An electron is moving round the nucleus of a hydrogen atom in a circular orbit of radius `r`. The coulomb force `vec(F)` between the two is (where `k=(1)/(4piepsilon_(0)))`

Options

A. `-k(e^(2))/(r^(3))hat(r )`
B. `k(e^(2))/(r^(3))vec(r )`
C. `-k(e^(2))/(r^(3))vec(r )`
D. `k(e^(2))/(r^(3))hat(r )`

Explanation

To find the Coulomb force between the electron and the nucleus of a hydrogen atom, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Charges**:
- The charge of the proton (nucleus) is \( +e \).
- The charge of the electron is \( -e \).

2. **Coulomb's Law**:
- The formula for the Coulomb force \( F \) between two point charges is given by:
\[ F = k \frac{|q_1 q_2|}{r^2} \]
- Here, \( k = \frac{1}{4 \pi \epsilon_0} \), \( q_1 = +e \), \( q_2 = -e \), and \( r \) is the distance between the charges (which is the radius of the orbit).

3. **Substituting the Values**:
- Substitute \( q_1 \) and \( q_2 \) into the formula:
\[ F = k \frac{|(+e)(-e)|}{r^2} \]
- This simplifies to:
\[ F = k \frac{e^2}{r^2} \]

4. **Direction of the Force**:
- The direction of the force is attractive because the charges are opposite. Therefore, the force vector can be expressed as:
\[ \vec{F} = -k \frac{e^2}{r^2} \hat{r} \]
- Here, \( \hat{r} \) is the unit vector pointing from the electron to the proton.

5. **Final Expression**:
- Thus, the Coulomb force acting between the electron and the nucleus is:
\[ \vec{F} = -k \frac{e^2}{r^2} \hat{r} \]