MHB Coulomb's Constant in electron energy formula.

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The discussion centers on the inclusion of Coulomb's constant, represented as \( \frac{1}{(4\pi\epsilon_0)^2} \), in the electron energy formula for the nth Bohr orbit. The formula \( E_n = -\frac{2\pi^2me^4Z^2}{n^2h^2} \) is derived by equating centripetal and electric forces, while incorporating quantum mechanical principles. The presence of \( (4\pi\epsilon_0)^2 \) arises from the mathematical manipulation of the equations governing electric and kinetic energy in the context of the Bohr model. This constant is essential for accurately reflecting the electrostatic interactions in atomic systems. Ultimately, the discussion emphasizes the importance of Coulomb's constant in deriving the energy levels of electrons in hydrogen-like atoms.
WMDhamnekar
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Hi,
If we multiply $En=-\frac{2\pi^2me^4Z^2}{ n^2h^2} $by $\frac{1}{(4\pi\epsilon_0)^2},$ it is the formula of electron energy in nth Bohr’s orbit. Why we should multiply it by $\frac{1}{ (4\pi\epsilon_0)^2}$ a Coulomb's constant in electrostatic force?

Where m=mass of electron, e= charge on electron h=Plank's constant, n=principal quantum number, Z= atomic mass number of element (Bohr'theory can only be applied to ions containing only one electron.$e.g. He^+, Li^{2+}, Be^{3+} $etc.
 
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Bohr's model is a classical mechanical model that treats the electron as a point particle that orbits the nucleus. Additionally it has the quantum rule that the angular momentum $L$ must be an integer multiple of $\hbar$.
Consequently we have:
\[ \begin{cases} F_{centripetal} = F_{electric} \\ E_{total} = E_{electric} + E_{kinetic} \\ L = n\hbar \end{cases} \implies
\begin{cases} \frac{m v^2}{r} = \frac{Ze\cdot e}{4\pi\epsilon_0 r^2} \\ E_n = -\frac{Ze\cdot e}{4\pi\epsilon_0 r} + \frac 12 m v^2 \\ m v r = n\hbar \end{cases} \]
Now eliminate $v$ and $r$ from those equations.

The result is:
$$E_n = -\frac 12 \frac{Z^2e^4 m}{(4\pi \epsilon_0)^2 n^2 \hbar^2}
=-\frac{2\pi^2Z^2e^4 m}{(4\pi \epsilon_0)^2 n^2 h^2}$$

Why do we see $(4\pi\epsilon_0)^2$ in this formula?
As I see it, it's the consequence of combining the given formulas that happen to contain some squares.
 
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