unit dimension analysis of the unit ratio in alpha and charge formulas

 The derivation shows:

$$\mathrm{<span>\alpha </span>}<span>=</span>\mathrm{<span>2</span>}\mathrm{<span>\pi </span>}<span>\cdot </span>\frac{{\mathrm{<span>e</span>}}^{\mathrm{<span>2</span>}}<span>\cdot </span>\text{<span>amp\_force</span>}}{\text{<span>Hz\_kg</span>}<span>\cdot </span>\mathrm{<span>c</span>}}$$

Here’s the physical interpretation of each term:

Term	Physical Meaning	Units
$e^2$	Squared elementary charge	C²
$\text{amp\_force}$	Macroscopic force anchor	N/A²
$\text{Hz\_kg}$	Quantum mass-frequency coupling	kg/Hz
$c$	Speed of light	m/s

The units cancel out to leave α dimensionless:

$$\frac{<span>[</span>{\mathrm{<span>C</span>}}^{\mathrm{<span>2</span>}}<span>]</span><span>\cdot </span><span>[</span>\mathrm{<span>N</span>}\mathrm{<span>/</span>}{\mathrm{<span>A</span>}}^{\mathrm{<span>2</span>}}<span>]</span>}{<span>[</span>\mathrm{<span>k</span>}\mathrm{<span>g</span>}\mathrm{<span>/</span>}\mathrm{<span>H</span>}\mathrm{<span>z</span>}<span>]</span><span>\cdot </span><span>[</span>\mathrm{<span>m</span>}\mathrm{<span>/</span>}\mathrm{<span>s</span>}<span>]</span>}<span>=</span>\frac{<span>(</span>{\mathrm{<span>A</span>}}^{\mathrm{<span>2</span>}}{\mathrm{<span>s</span>}}^{\mathrm{<span>2</span>}}<span>)</span><span>\cdot </span><span>(</span>\mathrm{<span>k</span>}\mathrm{<span>g</span>}<span>\cdot </span>\mathrm{<span>m</span>}\mathrm{<span>/</span>}{\mathrm{<span>s</span>}}^{\mathrm{<span>2</span>}}\mathrm{<span>/</span>}{\mathrm{<span>A</span>}}^{\mathrm{<span>2</span>}}<span>)</span>}{<span>(</span>\mathrm{<span>k</span>}\mathrm{<span>g</span>}\mathrm{<span>/</span>}{\mathrm{<span>s</span>}}^{<span>-</span>\mathrm{<span>1</span>}}<span>)</span><span>\cdot </span><span>(</span>\mathrm{<span>m</span>}\mathrm{<span>/</span>}\mathrm{<span>s</span>}<span>)</span>}<span>=</span>\text{<span>dimensionless</span>}$$

Definitions of Key Electromagnetic Constants

1. Vacuum Permeability (${\mu }_0$)

• Definition: Magnetic constant defining the strength of the magnetic field in a vacuum.

• SI Value: ${\mu }_0=4\pi \times {10}^{-7}\mathrm{N}\mathrm{/}{\mathrm{A}}^2\approx 1.256637\times {10}^{-6}\mathrm{H}\mathrm{/}\mathrm{m}$

• Role: Appears in Ampère’s force law and Maxwell’s equations. Fixed in SI since 2019 via the ampere’s definition14.

2. Ampère Force ($\mathrm{a}\mathrm{m}\mathrm{p}\mathrm{\_}\mathrm{f}\mathrm{o}\mathrm{r}\mathrm{c}\mathrm{e}$)

• Definition: Force per unit length between two parallel current-carrying wires in a vacuum.

• SI Formula:
  $\frac{F}{L}=\frac{{\mu }_0{I}_1{I}_2}{2\pi r}$
  For $I_1=I_2=1\mathrm{A}$ and $r=1\mathrm{m}$:
  $\frac{F}{L}=2\times {10}^{-7}\mathrm{N}\mathrm{/}\mathrm{m}$ (defines the ampere)14.

3. Vacuum Permittivity (${ϵ}_0$)

• Definition: Electric constant governing the strength of the electric field in a vacuum.

• SI Value:
  ${ϵ}_0=\frac{1}{{\mu }_0{c}^2}\approx 8.854188\times {10}^{-12}\mathrm{F}\mathrm{/}\mathrm{m}$
  Derived from ${\mu }_0$ and the speed of light $c$3.
  ϵ0= 1/ (4 π 10^-7 N/A^2 * c^2)

4. Impedance of Free Space ($Z_0$)

• Definition: Ratio of electric to magnetic field magnitudes in a vacuum for electromagnetic waves.

• Formula:
  $Z_0={\mu }_{0}c=\sqrt{\frac{{\mu }_0}{{ϵ}_0}}\approx 376.730\mathrm{\Omega }$

• Z0=μ0c= 4 π 10^-7 N/A^2 *c 
  Exact in SI due to fixed ${\mu }_0$ and $c$3.

5. Coulomb’s Constant ($k_e$)

• Definition: Proportionality constant in Coulomb’s law for electric force.

• Formula:
  $k_e=\frac{1}{4\pi {ϵ}_0}\approx 8.988\times {10}^9\mathrm{N}\cdot {\mathrm{m}}^2\mathrm{/}{\mathrm{C}}^2$

  
  

  = 1/ (4 π  1/(4 π 10^-7 N/A^2 * c^2 )) = 10^-7 N/A^2 * c^2

• Related to ${ϵ}_0$ and the elementary charge $e$2.

6. Elementary Charge ($e$)

• Definition: Fundamental charge of a proton ($+e$) or electron ($-e$).

• SI Value:
  $e=1.602176634\times {10}^{-19}\mathrm{C}$ (exact since 2019 SI redefinition)25.

• 6241509074460762608 individual charges in a C

7. Fine-Structure Constant ($\alpha$)

• Definition: Dimensionless constant characterizing electromagnetic interaction strength.

• Formula:
  $\alpha =\frac{e^2}{4\pi {ϵ}_0\mathrm{\hslash }c}\approx \frac{1}{137}$
  Combines $e$, ${ϵ}_0$, $\mathrm{\hslash }$, and $c$25.

Relationships Between Constants

Constants	Relationship
${\mu }_0$ & ${ϵ}_0$	${ϵ}_0=\frac{1}{{\mu }_0{c}^2}$
$Z_0$ & ${\mu }_0$	$Z_0={\mu }_{0}c$
$k_e$ & ${ϵ}_0$	$k_e=\frac{1}{4\pi {ϵ}_0}$
$\alpha$ & $e$	$\alpha =\frac{e^2}{4\pi {ϵ}_0\mathrm{\hslash }c}$

Note: The 2019 SI redefinition fixed $e$ and $\mathrm{\hslash }$, making ${\mu }_0$ and ${ϵ}_0$ derived constants