How does magnetic moment of electron reducing around the nucleus varies with principal quantum number:

Options

A. `mu prop n^(2)`
B. `mu prop sqrt(n)`
C. `mu prop n`
D. `mu = (1)/(n)`

Explanation

To determine how the magnetic moment of an electron revolving around a nucleus varies with the principal quantum number, we can follow these steps:

### Step 1: Understand the Magnetic Moment
The magnetic moment (M) of an electron in a circular orbit can be expressed using the formula:
\[ M = I \cdot A \]
where:
- \( I \) is the current due to the electron's motion,
- \( A \) is the area of the orbit.

### Step 2: Calculate the Current (I)
The current \( I \) can be defined as the charge (Q) passing through a point in a given time period (T). For an electron, the charge is \( e \) (approximately \( 1.6 \times 10^{-19} \) C), and the time period \( T \) for one complete revolution is given by:
\[ T = \frac{2\pi r}{v} \]
Thus, the current can be expressed as:
\[ I = \frac{Q}{T} = \frac{e}{T} = \frac{e \cdot v}{2\pi r} \]

### Step 3: Calculate the Area (A)
The area \( A \) of the circular orbit can be calculated as:
\[ A = \pi r^2 \]

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

### Step 5: Relate Velocity (v) and Principal Quantum Number (n)
The velocity \( v \) of the electron in the nth orbit can be related to the principal quantum number \( n \) using the Bohr model:
\[ v \propto \frac{1}{n} \]
Thus, we can express \( v \) as:
\[ v = \frac{k}{n} \]
where \( k \) is a constant.

### Step 6: Substitute Velocity into the Magnetic Moment Equation
Substituting \( v \) into the magnetic moment equation gives:
\[ M \propto \frac{e \cdot \left(\frac{k}{n}\right) \cdot r}{2} \]

### Step 7: Relate Radius (r) and Principal Quantum Number (n)
From the Bohr model, the radius \( r \) of the nth orbit is given by:
\[ r \propto n^2 \]
Thus, substituting this into the magnetic moment equation:
\[ M \propto \frac{e \cdot \left(\frac{k}{n}\right) \cdot (n^2)}{2} \]
This simplifies to:
\[ M \propto \frac{e \cdot k \cdot n}{2} \]

### Conclusion
From the above derivation, we can conclude that the magnetic moment \( M \) of an electron revolving around a nucleus varies directly with the principal quantum number \( n \):
\[ M \propto n \]