Why charges pull, and push

A photon is the smallest piece of light; this page follows one photon closed into a loop, and a second one standing in its wave. A charged loop fills the space around it with its null gravity wave, spreading at c and fading with the density deficit δ. A second loop stands in that wave with its two sides at different depths: the near side always rides a stronger wave than the far side, so the near side sets the size of the push or pull. Which way it goes is the facing: when the two loops turn opposite ways they pull together; when they turn the same way they push apart. Each particle’s boundary is painted by route length: darkest green at the point nearest the source, darkest yellow at the farthest, pale at the perpendicular crossover where the two sides are equal.

δ = 4.2×10⁻³δ = 3.3×10⁻³δ = 4.2×10⁻³δ = 3.3×10⁻³net drift: pullnet drift: pushopposite winding — toward the source: pullsame winding — away from the source: pushthe source loop and its null gravity waveR = 16.5 ƛₑ from the source1 ƛₑ = 386 fm (drawn radius)
δ = ƛe2/(R2 + ƛe2) → ƛe2/R2
(far from the loop, the inverse square; the wave fades with it)
Q ∝ ∮dφ = 2πn,  so  q = n·q0,  σ = ±1
(n the count of aligned turns per cycle; σ which way the loop turns)
Fσ1σ2 / r2,  σi ∈ {+1, −1}
(opposite facing pulls, same facing pushes; the size from the wave, the sign from the two orientations)
F = kq1q2 / r2,  E = kq / r2,  F = qE
(the same line in the textbook’s letters, with q = σe: Coulomb’s law and the field of a point charge; k = 1/(4πε0) is a lock on the scale, not an input)

Every element computed. The wave is the source loop’s null gravity wave, drawn by the same rule as the previous page (release front dr/du = 1/(1−δ), hide front dr/du = 1, the drawing’s own; contrast eased as √δ for printability); the base shading runs to the page edge. All three loops are at true size, radius ƛₑ. Both particles stand R = 16.5 ƛₑ from the source. The paired arrows are the local wave strength at each particle’s two edges, δ = 4.2×10⁻³ near against 3.3×10⁻³ far, a ratio of 1.27, on one scale along the loop to source line, each pointing in that side’s drift direction. The facing sets the sign, opposite windings toward, same windings away (Mathematical Bridge, section 4); the near side, on the stronger wave, sets the size. Boundary hue runs by the angle to the source line, ordered by computed route length and normalized per loop: darkest green at the nearest point (shortest routes, drawn heavier), darkest yellow at the farthest, pale at the two perpendicular points, which stand at equal distance from the source; the printed δ values carry the magnitudes. The net-drift arrows are summary indicators; the edge gauges lie on the true loop to source line. The impacted particles’ own waves are omitted for clarity.