The relation between the orbit radius and the electron velocity for a dynamically stable orbit in a hydrogen atom is (where, all notations have their usual meanings)

Options

A. `v=sqrt((4piepsilon_0)/(me^2r))`
B. `r=sqrt((e^2)/(4piepsilon_0v))`
C. `v=sqrt((e^2)/(4piepsilon_0mr))`
D. `r=sqrt((ve^2)/(4piepsilon_0m))`

Explanation

To derive the relation between the orbit radius (r) and the electron velocity (v) for a dynamically stable orbit in a hydrogen atom, we can follow these steps:

### Step 1: Understand the Forces Acting on the Electron
In a hydrogen atom, the electron revolves around the nucleus (proton) due to the electrostatic force of attraction between them. This force acts as the centripetal force required for the electron's circular motion.

### Step 2: Write the Expression for Electrostatic Force
The electrostatic force (F) between the electron and the nucleus can be expressed using Coulomb's Law:
\[ F = \frac{k \cdot e^2}{r^2} \]
where:
- \( k \) is Coulomb's constant (\( k = \frac{1}{4\pi \epsilon_0} \)),
- \( e \) is the charge of the electron,
- \( r \) is the radius of the orbit.

### Step 3: Write the Expression for Centripetal Force
The centripetal force required to keep the electron in a circular orbit is given by:
\[ F_c = \frac{m v^2}{r} \]
where:
- \( m \) is the mass of the electron,
- \( v \) is the velocity of the electron.

### Step 4: Set the Electrostatic Force Equal to the Centripetal Force
For a dynamically stable orbit, the electrostatic force must equal the centripetal force:
\[ \frac{k \cdot e^2}{r^2} = \frac{m v^2}{r} \]

### Step 5: Rearrange the Equation
To find the relationship between \( v \) and \( r \), we can rearrange the equation:
\[ k \cdot e^2 = m v^2 \cdot r \]
\[ v^2 = \frac{k \cdot e^2}{m} \cdot \frac{1}{r} \]

### Step 6: Solve for Velocity
Taking the square root of both sides gives us the expression for the velocity:
\[ v = \sqrt{\frac{k \cdot e^2}{m} \cdot \frac{1}{r}} \]

### Step 7: Substitute the Value of \( k \)
Substituting \( k = \frac{1}{4\pi \epsilon_0} \) into the equation:
\[ v = \sqrt{\frac{1}{4\pi \epsilon_0} \cdot \frac{e^2}{m} \cdot \frac{1}{r}} \]

### Final Expression
Thus, the final relation between the orbit radius \( r \) and the electron velocity \( v \) in a hydrogen atom is:
\[ v = \sqrt{\frac{e^2}{4\pi \epsilon_0 m r}} \]