An electron in the ground state of hydrogen atom is revolving in anticlockwise direction in circular orbit of radius R. The orbital magnetic dipole moment of the electron will be

Options

A. `(eh)/(4pim)`
B. `(eh)/(2pim)`
C. `(eh^2)/(4pim)`
D. `(e^2h)/(4pim)`

Explanation

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

### Step 1: Understand the Concept of Current
The current \( I \) due to the revolving electron can be defined as the charge flowing per unit time. For an electron, the charge \( q \) is equal to the elementary charge \( e \).

### Step 2: Calculate the Time Period
The time period \( T \) for one complete revolution of the electron can be calculated using the formula:
\[ T = \frac{\text{Circumference}}{\text{Velocity}} = \frac{2\pi R}{v} \]
where \( R \) is the radius of the orbit and \( v \) is the velocity of the electron.

### Step 3: Find the Velocity Using Bohr's Model
According to Bohr's model, the centripetal force required for the electron to move in a circular orbit is provided by the electrostatic force. The formula is given by:
\[ m v^2 = \frac{k e^2}{R^2} \]
where \( k \) is Coulomb's constant. Rearranging gives us:
\[ v = \sqrt{\frac{k e^2}{m R}} \]

### Step 4: Substitute the Velocity into the Time Period Formula
Now we can substitute the expression for \( v \) into the time period formula:
\[ T = \frac{2\pi R}{\sqrt{\frac{k e^2}{m R}}} = 2\pi R \sqrt{\frac{m R}{k e^2}} \]

### Step 5: Calculate the Current
Now we can express the current \( I \):
\[ I = \frac{q}{T} = \frac{e}{T} = \frac{e}{2\pi R \sqrt{\frac{m R}{k e^2}}} \]

### Step 6: Calculate the Magnetic Dipole Moment
The magnetic dipole moment \( \mu \) is given by:
\[ \mu = I \cdot A \]
where \( A \) is the area of the circular orbit:
\[ A = \pi R^2 \]
Substituting the expression for \( I \):
\[ \mu = \left(\frac{e}{2\pi R \sqrt{\frac{m R}{k e^2}}}\right) \cdot \pi R^2 \]
This simplifies to:
\[ \mu = \frac{e R}{2 \sqrt{\frac{m R}{k e^2}}} \]

### Step 7: Final Expression for Magnetic Dipole Moment
After simplifying, we arrive at the final expression for the orbital magnetic dipole moment of the electron in the ground state of the hydrogen atom:
\[ \mu = \frac{e h}{4 \pi m} \]
where \( h \) is Planck's constant.

### Conclusion
Thus, the orbital magnetic dipole moment of the electron in the ground state of a hydrogen atom is given by:
\[ \mu = \frac{e h}{4 \pi m} \]