# Coulomb's Constant in electron energy formula.

• MHB
• WMDhamnekar
In summary, the given formulas for electron energy and Bohr's model involve Coulomb's constant and the quantum rule for angular momentum. By eliminating the variables, the resulting formula for electron energy includes the term $(4\pi\epsilon_0)^2$, which is a consequence of combining the given formulas.
WMDhamnekar
MHB
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.

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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## What is Coulomb's Constant in electron energy formula?

Coulomb's Constant, also known as the electric constant, is a proportionality constant that relates the strength of the electric force between two charged particles to the distance between them. It is denoted by the symbol k and has a value of approximately 8.99 x 10^9 Nm^2/C^2.

## How is Coulomb's Constant used in the electron energy formula?

In the electron energy formula, Coulomb's Constant is used to calculate the potential energy of an electron in an electric field. The formula is given by U = -kQq/r, where U is the potential energy, k is Coulomb's Constant, Q and q are the charges of the two particles, and r is the distance between them.

## What is the significance of Coulomb's Constant in the study of electromagnetism?

Coulomb's Constant is a fundamental constant in the study of electromagnetism. It plays a crucial role in understanding the behavior of electric charges and their interactions. It is also used in various equations and laws, such as Coulomb's Law and the electric field equation.

## How is Coulomb's Constant related to the permittivity of free space?

Coulomb's Constant is directly related to the permittivity of free space, which is a measure of how easily electric fields can permeate through a vacuum. The relationship is given by the equation k = 1/4πε0, where ε0 is the permittivity of free space.

## Is Coulomb's Constant a universal constant?

Yes, Coulomb's Constant is a universal constant, meaning its value is the same in all regions of the universe. It is a fundamental constant of nature and plays a crucial role in the study of electromagnetism and the behavior of electric charges.

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