If `epsilon_(0)` be the permittivity of vacuum and r be the radius of orbit of H- atom in which electron is revolving, then velocity of electron is given by :

Options

A. `v=(e)/(sqrt(4piepsilon_(0)rm)`
B. `v=exxsqrt(4piepsilon_(0)rm)`
C. `v=(4piepsilon_(0)rm)/(e)`
D. `v=(4piepsilon_(0)rm)/(e^(2))`

Answer

To find the velocity of the electron in the hydrogen atom, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Forces Acting on the Electron**:
- In a hydrogen atom, the electron is attracted to the nucleus (proton) by the electrostatic force. This force can be expressed using Coulomb's law.

2. **Electrostatic Force Calculation**:
- The electrostatic force \( F \) between the electron and the nucleus is given by:
\[ F = \frac{e^2}{4 \pi \epsilon_0 r^2} \]
where:
- \( e \) is the charge of the electron (and also the charge of the nucleus for hydrogen, since \( Z = 1 \)),
- \( \epsilon_0 \) is the permittivity of vacuum,
- \( r \) is the radius of the orbit.

3. **Centrifugal Force**:
- When the electron is in circular motion, it experiences a centrifugal force, which can be expressed as:
\[ F_{\text{centrifugal}} = \frac{mv^2}{r} \]
where:
- \( m \) is the mass of the electron,
- \( v \) is the velocity of the electron.

4. **Equate the Forces**:
- For the electron to be in stable orbit, the electrostatic force must equal the centrifugal force:
\[ \frac{mv^2}{r} = \frac{e^2}{4 \pi \epsilon_0 r^2} \]

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

6. **Solving for \( v^2 \)**:
- Rearranging gives:
\[ v^2 = \frac{e^2}{4 \pi \epsilon_0 m r} \]

7. **Finding Velocity \( v \)**:
- Taking the square root of both sides to find \( v \):
\[ v = \sqrt{\frac{e^2}{4 \pi \epsilon_0 m r}} = \frac{e}{\sqrt{4 \pi \epsilon_0 m r}} \]

### Final Result:
Thus, the velocity of the electron in the hydrogen atom is given by:
\[ v = \frac{e}{\sqrt{4 \pi \epsilon_0 m r}} \]