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 determine the proportionality of the angular momentum of an electron in a hydrogen atom, we can follow these steps based on Bohr's model of the hydrogen atom.

### Step-by-Step Solution:

1. **Understanding Angular Momentum**:
According to Bohr's model, the angular momentum \( L \) of an electron in a hydrogen atom is given by the formula:
\[ L = mvr \]
where \( m \) is the mass of the electron, \( v \) is its velocity, and \( r \) is the radius of the electron's orbit.

2. **Using Centripetal Force**:
The centripetal force acting on the electron 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 for Velocity**:
From the centripetal force equation, we can rearrange it to find \( v^2 \):
\[ mv^2 = \frac{k e^2}{r} \]
Therefore,
\[ v^2 = \frac{k e^2}{mr} \]

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

5. **Finding Proportionality**:
From the expression \( L = \sqrt{m k e^2 r} \), we can see that \( L \) is proportional to \( \sqrt{r} \):
\[ L \propto \sqrt{r} \]

6. **Conclusion**:
Thus, the angular momentum of an electron in a hydrogen atom is proportional to the square root of the radius of its orbit.

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