CA Wave Equation – Charge Admittance

Temporal Structure of Electromagnetic Emission in CA: Wavelets in Energy Events

Abstract

This work proposes a refined mechanism for electromagnetic wave emission based on smooth charge acceleration constrained by the admittance of the vacuum field (as described by ε₀μ₀). The framework departs from mass-centered interpretations, focusing instead on temporally structured energy events. Each emission event is modeled as a smooth, bounded acceleration (or deceleration) process governed by jerk-limited motion — akin to a Taylor-series envelope or sinusoidal arc. The emission creates an asymmetric wavelet with a leading edge formed by near-field displacement (∼¼ wavelength prior to the main energy release) and a trailing edge defined by the field’s quality factor (Q). This model offers a physically grounded explanation for the structure of photon-like emissions and provides testable predictions for interactions with media and field discontinuities.

Introduction

In classical electrodynamics, radiation from an accelerating charge is well established. However, standard treatments often abstract away the fine temporal structure of the charge’s motion. Here, we propose a structured, time-asymmetric model for discrete electromagnetic emission events within the Charge Admittance (CA) framework — a causal interaction between charge motion and the vacuum field lattice characterized by ε₀ and μ₀.

The central insight is that energy release occurs through bounded, smooth transitions rather than idealized delta or sinusoidal functions. This nuance directly shapes wave morphology, the emission spectrum, and impedance-matched interactions.

Charge Motion and Wave Initiation

The emission process begins with a localized charge displacement — whether spontaneous (e.g., atomic decay) or externally driven (e.g., RF antenna). This displacement follows a smooth time-dependent trajectory minimizing jerk (the derivative of acceleration). A Taylor-series-derived or half-sinusoidal form ensures spectral compactness and coupling efficiency:

Generalized Position Equation (Extended Kinematics):

$x(t) = x_0 + v_0 t + \frac{1}{2} a t^2 + \frac{1}{6} j t^3 + \cdots$

Such motion promotes coherent energy transfer and minimal high-frequency sidebands.

$\vec{S}(r, t) = \frac{1}{4\pi r^2} \cdot \frac{1}{Z_0} \cdot \left[ \int_0^t \vec{a}(t') \, dt' \right]^2 \cdot \hat{r}$

Where:

$\vec{S}(r, t)$ — Poynting vector (radiated energy flux at distance $r$ and time $t$ )
$r$ — radial distance from the source (center of emission)
$t$ — observation time
$Z_0$ — impedance of free space $(\approx 376.73 \, \Omega)$
$\vec{a}(t')$ — time-dependent acceleration of the radiating charge
$\int_0^t \vec{a}(t') \, dt'$ — net velocity change (impulse) delivered to the charge over the interval
$\left[ \cdots \right]^2$ — energy radiated is proportional to the square of the momentum change
$\hat{r}$ — unit vector in the direction of propagation

Phase Timeline for a Single-Frequency Waveform / Pulse Envelope

Time 0 – Start of Emission:
• Charge acceleration begins (Taylor-like rise)
• Electric field $\vec{E}$ accelerates
• Magnetic field $\vec{B} \begins to form$

Phase 1 – First 1/4 Wavelength:
• Acceleration continues, smooth and band-limited
• $\vec{E}$ increases, $\vec{B}$ begins to rise
• Charge motion ends (no further energy input)

Phase 2 – Quarter to Half Wavelength:
• $\vec{B}$ peaks, lags $\vec{E}$ by $\frac{\pi}{2}$
• Poynting vector $\vec{S} = \frac{1}{\mu_0} \vec{E} \times \vec{B}$ emerges
• Fields propagate as a free wave

Phase 3 – Half to 3/4 Wavelength:
• Energy self-propagates in fields
• Fields begin decay without new input

Phase 4 – Final 1/4 Wavelength:
• Smooth decay of $\vec{E}$ , $\vec{B}$ (Taylor fall)
• Radiated energy decays via $\frac{1}{r^2}$ in $\vec{S}(r, t)$

Phase Relationships Summary

Component	Phase	Notes
$\vec{E}$	Leads $\vec{B}$ by $\frac{\pi}{2}$	Generated by charge acceleration
$\vec{B}$	Follows $\vec{E}$	Induced by moving charge
$\vec{S}$	Peaks when both $\vec{E}$ and $\vec{B}$ exist	Indicates power flow direction and amplitude
Charge Motion	Exists during first 1/4 λ	Only active during emission impulse
Pulse Envelope	1/4 λ rise + 1/4 λ fall	Sets the fundamental frequency of the photon

Propagation and Temporal Advance

The electromagnetic field responds with a wave that propagates at speed c=1/√ε₀μ₀. However, due to the smooth onset of the charge’s acceleration and the displacement current effects in the near field, a leading edge forms prior to the main energy transfer event, approximately ¼ wavelength ahead.

This early onset reflects the vacuum’s ability to respond causally to charge acceleration prior to peak energy injection.

Impedance Interaction and Wave Transfer

The interaction of this structured wavelet with media depends on the impedance differential between the propagation path and the interacting medium. For example:

Absorption: Occurs when the wave enters a medium of lower impedance, allowing current induction and energy transfer.
Reflection: Arises when the wave encounters higher impedance or open-field boundaries.
Phase Reversal: Happens at impedance mismatches where reactive components dominate, re-radiating energy with phase delay (∼π).

Experimental Implications

High-resolution experiments (e.g., single-photon interferometry, cavity QED) could detect asymmetric emission envelopes.
Field-based simulations using variable ε0(x,t)μ0(x,t) can model how field gradients influence the formation of leading and trailing wavelets.
Antenna design may benefit from this by shaping input pulses to match field admittance profiles for efficient radiation or absorption.

Conclusion

This CA framework reframes electromagnetic radiation as localized, asymmetric wavelet events shaped by charge motion and vacuum admittance. Rooted in Maxwell’s equations and informed by impedance-centric reasoning, it offers new insights into photon emission, radiative coupling, and wave-medium interactions, paving the way for advanced experimental and theoretical explorations.