Coulomb Interaction: Distances and Physical Quantities

01Fix a static interaction that can be compared

Before asking how magnetic interaction appears with motion, specify the baseline without common motion. Consider exterior charges $q_1,q_2$ in a local homogeneous background, separated by reference coordinate distance $r>0$. Neglect finite charge size, external field gradients and boundary corrections. The static Coulomb interaction magnitude is:

$F_0\geq0$ is a magnitude, and $\varepsilon_0$ is the vacuum permittivity in the electromagnetic normalization. The sign of $q_1q_2$ determines direction separately: like charges repel and unlike charges attract. Charge is a normalized source-strength input at this level. The origin of its sign and quantization belongs to the subsequent matter module.

The reference distance is the coordinate input for this comparison. Relating it to laboratory rods, optical paths or trajectories requires a background, standards and calibration. ASD's true length is the spatial length assigned by the model to the relevant physical process; its name does not turn it into a direct reading independent of a measurement model.

02Which quantity changes under common motion?

Give both charges the same constant parallel velocity $\mathbf v$, with their separation perpendicular to it. Velocity is defined relative to the local homogeneous spatial background. With characteristic speed $c$, write $\beta=v/c$ and $\gamma=(1-\beta^2)^{-1/2}$. The comparison fixes $q_1,q_2$ and reference transverse separation $r$, changing their common motion state.

Upstream LVE connects reference and true lengths through $K(\phi)=\gamma(1-\beta\cos\phi)$. The angle between the directed length and motion is $\phi=\pi/2$ here, giving the distance used in this chapter's interaction law: $r_T=\gamma r$. This specifies ASD interaction geometry; it does not assert that every apparatus reading grows by $\gamma$.

Geometry alone is insufficient to specify force. Chapter 3 separately chooses the true distance as the actual interaction distance and applies the exterior inverse-square law there. Let $\hat{\mathbf r}$ point from $q_1$ to $q_2$ and retain the charge signs in vectors:

$\mathbf F_T$ is the true physical force supplied by this constitutive law; $\mathbf F_0$ is the static baseline at the same source strengths and reference separation. Their magnitudes are $F_T,F_0$. True-distance inverse-square interaction is the dynamical input, while LVE is the geometrical input. The former is not proved by the definition of the length unit. At rest $r_T=r$, recovering the same interaction.

03Pair potential energy with its differentiation variable

An interaction potential represents the inverse-square law. Fix the motion state and charges and vary the true radial distance. Taking the zero at infinity gives:

$U_T$ is measured in joules and $r_T$ in metres. In this potential relation, scalar $F_T$ denotes the signed radial component along $\hat{\mathbf r}$: positive for $q_1q_2>0$ and negative for $q_1q_2<0$. Its absolute value is the interaction magnitude above. Differentiating $1/r_T$ introduces a minus sign which cancels the one in $-\partial U_T/\partial r_T$, recovering the vector interaction law.

If the same energy is written as $U_T(\gamma r)$ and differentiated with respect to reference coordinate $r$, the chain rule introduces a further $\gamma$. This gives the coordinate generalized force paired with $dr$, not another physical force that can replace $F_T$. At fixed common velocity $dr_T=\gamma\,dr$, so virtual work must agree in both representations. The magnetic-force virtual-work relation develops this conversion alongside the motion-dependent correction.

04Using these inputs in the magnetic calculation

Charges, reference separation, true interaction distance, force direction and potential energy are now specified. The next calculation needs no magnetic field premise: insert $r_T=\gamma r$ into the inverse-square law, then subtract the same static baseline to obtain the constitutive model's magnetic correction.

These inputs apply to the common constant-velocity, transverse exterior configuration. Acceleration, finite bunch size, external boundaries and different source velocities require interaction momentum, field exchange and measurement response. Using the same charge unit does not supply an otherwise unspecified apparatus formula.