Faraday's Law Derivation

Derivation Goal

Prove rigorously that changing rotating nucleon configurations (∂B/∂t) mechanically induce circulation in bonding shell orbitron flow (∇ × E), with the negative sign emerging from Lenz's law.

Physical Setup and Definitions

What B Represents Mechanically

The magnetic field B measures the collective response of aligned rotating nucleon pairs:

Where:

• β = coupling constant (dimensionally includes μ₀)
• f_aligned = fraction of atoms with aligned nucleon rotation axes (dimensionless)
• ρ_nucleon = nucleon density (nucleons/m³)
• ω_nucleon = angular velocity vector of rotating nucleon pair (rad/s)
• r_nucleon = position vector from rotation axis (m)

Physical picture: Each atom has internal nucleons rotating like a gyroscope. When many atoms align their gyroscopes, we measure this as B-field.

What E Represents Mechanically

The electric field E measures the collective response of bonding valence shells:

Where:

• α = coupling constant (dimensionally includes ε₀)
• ρ_aligned = density of atoms with aligned bonding shells (atoms/m³)
• ⟨v_orbitron⟩ = average orbitron velocity in aligned shells (m/s)

Physical picture: Aligned bonding shells allow coordinated orbitron flow, which we measure as E-field.

The Geometric Constraint

From Axiom 1 and Axiom 8, the atomic structure dictates:

The Physical Mechanism of Induction

Starting Point: Changing External B-Field

When an external B-field changes (∂B_ext/∂t ≠ 0):

• External pressure wave pattern is changing
• This creates changing torque on rotating nucleons in nearby atoms
• Nucleons are gyroscopes — they respond to changing torque by precessing

Step 1: Changing Torque on Nucleons

From pressure wave theory, external changing B represents changing pressure wave pattern. This creates time-varying torque on nucleons:

where P(x,t) is the pressure wave oscillation.

For a gyroscope experiencing changing external torque:

Since L = Iω (angular momentum = moment of inertia × angular velocity):

Key point: Changing external field → changing angular velocity of nucleons

Step 2: Precession Changes Nucleon Axis Orientation

When a gyroscope experiences torque perpendicular to spin axis, it precesses:

The precession rate (how fast the axis orientation changes) is:

where n̂_nucleon is the unit vector along nucleon rotation axis.

Key point: Changing torque → precessing nucleon axis → changing orientation

Step 3: Nucleon Reorientation Forces Atomic Reorientation

The rotating nucleons are the gyroscopic core of each atom. They set the atomic orientation. When they reorient:

The atom must follow the nucleon axis orientation.

Key point: Since nucleons ARE the atom's gyroscope, the whole atom reorients when nucleons precess.

Step 4: Atomic Reorientation Changes Bonding Shell Alignment

Remember the geometric constraint: bonding shell ⊥ nucleon rotation axis.

If nucleon axis is n̂_nucleon, then bonding shell orientation is in the plane perpendicular to this.

When nucleon axis changes:

where the shell orientation vector rotates to maintain perpendicularity.

Key point: Nucleon reorientation → bonding shell reorientation (geometric coupling)

Step 5: Bonding Shell Reorientation Alters Orbitron Flow Paths

When bonding shells reorient:

• Shell-to-shell transfer paths change
• Orbitrons must flow in new directions
• This creates circulation in orbitron velocity field
• We measure this circulation as ∇ × E

Key point: Shell reorientation → changes in orbitron flow pattern → induced E-field circulation

From Mechanism to Mathematics

Time Derivative of B-Field

Starting from our definition:

Taking time derivative:

Two contributions:

1. Changing alignment fraction: More/fewer atoms aligning their nucleon axes
2. Changing rotation rate: Individual nucleon angular velocities changing

Both create changing torques that drive atomic reorientation.

Relating ∂B/∂t to Atomic Reorientation Rate

From gyroscopic precession:

But changing B-field IS the changing external torque environment:

Therefore:

This is the key relationship: Rate of nucleon axis change is proportional to rate of B-field change.

The Curl Relationship

The curl operator measures circulation per unit area. When we have:

• Changing nucleon orientations throughout space
• These create changing shell orientations
• Shell orientations vary spatially in a circulating pattern
• This spatial circulation is exactly what ∇ × E measures

For a single atom at position r, if its shell orientation changes at rate:

Then the contribution to E-field circulation at that location is:

Summing over all atoms in a volume (taking continuum limit):

The Negative Sign: Lenz's Law Mechanically

Why Negative?

The equation is:

The negative sign represents opposition — the induced effect opposes the change. This is Lenz's law.

Mechanical Explanation

Consider increasing external B-field (∂B/∂t > 0, pointing up):

Step 1: External B increasing

• More pressure waves with specific orientation
• Torque on nucleons trying to align them with external field
• Nucleons begin precessing toward alignment

Step 2: Atoms reorient

• Following their nucleon gyroscopes
• Bonding shells rotate (perpendicular to nucleon axes)
• Shell orientations changing throughout material

Step 3: Induced current opposes change

• Reoriented shells create new orbitron flow pattern
• This flow creates its own B-field (from current)
• The induced B-field opposes the increasing external B
• This is electromagnetic induction fighting the change

Mathematical Origin of Negative Sign

The negative sign comes from the cross product relationship in gyroscopic precession.

