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 Concept of Magnetic Moment
The magnetic moment (\( \mu \)) of a current loop is given by the product of the current (\( I \)) flowing through the loop and the area (\( A \)) of the loop:
\[ \mu = I \cdot A \]

### Step 2: Determine the Current
The current due to the electron moving in a circular orbit can be expressed as:
\[ I = \frac{E}{T} \]
where \( E \) is the charge of the electron and \( T \) is the time period of the electron's orbit.

### Step 3: Calculate the Time Period
The time period \( T \) can be calculated using the formula:
\[ T = \frac{2\pi r}{v} \]
where \( r \) is the radius of the orbit and \( v \) is the velocity of the electron.

### Step 4: Substitute Time Period into Current Equation
Substituting the expression for \( T \) into the current equation gives:
\[ I = \frac{E \cdot v}{2\pi r} \]

### Step 5: Calculate the Area of the Orbit
The area \( A \) of the circular orbit is:
\[ A = \pi r^2 \]

### Step 6: Substitute Current and Area into Magnetic Moment Equation
Now, substituting \( I \) and \( A \) into the magnetic moment equation:
\[ \mu = I \cdot A = \left(\frac{E \cdot v}{2\pi r}\right) \cdot (\pi r^2) = \frac{E \cdot v \cdot r}{2} \]

### Step 7: Relate Velocity to Angular Momentum
The angular momentum \( L \) of the electron can be expressed as:
\[ L = m \cdot v \cdot r \]
where \( m \) is the mass of the electron. Rearranging gives:
\[ v = \frac{L}{m \cdot r} \]

### Step 8: Substitute Velocity into Magnetic Moment Equation
Substituting \( v \) back into the magnetic moment equation:
\[ \mu = \frac{E \cdot \left(\frac{L}{m \cdot r}\right) \cdot r}{2} = \frac{E \cdot L}{2m} \]

### Step 9: Substitute Angular Momentum for nth Orbit
For the nth orbit, the angular momentum is quantized as:
\[ L = n \cdot \frac{h}{2\pi} \]
Substituting this into the magnetic moment equation gives:
\[ \mu = \frac{E \cdot n \cdot \frac{h}{2\pi}}{2m} = \frac{n \cdot e \cdot h}{4 \pi m} \]

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