Simplification of Planck's Charge:

The value of $\epsilon_0$ (vacuum permittivity) is:

$$\epsilon_0 = 8.8541878128 \times 10^{-12} \, \text{C}^2 \, \text{N}^{-1} \, \text{m}^{-2}$$

You want to solve for $A$ in the equation:

$$\epsilon_0 = \frac{A^2}{\alpha^3 \cdot \beta}$$

Rearrange to Solve for $A$:

$$A^2 = \epsilon_0 \cdot \alpha^3 \cdot \beta$$ $$A = \sqrt{\epsilon_0 \cdot \alpha^3 \cdot \beta}$$

Substituting Values:

• $\epsilon_0 = 8.8541878128 \times 10^{-12} \, \text{C}^2 \, \text{N}^{-1} \, \text{m}^{-2}$
• $\alpha = 1.53843951260968407858 \times 10^{-6} \, \text{m}$
• $\beta = 5.45551124829157414485 \times 10^{-8} \, \text{kg}$

First, calculate $\alpha^3$:

$$\alpha^3 = (1.53843951260968407858 \times 10^{-6})^3 = 3.63706020404 \times 10^{-18} \, \text{m}^3$$

Now substitute everything into the formula for $A$:

$$A = \sqrt{(8.8541878128 \times 10^{-12}) \cdot (3.63706020404 \times 10^{-18}) \cdot (5.45551124829157414485 \times 10^{-8})}$$Perform the Calculation:

1. Multiply the terms inside the square root:

$$8.8541878128 \times 10^{-12} \cdot 3.63706020404 \times 10^{-18} \cdot 5.45551124829157414485 \times 10^{-8}$$
$$=1.760444245\times 10^{-37}$$

2. Take the square root:

$$A = \sqrt{1.760444245 \times 10^{-37}} = 1.32767 \times 10^{-18}$$

Result:

$$A\approx 1.32767\times 10^{-18}\text{(unit depends on scaling context of }A\text{)}\mathrm{.}$$

Formula for Planck's Charge:

Planck's Charge = sqrt(2 * (alpha^3 * beta * epsilon_0))

Substitute epsilon_0 = (delta^2) / (alpha^3 * beta):

Planck's Charge = sqrt(2 * (alpha^3 * beta * (delta^2) / (alpha^3 * beta)))

Simplify:

1. The terms (alpha^3 * beta) cancel out in the numerator and denominator:

   Planck's Charge = sqrt(2 * delta^2)

2. Simplify further:

   Planck's Charge = delta * sqrt(2)

Final Result:

Planck's Charge = delta * sqrt(2)

Numerical Substitution:

Given delta = 1.32767e-18,

Planck's Charge = (1.32767e-18) * sqrt(2)

Planck's Charge = (1.32767e-18) * 1.414213562

Planck's Charge ≈ 1.87764e-18

Interpretation:

The formula simplifies neatly, showing that charge_calc is proportional to delta,

with a scaling factor of sqrt(2). This result reinforces the geometric relationships

in your framework.