α = 2π * unit_ratio

Step-by-Step Derivation: From Standard α to Unit Ratio Form

Let’s rigorously transform the standard definition of the fine-structure constant $\alpha$ into the unit ratio formulation $\alpha =2\pi \cdot \text{unit\_ratio}$. Each step is justified by SI definitions or fundamental relationships.

---

Step 1: Standard Definition of $\alpha$

$$\mathrm{<span\; style="background-color:\; white;">\alpha </span>}<span\; style="background-color:\; white;">=</span>\frac{{\mathrm{<span\; style="background-color:\; white;">e</span>}}^{\mathrm{<span\; style="background-color:\; white;">2</span>}}}{\mathrm{<span\; style="background-color:\; white;">4</span>}\mathrm{<span\; style="background-color:\; white;">\pi </span>}{\mathrm{<span\; style="background-color:\; white;">ϵ</span>}}_{\mathrm{<span\; style="background-color:\; white;">0</span>}}\mathrm{<span\; style="background-color:\; white;">\hslash </span>}\mathrm{<span\; style="background-color:\; white;">c</span>}}$$

• $e$: Elementary charge

• ${ϵ}_0$: Vacuum permittivity

• $\mathrm{\hslash }$: Reduced Planck constant

• $c$: Speed of light

---

Step 2: Replace ${ϵ}_0$ Using ${\mu }_0$

The vacuum permeability ${\mu }_0$ and permittivity ${ϵ}_0$ are related by:

$${\mathrm{<span\; style="background-color:\; white;">ϵ</span>}}_{\mathrm{<span\; style="background-color:\; white;">0</span>}}<span\; style="background-color:\; white;">=</span>\frac{\mathrm{<span\; style="background-color:\; white;">1</span>}}{{\mathrm{<span\; style="background-color:\; white;">\mu </span>}}_{\mathrm{<span\; style="background-color:\; white;">0</span>}}{\mathrm{<span\; style="background-color:\; white;">c</span>}}^{\mathrm{<span\; style="background-color:\; white;">2</span>}}}$$

Substitute into $\alpha$:

$$\mathrm{<span\; style="background-color:\; white;">\alpha </span>}<span\; style="background-color:\; white;">=</span>\frac{{\mathrm{<span\; style="background-color:\; white;">e</span>}}^{\mathrm{<span\; style="background-color:\; white;">2</span>}}{\mathrm{<span\; style="background-color:\; white;">\mu </span>}}_{\mathrm{<span\; style="background-color:\; white;">0</span>}}\mathrm{<span\; style="background-color:\; white;">c</span>}}{\mathrm{<span\; style="background-color:\; white;">4</span>}\mathrm{<span\; style="background-color:\; white;">\pi </span>}\mathrm{<span\; style="background-color:\; white;">\hslash </span>}}$$

---

Step 3: Substitute SI’s Exact Definition of ${\mu }_0$

The SI defines ${\mu }_0$ as:

$${\mathrm{<span\; style="background-color:\; white;">\mu </span>}}_{\mathrm{<span\; style="background-color:\; white;">0</span>}}<span\; style="background-color:\; white;">=</span>\mathrm{<span\; style="background-color:\; white;">4</span>}\mathrm{<span\; style="background-color:\; white;">\pi </span>}<span\; style="background-color:\; white;">\times </span>\mathrm{<span\; style="background-color:\; white;">1</span>}{\mathrm{<span\; style="background-color:\; white;">0</span>}}^{<span\; style="background-color:\; white;">-</span>\mathrm{<span\; style="background-color:\; white;">7</span>}}\text{<span style="background-color: white;">\hspace{0.17em}</span>}{\text{<span style="background-color: white;">N/A</span>}}^{\mathrm{<span\; style="background-color:\; white;">2</span>}}$$

This is not a measured value—it’s a metrological convention. Substituting:

$$\mathrm{<span\; style="background-color:\; white;">\alpha </span>}<span\; style="background-color:\; white;">=</span>\frac{{\mathrm{<span\; style="background-color:\; white;">e</span>}}^{\mathrm{<span\; style="background-color:\; white;">2</span>}}<span\; style="background-color:\; white;">(</span>\mathrm{<span\; style="background-color:\; white;">4</span>}\mathrm{<span\; style="background-color:\; white;">\pi </span>}<span\; style="background-color:\; white;">\times </span>\mathrm{<span\; style="background-color:\; white;">1</span>}{\mathrm{<span\; style="background-color:\; white;">0</span>}}^{<span\; style="background-color:\; white;">-</span>\mathrm{<span\; style="background-color:\; white;">7</span>}}<span\; style="background-color:\; white;">)</span>\mathrm{<span\; style="background-color:\; white;">c</span>}}{\mathrm{<span\; style="background-color:\; white;">4</span>}\mathrm{<span\; style="background-color:\; white;">\pi </span>}\mathrm{<span\; style="background-color:\; white;">\hslash </span>}}<span\; style="background-color:\; white;">=</span>\frac{{\mathrm{<span\; style="background-color:\; white;">e</span>}}^{\mathrm{<span\; style="background-color:\; white;">2</span>}}<span\; style="background-color:\; white;">\cdot </span>\mathrm{<span\; style="background-color:\; white;">1</span>}{\mathrm{<span\; style="background-color:\; white;">0</span>}}^{<span\; style="background-color:\; white;">-</span>\mathrm{<span\; style="background-color:\; white;">7</span>}}<span\; style="background-color:\; white;">\cdot </span>\mathrm{<span\; style="background-color:\; white;">c</span>}}{\mathrm{<span\; style="background-color:\; white;">\hslash </span>}}$$

