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 find the relation between the orbit radius and the electron velocity for a dynamically stable orbit in a hydrogen atom, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding Forces in the Atom**:
- In a hydrogen atom, the electron orbits the nucleus (proton) due to the electrostatic force of attraction between them. For a stable orbit, this electrostatic force must provide the necessary centripetal force to keep the electron in circular motion.

2. **Identify the Forces**:
- The **electrostatic force (Fe)** between the electron and the nucleus is given by Coulomb's law:
\[ F_e = \frac{1}{4 \pi \epsilon_0} \frac{e^2}{r^2} \]
where \( e \) is the charge of the electron, \( r \) is the radius of the orbit, and \( \epsilon_0 \) is the permittivity of free space.

- The **centripetal force (Fc)** required to keep the electron in circular motion is given by:
\[ F_c = \frac{mv^2}{r} \]
where \( m \) is the mass of the electron and \( v \) is its velocity.

3. **Equating the Forces**:
- For a dynamically stable orbit, we set the centripetal force equal to the electrostatic force:
\[ \frac{mv^2}{r} = \frac{1}{4 \pi \epsilon_0} \frac{e^2}{r^2} \]

4. **Rearranging the Equation**:
- Multiply both sides by \( r^2 \) to eliminate the denominator:
\[ mv^2 r = \frac{1}{4 \pi \epsilon_0} e^2 \]

5. **Solving for Velocity**:
- Rearranging the equation to solve for \( v^2 \):
\[ v^2 = \frac{1}{4 \pi \epsilon_0} \frac{e^2}{mr} \]

6. **Taking the Square Root**:
- Taking the square root of both sides gives us the velocity:
\[ v = \sqrt{\frac{e^2}{4 \pi \epsilon_0 m r}} \]

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