Magnetic moment of an electron in nth orbit of hydrogen atom is

Options

A. `(n eh)/(pim)`
B. `(n eh)/(4pim)`
C. `(meh)/(2pim)`
D. `(n eh)/(4pin)`

Explanation

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

### Step 1: Understand the Magnetic Moment Formula
The magnetic moment (M) produced by a moving charge can be expressed as:
\[ M = I \cdot A \]
where \( I \) is the current and \( A \) is the area of the orbit.

### Step 2: Determine the Current (I)
The current \( I \) due to the moving electron can be calculated using the charge of the electron \( e \) and the time period \( T \) of its motion:
\[ I = \frac{e}{T} \]

### Step 3: Calculate the Area (A)
The area \( A \) of the circular orbit of the electron is given by:
\[ A = \pi r^2 \]
where \( r \) is the radius of the orbit.

### Step 4: Substitute into the Magnetic Moment Formula
Now, substituting the expressions for \( I \) and \( A \) into the magnetic moment formula:
\[ M = \frac{e}{T} \cdot \pi r^2 \]

### Step 5: Find the Time Period (T)
The time period \( T \) can be expressed in terms of the radius \( r \) and the speed \( v \) of the electron. The distance traveled in one complete orbit is the circumference:
\[ T = \frac{2\pi r}{v} \]

### Step 6: Substitute T into the Magnetic Moment Formula
Now substituting \( T \) into the magnetic moment equation:
\[ M = \frac{e \cdot \pi r^2 \cdot v}{2\pi r} \]
This simplifies to:
\[ M = \frac{e v r}{2} \]

### Step 7: Relate Velocity and Radius to Angular Momentum
From Bohr's model, the angular momentum \( L \) of the electron in the nth orbit is given by:
\[ L = mvr = n \frac{h}{2\pi} \]
where \( m \) is the mass of the electron and \( n \) is the principal quantum number. Rearranging gives:
\[ vr = \frac{n h}{2\pi m} \]

### Step 8: Substitute \( vr \) into the Magnetic Moment Equation
Now substitute \( vr \) back into the magnetic moment equation:
\[ M = \frac{e}{2} \cdot \left(\frac{n h}{2\pi m}\right) \]
This simplifies to:
\[ M = \frac{n e h}{4 \pi m} \]

### Conclusion
Thus, the magnetic moment of an electron in the nth orbit of a hydrogen atom is:
\[ M = \frac{n e h}{4 \pi m} \]

### Final Answer
The correct option is:
\[ \text{Magnetic moment } M = \frac{n e h}{4 \pi m} \]

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