When torque τ acts on gyroscope with angular momentum L:

The precession is perpendicular to both τ and L, and the direction follows the right-hand rule.

When this precession changes atomic orientation, the induced effect (from geometric constraints) naturally opposes the causative change.

In vector calculus terms:

• Changing B creates torque in one direction
• Nucleon precession perpendicular to torque
• Atomic reorientation perpendicular to precession
• Induced E circulation perpendicular to reorientation
• Net result: induced effect opposes ∂B/∂t

Concrete Example: Solenoid

Setup

Consider a solenoid with:

• N turns of wire
• Radius R
• Length L
• Current I(t) increasing linearly

Inside solenoid:

where n = N/L (turns per unit length).

Time Derivative

This is uniform inside solenoid, zero outside.

Induced E-Field

By Faraday's law:

Using cylindrical symmetry, E-field forms circles around axis:

for r < R (inside solenoid).

AAM Interpretation

What's happening mechanically:

1. Increasing current in solenoid wire:

• Creates increasing B-field inside
• Pressure wave pattern strengthening

2. Atoms inside solenoid:

• Rotating nucleons experience increasing torque (from increasing B)
• Nucleons precess, changing orientation
• ∂B/∂t uniform → all nucleons changing at same rate

3. Spatial pattern creates circulation:

• At radius r from axis: certain precession rate
• Different r → different geometric relationship to axis
• This creates azimuthal pattern (circulating around axis)
• Bonding shells develop circular alignment pattern

4. Induced E-field:

• Circularly aligned shells create circular orbitron flow
• Flow opposes the change (Lenz's law)
• Measured as E_φ(r) ∝ r (linear with radius)

5. Why E ∝ r:

• Larger radius → longer path around circle
• More cumulative effect of aligned shells
• Total circulation increases linearly with area
• By geometry: E ∝ r

Verification

The induced E-field creates circulating current (if conducting loop present):

where induced EMF:

This induced current creates B-field opposing the increasing external B — exactly Lenz's law!

Quantitative Coefficient

Getting the Exact Relationship

We've established:

The proportionality constant should be exactly 1 (dimensionless).

Why Should It Be Unity?

This comes down to consistency with how we've defined E and B.

If we define:

• E as measuring bonding shell response in specific units (V/m)
• B as measuring nucleon rotation response in specific units (T)

Then the coupling constants α (for E) and β (for B) must be chosen such that:

exactly, with no additional numerical factor.

Dimensional Consistency Check

• [∇ × E] = [E]/[length] = (V/m)/m = V/m²
• [∂B/∂t] = [B]/[time] = T/s = (Wb/m²)/s = V/m²

The dimensions match!

This means the equation ∇ × E = -∂B/∂t is dimensionally consistent with coefficient of unity.

Physical Reasoning for Unity Coefficient

The coefficient being unity (no extra numerical factors) reflects:

1. Energy conservation: Induced E-field stores exactly the right energy to account for changing B-field
2. Geometric consistency: The perpendicular relationship between shells and nucleons is exact (90°)
3. Definition consistency: E and B defined to make this relation clean

In other words, we define the units of E and B such that this equation works with coefficient 1. The constants α and β absorb all the atomic-scale factors.

What Determines α and β?

The Coupling Constants

From our definitions:

The constants α and β must be chosen to:

1. Give correct dimensions for E (V/m) and B (T)
2. Make Faraday's law work with coefficient 1
3. Be consistent with measured values of ε₀ and μ₀

Relationship to ε₀ and μ₀

From Gauss's law: ∇ · E = ρ/ε₀

This means α must involve ε₀:

From Ampere's law: ∇ × B = μ₀J

This means β must involve μ₀:

Connection to Challenge 1.9

To fully determine α and β, we need:

• Derive ε₀ from bonding shell properties and aether coupling
• Derive μ₀ from nucleon rotation properties
• Show these give the right relationship for Faraday's law

This is why the challenges are coupled — we can't fully solve Maxwell's equations without deriving the fundamental constants.

Summary: Faraday's Law Derivation

Physical Mechanism

Mathematical Form

• Dimensional consistency: Both sides have units V/m²
• Coefficient unity: Follows from consistent definitions of E and B
• Vector relationship: Curl (circulation) on left, time derivative on right

Examples Verified

Solenoid:

• Increasing current → increasing B → induced circular E-field
• E_φ ∝ r (follows from geometry)
• Creates opposing current (Lenz's law)
• Quantitatively matches observed behavior

Transformer:

• Changing B in core → induced E in secondary coil
• Mechanism: precessing nucleons → reorienting shells → induced current
• Opposes change (Lenz's law)
• Standard electromagnetic induction

Confidence Assessment

Connections to Other AAM Principles

Related Axioms

• Axiom 1: All phenomena as space, matter, motion. Induction is wave motion affecting matter.
• Axiom 8: Constant motion. Dual valence cloud structure explains perpendicularity constraint.

Related Derivations

• Displacement Current: The symmetric reverse process — changing shells torque nucleons.
• Gauss's Law: Establishes the ε₀ constant needed for α.