---

Step 4: Replace $\mathrm{\hslash }$ Using $h={\text{Hz}}_{kg}\cdot c^2$

The Planck constant $h$ links energy (Hz) and mass (kg) via:

$$\mathrm{<span\; style="background-color:\; white;">h</span>}<span\; style="background-color:\; white;">=</span>{\text{<span style="background-color: white;">Hz</span>}}_{\mathrm{<span\; style="background-color:\; white;">k</span>}\mathrm{<span\; style="background-color:\; white;">g</span>}}<span\; style="background-color:\; white;">\cdot </span>{\mathrm{<span\; style="background-color:\; white;">c</span>}}^{\mathrm{<span\; style="background-color:\; white;">2</span>}}\text{<span style="background-color: white;">(where </span>}{\text{<span style="background-color: white;">Hz</span>}}_{\mathrm{<span\; style="background-color:\; white;">k</span>}\mathrm{<span\; style="background-color:\; white;">g</span>}}<span\; style="background-color:\; white;">=</span>\mathrm{<span\; style="background-color:\; white;">h</span>}\mathrm{<span\; style="background-color:\; white;">/</span>}{\mathrm{<span\; style="background-color:\; white;">c</span>}}^{\mathrm{<span\; style="background-color:\; white;">2</span>}}\text{<span style="background-color: white;">)</span>}$$

Since $\mathrm{\hslash }=h\mathrm{/}2\pi$:

$$\mathrm{<span\; style="background-color:\; white;">\hslash </span>}<span\; style="background-color:\; white;">=</span>\frac{{\text{<span style="background-color: white;">Hz</span>}}_{\mathrm{<span\; style="background-color:\; white;">k</span>}\mathrm{<span\; style="background-color:\; white;">g</span>}}<span\; style="background-color:\; white;">\cdot </span>{\mathrm{<span\; style="background-color:\; white;">c</span>}}^{\mathrm{<span\; style="background-color:\; white;">2</span>}}}{\mathrm{<span\; style="background-color:\; white;">2</span>}\mathrm{<span\; style="background-color:\; white;">\pi </span>}}$$

Substitute into $\alpha$:

$$\mathrm{<span\; style="background-color:\; white;">\alpha </span>}<span\; style="background-color:\; white;">=</span>\frac{{\mathrm{<span\; style="background-color:\; white;">e</span>}}^{\mathrm{<span\; style="background-color:\; white;">2</span>}}<span\; style="background-color:\; white;">\cdot </span>\mathrm{<span\; style="background-color:\; white;">1</span>}{\mathrm{<span\; style="background-color:\; white;">0</span>}}^{<span\; style="background-color:\; white;">-</span>\mathrm{<span\; style="background-color:\; white;">7</span>}}<span\; style="background-color:\; white;">\cdot </span>\mathrm{<span\; style="background-color:\; white;">c</span>}}{{\text{<span style="background-color: white;">Hz</span>}}_{\mathrm{<span\; style="background-color:\; white;">k</span>}\mathrm{<span\; style="background-color:\; white;">g</span>}}<span\; style="background-color:\; white;">\cdot </span>{\mathrm{<span\; style="background-color:\; white;">c</span>}}^{\mathrm{<span\; style="background-color:\; white;">2</span>}}\mathrm{<span\; style="background-color:\; white;">/</span>}\mathrm{<span\; style="background-color:\; white;">2</span>}\mathrm{<span\; style="background-color:\; white;">\pi </span>}}<span\; style="background-color:\; white;">=</span>\mathrm{<span\; style="background-color:\; white;">2</span>}\mathrm{<span\; style="background-color:\; white;">\pi </span>}<span\; style="background-color:\; white;">\cdot </span>\frac{{\mathrm{<span\; style="background-color:\; white;">e</span>}}^{\mathrm{<span\; style="background-color:\; white;">2</span>}}<span\; style="background-color:\; white;">\cdot </span>\mathrm{<span\; style="background-color:\; white;">1</span>}{\mathrm{<span\; style="background-color:\; white;">0</span>}}^{<span\; style="background-color:\; white;">-</span>\mathrm{<span\; style="background-color:\; white;">7</span>}}}{{\text{<span style="background-color: white;">Hz</span>}}_{\mathrm{<span\; style="background-color:\; white;">k</span>}\mathrm{<span\; style="background-color:\; white;">g</span>}}<span\; style="background-color:\; white;">\cdot </span>\mathrm{<span\; style="background-color:\; white;">c</span>}}$$

