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

Options

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

Answer

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

### Step 1: Understand the Forces Acting on the Electron
In a hydrogen atom, the electron is bound to the nucleus (proton) by the electrostatic force. According to Bohr's theory, the centripetal force required to keep the electron in a circular orbit is provided by the Coulombic force between the electron and the proton.

### Step 2: Write the Equation for Centripetal Force
The centripetal force can be expressed as:
\[ F_c = \frac{mv^2}{r} \]
where \( m \) is the mass of the electron, \( v \) is its velocity, and \( r \) is the radius of the orbit.

### Step 3: Write the Equation for Coulombic Force
The Coulombic force between the electron and proton is given by:
\[ F_e = \frac{k \cdot e^2}{r^2} \]
where \( k \) is Coulomb's constant and \( e \) is the charge of the electron.

### Step 4: Set the Forces Equal
According to Bohr's theory, these two forces are equal:
\[ \frac{mv^2}{r} = \frac{k \cdot e^2}{r^2} \]

### Step 5: Solve for Velocity \( v \)
Rearranging the equation gives:
\[ mv^2 = \frac{k \cdot e^2}{r} \]
\[ v^2 = \frac{k \cdot e^2}{mr} \]
Taking the square root:
\[ v = \sqrt{\frac{k \cdot e^2}{mr}} \]

### Step 6: Write the Expression for Angular Momentum \( L \)
The angular momentum \( L \) of the electron is given by:
\[ L = mvr \]
Substituting the expression for \( v \):
\[ L = m \cdot \sqrt{\frac{k \cdot e^2}{mr}} \cdot r \]

### Step 7: Simplify the Expression
This simplifies to:
\[ L = r \cdot \sqrt{mk \cdot e^2} \]
Thus, we can see that the angular momentum \( L \) is proportional to:
\[ L \propto \sqrt{r} \]

### Conclusion
Therefore, the angular momentum of an electron in a hydrogen atom is proportional to the square root of the radius of the orbit.

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

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