If potential energy between a proton and an electron is given by `| U | = k e^(2)//2 R^(3)`, where `e` is the charge of electron and `R` is the radius of atom , that radius of Bohr's orbit is given by `(h = Plank's constant, k = constant)`

Options

A. `(k e^(2) m)/(h^(2))`
B. `(6 pi^(2))/(n^(2)) (k e^(2) m)/(h^(2))`
C. `(2 pi)/(n) (k e^(2) m)/(h^(2))`
D. `(4 pi^(2) k e^(2) m)/(n^(2) h^(2))`

Explanation

To find the radius of Bohr's orbit given the potential energy between a proton and an electron as \( |U| = \frac{k e^2}{2 R^3} \), we will follow these steps:

### Step 1: Understand the Potential Energy Expression
The potential energy \( U \) between a proton and an electron is given by:
\[ |U| = \frac{k e^2}{2 R^3} \]
where \( k \) is a constant, \( e \) is the charge of the electron, and \( R \) is the radius of the atom.

### Step 2: Find the Force from Potential Energy
The force \( F \) between the proton and electron can be derived from the potential energy using the relation:
\[ F = -\frac{dU}{dR} \]
We need to differentiate \( U \) with respect to \( R \):
\[ F = -\frac{d}{dR} \left( \frac{k e^2}{2 R^3} \right) \]
Using the power rule of differentiation:
\[ F = -\left( \frac{k e^2}{2} \cdot (-3 R^{-4}) \right) = \frac{3 k e^2}{2 R^4} \]

### Step 3: Relate Force to Centripetal Force
In a circular orbit, the centripetal force \( F_c \) acting on the electron is given by:
\[ F_c = \frac{m v^2}{R} \]
where \( m \) is the mass of the electron and \( v \) is its velocity. Setting the electrostatic force equal to the centripetal force:
\[ \frac{3 k e^2}{2 R^4} = \frac{m v^2}{R} \]
Multiplying both sides by \( R \):
\[ \frac{3 k e^2}{2 R^3} = m v^2 \]

### Step 4: Use Bohr's Postulate for Angular Momentum
According to Bohr's postulate, the angular momentum \( L \) of the electron is quantized:
\[ L = m v R = n \frac{h}{2 \pi} \]
where \( n \) is the principal quantum number and \( h \) is Planck's constant. From this, we can express \( v \):
\[ v = \frac{n h}{2 \pi m R} \]

### Step 5: Substitute Velocity into the Force Equation
Substituting the expression for \( v \) into the centripetal force equation:
\[ m \left( \frac{n h}{2 \pi m R} \right)^2 = \frac{3 k e^2}{2 R^3} \]
This simplifies to:
\[ \frac{n^2 h^2}{4 \pi^2 m R^2} = \frac{3 k e^2}{2 R^3} \]

### Step 6: Rearranging to Find R
Multiplying both sides by \( 4 \pi^2 m R^3 \):
\[ n^2 h^2 R = 6 \pi^2 m k e^2 \]
Now, solving for \( R \):
\[ R = \frac{6 \pi^2 m k e^2}{n^2 h^2} \]

### Final Expression for Bohr's Radius
The radius of Bohr's orbit is given by:
\[ R = \frac{6 k e^2 \pi^2 m}{n^2 h^2} \]