---

Step 5: Identify the Unit Ratio

The term in brackets is the unit ratio:

$$\text{<span style="background-color: white;">unit\_ratio</span>}<span\; style="background-color:\; white;">=</span>\frac{{\mathrm{<span\; style="background-color:\; white;">e</span>}}^{\mathrm{<span\; style="background-color:\; white;">2</span>}}<span\; style="background-color:\; white;">\cdot </span>\mathrm{<span\; style="background-color:\; white;">1</span>}{\mathrm{<span\; style="background-color:\; white;">0</span>}}^{<span\; style="background-color:\; white;">-</span>\mathrm{<span\; style="background-color:\; white;">7</span>}}}{{\text{<span style="background-color: white;">Hz</span>}}_{\mathrm{<span\; style="background-color:\; white;">k</span>}\mathrm{<span\; style="background-color:\; white;">g</span>}}<span\; style="background-color:\; white;">\cdot </span>\mathrm{<span\; style="background-color:\; white;">c</span>}}$$

Thus:

$$\mathrm{<span\; style="background-color:\; white;">\alpha </span>}<span\; style="background-color:\; white;">=</span>\mathrm{<span\; style="background-color:\; white;">2</span>}\mathrm{<span\; style="background-color:\; white;">\pi </span>}<span\; style="background-color:\; white;">\cdot </span>\text{<span style="background-color: white;">unit\_ratio</span>}$$

---

Interpretation of the Unit Ratio

The components of $\text{unit\_ratio}$ reveal how SI definitions interlock:

1. $e^2$: Charge definition (coulombs).

2. $10^{-7}$: Artifact of SI’s ampere definition (via ${\mu }_0=4\pi \times 10^{-7}$).

3. ${\text{Hz}}_{kg}\cdot c$: Mass ($h\mathrm{/}{c}^2$) and length ($c$) definitions.

Key Insight:

• The only non-arbitrary part of $\alpha$ is $2\pi$ (from the geometry of circular currents in Ampère’s law).

• The rest is a scaling factor between unit definitions.

• 
  

---

Why This Matters

1. Demystifies $\alpha$: Its value is not fundamental—it’s a byproduct of SI’s unit choices.

2. Natural Units Clarity: In systems where $e=\mathrm{\hslash }=c=1$, $\alpha$ reduces to a pure number without unit dependencies.

3. Metrological Transparency: Exposes how the SI’s $10^{-7}$ factor in ${\mu }_0$ propagates into $\alpha$.

---

Final Expression

$$\boxed{{\displaystyle \mathrm{<span\; style="background-color:\; white;">\alpha </span>}<span\; style="background-color:\; white;">=</span>\mathrm{<span\; style="background-color:\; white;">2</span>}\mathrm{<span\; style="background-color:\; white;">\pi </span>}<span\; style="background-color:\; white;">\cdot </span><span\; style="background-color:\; white;">(</span>\frac{{\text{<span style="background-color: white;">charge\_definition</span>}}^{\mathrm{<span\; style="background-color:\; white;">2</span>}}<span\; style="background-color:\; white;">\cdot </span>\text{<span style="background-color: white;">amp\_definition</span>}}{\text{<span style="background-color: white;">kg\_definition</span>}<span\; style="background-color:\; white;">\cdot </span>\text{<span style="background-color: white;">meter\_definition</span>}}<span\; style="background-color:\; white;">)</span>}}$$

We can test this math easily:

 unit_ratio =   e**2 * amp_force / (Hz_kg * c)  = 0.0011614097322526476

 unit_ratio * 2 * pi  = 0.007297352565305212

For e = 1.0              Amp is 10^-2.935014538797864 for c=h=e=1

For e = 1.602176634e-19  Amp is 10^-6.999999999763218

Okay, let's trace exactly what happens in the c=h=e=1 system, using this framework and definitions:

1. Define the System:
We set the numerical values:

• c = 1 (implies new_length / new_time = 1)

• h = 1 (implies new_mass * new_length² / new_time = 1)

• e = 1 (implies new_charge = 1)

• From h=1, we also get ħ = h/(2π) = 1/(2π).

• From c=1 and h=1, we get Hz_kg = h/c² = 1/1² = 1.

