The minimum magnetic dipole moment of electron in hydrogen atom is

Options

A. `(eh)/(2pim)`
B. `(eh)/(4pim)`
C. `(eh)/(pim)`
D. 0

Answer

To find the minimum magnetic dipole moment of an electron in a hydrogen atom, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding Magnetic Dipole Moment**:
The magnetic dipole moment (μ) for a current-carrying loop is given by the formula:
\[ \mu = I \times A \]
where \(I\) is the current and \(A\) is the area of the loop.

2. **Area of the Circular Path**:
For an electron revolving in a circular path of radius \(r\), the area \(A\) is:
\[ A = \pi r^2 \]

3. **Calculating the Current**:
The current \(I\) due to the electron can be expressed as the charge \(e\) divided by the time \(T\) it takes to complete one revolution:
\[ I = \frac{e}{T} \]
The time period \(T\) can be related to the angular speed \(\omega\) by:
\[ T = \frac{2\pi}{\omega} \]

4. **Angular Momentum**:
According to Bohr's model, the angular momentum \(L\) of the electron is quantized:
\[ L = n \frac{h}{2\pi} \]
where \(n\) is the principal quantum number and \(h\) is Planck's constant.

5. **Relating Angular Momentum to Current**:
The angular momentum can also be expressed as:
\[ L = I \cdot \omega \cdot r^2 \]
Substituting \(I\) from Step 3:
\[ L = \left(\frac{e}{T}\right) \cdot \omega \cdot r^2 \]

6. **Substituting for Time Period**:
We can substitute \(T\) from Step 3 into the angular momentum equation:
\[ L = \left(\frac{e \cdot \omega}{2\pi}\right) \cdot \frac{2\pi}{\omega} \cdot r^2 = e \cdot r^2 \cdot \frac{\omega}{2\pi} \]

7. **Finding the Expression for Magnetic Moment**:
Now substituting \(L\) into the magnetic dipole moment equation:
\[ \mu = I \cdot A = \frac{e}{T} \cdot \pi r^2 = \frac{e \cdot \omega}{2\pi} \cdot \pi r^2 = \frac{e \cdot r^2 \cdot \omega}{2} \]

8. **Using the Angular Momentum Expression**:
From the angular momentum expression, we can express \(r\) in terms of \(n\):
\[ r = \frac{n h}{2 \pi m \omega} \]
Substitute this back into the magnetic moment equation to find the minimum value when \(n=1\).

9. **Final Expression for Minimum Magnetic Moment**:
After simplification, we find:
\[ \mu = \frac{n h e}{4 \pi m} \]
For the minimum case \(n=1\):
\[ \mu_{min} = \frac{h e}{4 \pi m} \]

### Conclusion:
The minimum magnetic dipole moment of the electron in a hydrogen atom is:
\[ \mu_{min} = \frac{e h}{4 \pi m} \]