The angular momentum of an electron in a hydrogen atom is proportional to

Options

A. `1// sqrtr`
B. `1//r`
C. `sqrt r`
D. `r^(2)`

Explanation

To solve the question regarding the angular momentum of an electron in a hydrogen atom, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding Angular Momentum**:
The angular momentum (L) of an electron in a circular orbit can be expressed as:
\[ L = mvr \]
where \(m\) is the mass of the electron, \(v\) is its velocity, and \(r\) is the radius of the orbit.

2. **Using Centripetal Force**:
The centripetal force required to keep the electron in circular motion is provided by the electrostatic force between the electron and the proton. This can be expressed as:
\[ \frac{mv^2}{r} = \frac{k e^2}{r^2} \]
where \(k\) is Coulomb's constant and \(e\) is the charge of the electron.

3. **Rearranging the Equation**:
From the above equation, we can rearrange it to find \(v^2\):
\[ v^2 = \frac{k e^2}{m r} \]

4. **Substituting for Velocity in Angular Momentum**:
Now, substituting \(v\) back into the angular momentum equation:
\[ L = mvr = m \cdot r \cdot \sqrt{\frac{k e^2}{m r}} \]
Simplifying this gives:
\[ L = m \cdot r \cdot \sqrt{\frac{k e^2}{m}} \cdot \frac{1}{\sqrt{r}} = \sqrt{m k e^2} \cdot \sqrt{r} \]

5. **Establishing Proportionality**:
From the final expression, we can see that:
\[ L \propto \sqrt{r} \]
This indicates that the angular momentum of the electron in a hydrogen atom is directly proportional to the square root of the radius of its orbit.

6. **Conclusion**:
Therefore, the angular momentum of an electron in a hydrogen atom is proportional to \(\sqrt{r}\).

### Final Answer:
The angular momentum of an electron in a hydrogen atom is proportional to \(\sqrt{r}\).