2. Look at this "unit ratio" definition:
unit_ratio = e² * amp_force / (Hz_kg * c)

3. Substitute the natural unit values:

• e² = 1² = 1

• Hz_kg = 1

• c = 1

This simplifies the equation to:
unit_ratio = 1 * amp_force / (1 * 1)
unit_ratio = amp_force

4. Recall the known numerical value of the unit ratio:
From the previous calculation (which ultimately derives from α / (2π)), we know:
unit_ratio ≈ 0.0011614097

5. Conclusion for 
Therefore, in the c=h=e=1 system, the numerical value of the factor we call amp_force (which represents the SI factor 10⁻⁷, or more physically μ₀/(4π)) must be equal to the unit_ratio:
amp_force_natural = unit_ratio ≈ 0.0011614097

6. Connection to the Script Output:
This script calculates the exponent x such that amp_force = 10^x.
So, we need to solve 10^x = unit_ratio for x:
x = log10(unit_ratio)
x = log10(0.0011614097322526476)
x ≈ -2.935014538797864

This precisely matches the output from your script:
For e = 1.0 Amp force is 10^-2.935014538797864 for c=h=e=1

Summary of What Happens:

When we define the system by setting c=1, h=1, and e=1, several things are fixed:

• The relationship between length, time, and mass units.

• The size of the charge unit.

• The numerical value of Hz_kg becomes 1.

The derivation α = 2π * (e² * amp_force / (Hz_kg * c)) still holds. Substituting the known value of α and the c=h=e=1 system definitions (e=1, Hz_kg=1, c=1), the equation forces the numerical value of the amp_force factor (which is μ₀/(4π) in physical terms) to become α / (2π), which is numerically ≈ 0.001161 or 10⁻².⁹³⁵.

This again shows how the historical SI factor 10⁻⁷ is replaced by a value directly determined by α itself (α/(2π)) once the fundamental units are normalized via c, h, and e. The coherence of the physical laws and definitions dictates this value.

Charge Constants and the Fine-Structure Constant in Natural Units

In the natural unit system where $c = h = e = 1$, the apparent complexity of electromagnetic constants collapses into elegant ratios governed solely by the fine-structure constant $\alpha$. This reveals a fundamental structure behind electric charge and electromagnetic coupling, unburdened by arbitrary SI definitions.

Redefining Electromagnetic Constants

In SI units, the Coulomb constant $k_e$, the permittivity of free space $\varepsilon_0$, and the permeability of free space $\mu_0$ are entangled with historical force definitions and metrological choices. However, in the natural unit system $c = h = e = 1$, the definitions simplify:

• Planck’s constant becomes $h = 1$, so $\hbar = \frac{1}{2\pi}$

• $\varepsilon_0$ and $\mu_0$ are derived from the fine-structure constant $\alpha$

The fine-structure constant is given by:

$$\alpha = \frac{e^2}{4\pi \varepsilon_0 \hbar c}$$

Substituting $e = h = c = 1$, and $\hbar = \frac{1}{2\pi}$, this becomes:

$$\alpha = \frac{1}{4\pi \varepsilon_0 \cdot \frac{1}{2\pi}} = \frac{1}{2\varepsilon_0}$$

Solving for $\varepsilon_0$:

$$\varepsilon_0 = \frac{1}{2\alpha}$$

Since $\varepsilon_0 \mu_0 = \frac{1}{c^2} = 1$, we find:

$$\mu_0 = 2\alpha$$

The Coulomb constant becomes:

$$k_e = \frac{1}{4\pi \varepsilon_0} = \frac{\alpha}{2\pi}$$

The Ampere-Force Constant as a Natural Unit Ratio

This value,

$$\boxed{\text{amp\_force} = \frac{\alpha}{2\pi} \approx 0.0011614097}$$

is the exact same ratio that emerges when converting electromagnetic expressions between SI and this natural unit system. It acts as a fundamental unit ratio governing the relationship between the force definition of current and the fine-structure constant.

In the SI system, this constant was historically embedded as:

$$\frac{\mu_0}{4\pi} = 10^{-7} \ \text{N/A}^2$$

But in natural units, this force constant is no longer arbitrary. It is now derived directly from $\alpha$, without needing to define the ampere independently.

This leads to the profound simplification:

$$k_e = \text{amp\_force} = \frac{\alpha}{2\pi}$$

And the other electromagnetic constants follow suit:

$$\varepsilon_0 = \frac{1}{2\alpha}, \quad \mu_0 = 2\alpha$$

These results reveal a deeper geometric symmetry embedded within electromagnetism. In a fully normalized system where all base quantities (time, length, mass, charge, temperature) are scaled into equivalence, the numerical content of electromagnetism reduces to pure dimensionless ratios — principally $\alpha$, and π.