Ampere-Maxwell Law (Displacement Current)

Derivation Goal

Prove rigorously that changing bonding shell configurations (∂E/∂t) create torques on internal rotating nucleons that produce the exact curl relationship:

Physical Setup

What E Represents Mechanically

In AAM, the electric field E measures the collective response of bonding valence shells:

Where:

• α = coupling constant (relates microscopic to macroscopic, includes ε₀)
• ρ_aligned = number density of atoms with aligned bonding shells (atoms/m³)
• ⟨v_orbitron⟩ = average orbitron velocity vector in aligned shells (m/s)

What B Represents Mechanically

In AAM, the magnetic field B measures aligned rotating nucleon response:

Where:

• β = coupling constant (relates microscopic to macroscopic, includes μ₀)
• f_aligned = fraction of atoms with aligned nucleon rotation axes (dimensionless)
• ρ_nucleon = nucleon number density (nucleons/m³)
• ⟨L_nucleon⟩ = average angular momentum vector of nucleon pair (kg·m²/s)

The Key Insight: Symmetry with Faraday

Maxwell's equations have beautiful symmetry:

• Faraday: ∇ × E = -∂B/∂t (changing B creates circulating E)
• Ampere-Maxwell: ∇ × B = μ₀J + μ₀ε₀ ∂E/∂t (changing E creates circulating B)

These are almost mirror images! The displacement current term μ₀ε₀ ∂E/∂t is the reverse coupling of what appears in Faraday's law.

AAM Interpretation

• Faraday: Changing nucleon rotation (∂B/∂t) → torques atoms → changes bonding shells → induces E circulation
• Displacement: Changing bonding shells (∂E/∂t) → reorients atoms → torques nucleons → induces B circulation

The Mechanical Coupling

Consider an atom with:

• Bonding valence shell at angle θ_shell relative to some reference
• Internal nucleon pair rotating with angular momentum L at angle θ_nucleon relative to reference
• Geometric constraint: θ_shell ⊥ θ_nucleon (perpendicular by atomic structure)

If the bonding shell orientation changes (∂θ_shell/∂t ≠ 0), the atom must reorient. Since the nucleons are rigidly connected to the atomic structure, they experience a torque.

Torque on Gyroscope from Forced Reorientation

Standard Gyroscopic Dynamics

For a gyroscope (our rotating nucleon pair) with angular momentum L:

If we force the gyroscope to precess (change its axis orientation) at angular velocity Ω:

Where:

• τ = required torque (N·m)
• Ω = angular velocity of precession (rad/s)
• L = angular momentum of gyroscope (kg·m²/s)

Application to Our Atom

When bonding shells change orientation at rate ∂θ_shell/∂t:

• Atom must rotate to accommodate this change
• Atomic rotation rate: Ω_atom ∼ ∂θ_shell/∂t
• Nucleons forced to precess with atom (they're inside it)
• Required torque on nucleons: τ ∼ Ω_atom × L_nucleon

Key equation:

But ∂θ_shell/∂t is related to ∂E/∂t (since E depends on shell orientation and orbitron velocity).

Connecting ∂E/∂t to Atomic Reorientation Rate

Time Derivative of E-Field

Starting from:

Taking the time derivative:

Two terms:

1. Changing alignment density: ∂ρ_aligned/∂t — atoms aligning/dealigning
2. Changing orbitron velocity: ∂⟨v_orbitron⟩/∂t — current accelerating/decelerating

Both require atomic motion/reorientation.

Alignment Density Change

If ρ_aligned is changing, atoms are reorienting from random to aligned (or vice versa).

Rate of alignment:

where f_aligned = fraction of aligned atoms.

Each atom that aligns must rotate through angle ~θ in time ~Δt, giving average angular velocity:

Orbitron Velocity Change

If orbitron velocity is changing in already-aligned shells, this represents acceleration of orbitrons.

In an antenna or capacitor, it's the applied electric potential. But mechanically, this means:

• Pressure wave applying force to orbitrons
• Orbitrons accelerating through bonding shells
• Shells must adjust geometry to accommodate changing flow
• This also requires atomic structural adjustment

So changing v_orbitron also involves atomic motion.

Spatial Distribution: From Torque to ∇ × B

Here's the crucial step: The torque on nucleons isn't uniform — it varies spatially.

Why Curl Appears

In a region where ∂E/∂t is happening:

• E-field changing means bonding shell configuration changing
• But this change spreads through space (it's not instantaneous everywhere)
• Near the source of change: high ∂E/∂t
• Far from source: low ∂E/∂t
• Gradient in ∂E/∂t creates spatial pattern

The spatial pattern of torqued nucleons creates circulation in the B-field.

The Curl-Generating Mechanism

Consider a simple case: parallel-plate capacitor charging.

Between plates:

• E-field increasing uniformly: ∂E/∂t pointing up (say)
• All atoms experiencing same ∂E/∂t
• All atoms reorienting at same rate
• All nucleons torqued similarly

But at the edges:

• E-field drops off
• Spatial gradient in E
• Spatial gradient in ∂E/∂t
• Different torques on different atoms

The circulation of B-field (∇ × B) arises from this spatial variation in nucleon torques.

