Charge Distribution and Field Line Topology in Curved Vacuum Geometries

Abstract

This paper explores the behavior of electric charge and electromagnetic field lines in regions of space where vacuum permittivity and spacetime curvature vary due to the presence of localized energy densities, as proposed by the Electromagnetic Permittivity Variation and Orbital Dynamics (EPVOD) framework. We investigate how traditional models of charge distribution are modified when electric field propagation is constrained by harmonic vacuum gradients, and how these constraints influence the topology and stability of field lines. The result is a reformulated vision of electric fields, not as abstract vector fields in flat space, but as emergent structures guided by the geometric and harmonic characteristics of the vacuum itself.

1. Introduction

Conventional models of electromagnetism assume a uniform, isotropic vacuum where electric and magnetic fields extend from charges in a predictable topology. However, in the EPVOD model, vacuum permittivity varies locally due to deformation induced by concentrated energy, particularly near atomic nuclei. This alters the behavior of field lines, creating asymmetries and non-Euclidean topologies that diverge from classical Coulombic symmetry.

2. Charge as Curvature Source

In EPVOD, an electric charge is not a singularity in flat space, but a localized curvature feature in a harmonically deformed vacuum. This implies:

• Charge modifies local permittivity
• Field lines follow geodesics in curved permittivity space
• Electric potential is a scalar representation of harmonic strain in the vacuum

We define the modified Poisson equation:

$$
\nabla \cdot \left(\frac{1}{\epsilon(x)} \nabla \phi(x)\right) = -\frac{\rho(x)}{\epsilon_0}
$$

Where ( \epsilon(x) ) varies with vacuum deformation.

3. Field Line Behavior in Non-Uniform Vacuum

Traditional field lines radiate spherically from a point charge. In curved permittivity geometry:

• Field lines may bend, converge, or diverge depending on spatial permittivity gradients
• Equipotential surfaces become anisotropic and distorted
• Field strengths are enhanced or suppressed in regions of steep permittivity change

Visual analog: Field lines as streamlines in a refractive optical medium with spatially varying index.

4. Dipole and Multipole Extensions

Dipoles exhibit:

• Asymmetric lobe formation due to differential curvature around each pole
• Suppressed cancellation fields between poles if curvature is non-linear

Multipoles induce complex topological patterns in field line configurations, with emergent nodal structures reflecting underlying curvature harmonics.

5. Implications for Charge Mobility and Screening

Curved field topologies affect charge interactions:

• Coulomb force becomes directionally modulated by gradient tension
• Debye shielding in plasmas must account for curvature-modulated screening lengths
• Charge migration (e.g., conduction) may prefer lower-tension harmonic channels

6. Predictive Consequences

• Dielectric behavior in anisotropic materials may trace back to micro-curvature geometries
• Enhanced field strengths in vacuum cavities could explain high-voltage anomalies
• Atomic-scale charge distributions deviate from Gaussian profiles

7. Conclusion

Charge and its associated fields are emergent phenomena governed by the harmonic deformation of vacuum. The EPVOD framework reveals that electromagnetic interactions are shaped not merely by the presence of charge, but by the geometric response of space itself to energy density. This transforms field lines into geodesic indicators of vacuum tension, enabling deeper insight into electrostatics, material polarization, and the coupling between charge and spacetime.

Next Paper

Title: "Polarization, Dielectrics, and Field Memory in Harmonic Vacuum Substrates"