Why Does Magnetic Interaction Arise?

01Keep motion and the static baseline in one comparison

The magnetic question begins with a specified change: hold charges and reference transverse separation fixed while giving both charges a common motion. Charges $q_1,q_2$ move with the same constant parallel velocity $\mathbf v$, their separation perpendicular to it, relative to a local homogeneous spatial background. Let $r>0$, let $\hat{\mathbf r}$ point from $q_1$ to $q_2$, and write $\beta=v/c$, $\gamma=(1-\beta^2)^{-1/2}$.

The Coulomb page fixes the exterior inverse-square law, source strengths and direction. Finite size, external gradients and boundary corrections remain neglected. $\mathbf F_0$ is the static baseline at the same charges and reference separation; $\mathbf F_T$ is the interaction paired with true physical displacement. In the magnitude formulas below, $F_0,F_T$ are nonnegative.

02How transverse distance enters the inverse-square interaction

LVE gives $K(\phi)=\gamma(1-\beta\cos\phi)$. At transverse angle $\phi=\pi/2$, $K=\gamma$, hence:

Now use Chapter 3's constitutive input: $r_T$ is the actual Coulomb interaction distance. Inserting it into $F_T=|q_1q_2|/(4\pi\varepsilon_0r_T^2)$ changes the denominator to $4\pi\varepsilon_0\gamma^2r^2$. Factoring out the static baseline and using $\gamma^{-2}=1-\beta^2$ yields:

The moving interaction still lies along the charge separation, but its strength changes. Subtract the same baseline using signed vectors:

This is how ASD organizes magnetic interaction in the specified configuration. $\Delta\mathbf F_m$ is the correction relative to the static baseline, while $\mathbf F_T$ is the total force. Adding the correction again to an interaction already calculated at $r_T$ would double-count the scale effect.

For like charges $\mathbf F_0$ points outward and the correction inward, reducing repulsion. For unlike charges the baseline points inward and the correction outward, reducing the attraction magnitude. Define the positive reduction separately:

$F_m^{\mathrm{red}}$ is a magnitude difference, not the directed vector $\Delta\mathbf F_m$. Using SI electromagnetic normalization $\varepsilon_0\mu_0=c^{-2}$ gives the equivalent correction:

$\mu_0$ is the vacuum permeability. This rewrites normalization without introducing another basic magnetic force law. The product relation does not determine the microscopic origin of either constant.

03Why this result needs no low-speed truncation

The length relation uses the complete $\gamma$, followed by an exact identity. The correction is exactly proportional to $\beta^2$; calling it second order does not mean that higher terms in a longer series were discarded. It applies for the stated constitutive law and common constant-velocity, parallel, transverse configuration throughout $0\leq v<c$.

As $v\to0$, $r_T\to r$, the interaction returns to the baseline and the correction vanishes. At fixed nonzero $r$ and charges, $v\to c^-$ gives $r_T\to\infty$, $F_T/F_0\to0$ and a reduction approaching $F_0$. These are limits of an ideal configuration, not a complete description of charges translating at $c$ or finite apparatuses. Unequal, opposite, oblique and accelerating velocities require a general source and response model.

The figure compares normalized like-charge interactions; lengths and arrows are schematic. Its velocity control tests the algebra, not experimental data. The later slow-source near-zone potential approximation does not retroactively make this configuration-specific result a low-speed truncation.

04True force and coordinate generalized force in the same work

An experimental comparison must specify which displacement is paired with force. At fixed common velocity, $r_T=\gamma r$ implies $dr_T=\gamma\,dr$. With interaction energy $U_T=q_1q_2/(4\pi\varepsilon_0r_T)$, virtual work in the two variables is:

In this virtual-work relation $F_T,F_0,F_r$ may be signed radial components along $\hat{\mathbf r}$; for like charges they also equal the above magnitudes. $F_T=-\partial U_T/\partial r_T$ pairs with true displacement; $F_r=-\partial U_T(\gamma r)/\partial r$ pairs with reference displacement. The chain rule gives $F_r=\gamma F_T$ rather than another physical interaction.

Chapter 2 relates force to mechanical momentum change and work–energy. Physical displacement, mechanical momentum and process parameter must be paired within the same representation; reference generalized force pairs with its coordinate's conjugate momentum. Trajectory curvature, bunch momentum exchange, support reaction or electro-optic signals each require an apparatus mapping. An instrument reading cannot simply be declared to be $F_T$ or $F_r$. Acceleration also requires the velocity dependence of scale and accompanying-field energy exchange.

05A same-configuration comparison retains quantity definitions

Introduce standard Maxwell–Lorentz electrodynamics separately as the comparison theory. For like charges with the same common velocity and transverse separation, its laboratory electric force, positive magnetic reduction and total transverse force are:

These relations were not premises of the ASD derivation. ASD's true force is $F_0/\gamma^2$ and its coordinate generalized force is $F_0/\gamma$. The latter matches the standard laboratory total numerically, but pairs with different variables. A low-speed expansion shows:

Matching the leading magnetic reduction is not matching the complete force at that order: the standard electric-field correction also contributes. ASD's subsequent quasistatic potential and minimal coupling reproduce the magnetic leading term without completing a general source coupling consistent with the true total force. Detectable differences require both models to calculate the same apparatus, reading and measurement process. The experimental matrix and high-speed discriminator task retain this requirement.