Mathematical Form

If torque on nucleons: τ ∼ (∂θ_shell/∂t) × L_nucleon

And ∂θ_shell/∂t ∼ ∂E/∂t (changing shells ~ changing E)

Then the spatial distribution of torques creates:

The proportionality constant is μ₀ε₀.

Concrete Example: Charging Capacitor

Setup

• Parallel plates separated by distance d
• Circular plates of radius R
• Charge Q(t) increasing linearly: dQ/dt = I = constant
• E-field between plates: E = Q/(ε₀A) where A = πR²

E-Field Time Derivative

This is uniform between plates (spatially constant), pointing perpendicular to plates.

What's Happening to Atoms?

As E increases:

• More bonding shells aligning vertically (parallel to E)
• Atoms between plates progressively aligning
• Each atom rotates from random orientation to aligned
• Rotation creates angular velocity Ω_atom for each atom

Nucleon Torque

Each atom's nucleons experience torque:

But Ω_atom is perpendicular to L_nucleon (by geometric constraint), so:

What Creates Circulation in B?

Here's the key: Even though ∂E/∂t is uniform between plates, the boundary conditions create circulation.

At the edge of the capacitor (r = R):

• E-field drops off
• Fringing field curves around edges
• Spatial variation in E
• This creates spatial variation in alignment
• Circulation in nucleon torques

The B-field wraps around the capacitor edges, forming closed loops.

The circulation integral:

By Stokes' theorem:

Therefore:

Why μ₀ε₀?

Dimensional Analysis

• [∇ × B] = T/m = kg/(A·s²·m)
• [∂E/∂t] = (V/m)/s = V/(m·s) = kg·m/(A·s³·m) = kg/(A·s³)

For these to be equal (up to constant):

Solving for C:

And indeed:

• [μ₀] = H/m = kg·m/A²·s²
• [ε₀] = F/m = A²·s&sup4;/(kg·m³)
• [μ₀ε₀] = (kg·m/A²·s²) × (A²·s&sup4;/kg·m³) = s²/m²

So μ₀ε₀ has the right dimensions.

The Constants Must Come From Atomic Properties

μ₀ should relate to nucleon gyroscopic properties:

• Nucleon mass (inertia)
• Nucleon rotation rate
• Nucleon separation distance
• How torque translates to B-field change

ε₀ should relate to bonding shell properties:

• Orbitron mass
• Orbitron density in shells
• Shell compressibility
• Aether coupling strength

The Product μ₀ε₀ = 1/c²

This is the most important relationship:

From pressure wave theory:

Where:

• K = bulk modulus of aether (~10^-11 Pa)
• ρ_aether = aether density

Therefore:

This is the constraint: Whatever values we derive for μ₀ and ε₀ separately, their product must equal ρ_aether/K.

The Current Term: μ₀J

Physical Mechanism

The first term in Ampere's law relates current to B-field circulation:

Current (J) in AAM:

• Orbitron flow through bonding shells
• Flow causes atomic alignment (shells orient with flow)
• Aligned atoms have aligned rotating nucleons
• Aligned nucleons create coherent B-field

Forward Cascade: Current Creates Magnetic Field

Current → Bonding Shell Alignment → Atomic Orientation → Nucleon Alignment → B-field

This explains the J term in Ampere-Maxwell law: ∇ × B = μ₀J + ...

Summary of Displacement Current Mechanism

Physical Chain

Consistency Checks

Symmetry with Faraday

• Faraday: Changing nucleons → changes shells → ∇ × E
• Displacement: Changing shells → torques nucleons → ∇ × B

Mechanically symmetric reverse processes.

The Full Ampere-Maxwell Equation

Two terms:

• μ₀J (current term): Orbitron flow → atomic alignment → nucleon rotation alignment
• μ₀ε₀∂E/∂t (displacement current): Changing shells → gyroscopic torque → nucleon precession

Both create circulation in B-field through the same underlying mechanism: aligned rotating nucleons.

What Remains to Be Done

Quantitative Derivation (Challenge 1.9)

• Exact relationship between ∂θ_shell/∂t and ∂E/∂t (involves α constant)
• Exact relationship between nucleon torque and ∇ × B (involves β constant)
• Show that these combine to give exactly μ₀ε₀ ∂E/∂t

Derive μ₀ and ε₀ Separately

• μ₀ from nucleon properties (mass, rotation rate, separation)
• ε₀ from shell properties (orbitron mass, density, aether coupling)
• Prove product = 1/c² = ρ_aether/K

General Proof

• Current derivation used capacitor example
• Need general proof for arbitrary ∂E/∂t configuration
• Vector calculus proof that curl relationship holds universally

Confidence Assessment

Connections to Other AAM Principles

Related Axioms

• Axiom 1: All phenomena as space, matter, motion. Displacement current is matter responding to changing shells.
• Axiom 8: Constant motion. Dual valence cloud structure provides the coupling.

Related Derivations

• Faraday's Law: The symmetric reverse process — changing nucleons affect shells.
• No Magnetic Monopoles: Both derive from rotating nucleon dynamics.