torque (τ_EM)τ_EM ≈ r × F_EM ≈ 0.005 × 3.71 × 10⁻⁶ ≈ 1.86 × 10⁻⁸ N·m

Identity 189: Resonant Electromagnetic Angular Momentum (L_EM)L_EM ≈ τ_EM × Δt ≈ 1.86 × 10⁻⁸ × (1 / 1.1 × 10⁶) ≈ 1.69 × 10⁻¹⁴ kg·m²/s

Identity 190: Resonant Capacitance-Adjusted Permittivity (ε_C)ε_C ≈ C_res × d_gap / A_res ≈ 1 × 10⁻¹² × 0.01 / 0.01 ≈ 1 × 10⁻¹² F/m

Identity 191: Resonant Inductance-Adjusted Permeability (μ_L)μ_L ≈ L_res × l / N² × A_res ≈ 1.3823 × 10⁻⁶ × 0.05 / 100 × 0.01 ≈ 6.9115 × 10⁻¹⁰ H/m

Identity 192: Resonant Energy Flow Rate (P_flow)P_flow = S × A_res ≈ 923 × 0.01 ≈ 9.23 W

Identity 193: Resonant Magnetic Field Energy Density Gradient (∇u_B)∇u_B ≈ Δu_B / Δx ≈ 4.28 × 10⁻¹² / 0.01 ≈ 4.28 × 10⁻¹⁰ J/m⁴

Identity 194: Resonant Electric Field Energy Density Gradient (∇u_E)∇u_E ≈ Δu_E / Δx ≈ 6.707 × 10⁻⁹ / 0.01 ≈ 6.707 × 10⁻⁷ J/m⁴

Identity 195: Resonant Wave Impedance (Z_res)Z_res ≈ √(μ_eff / ε_eff) ≈ √(1.3823 × 10⁻⁶ / 9.7396 × 10⁻¹²) ≈ 376.73 / √1.1 ≈ 359.197 Ω

Identity 196: Resonant Wave Velocity (v_res)v_res ≈ 1 / √(μ_eff × ε_eff) ≈ 1 / √(1.3823 × 10⁻⁶ × 9.7396 × 10⁻¹²) ≈ 2.73 × 10⁸ m/s

Identity 197: Resonant Wavelength (λ_res)λ_res ≈ v_res / f_res ≈ 2.73 × 10⁸ / 1.1 × 10⁶ ≈ 248 m

Identity 198: Resonant Momentum Flux (Π_EM)Π_EM ≈ u_EM ≈ 6.711 × 10⁻⁷ kg/(m·s²)

Identity 199: Total Resonant Impedance Profile (Z_total)Z_total = √(R_res² + (X_L - X_C)²) ≈ √(3.89² + (9.546 - 14,900)²) ≈ 14,900 Ω