The magntic moment `(mu)` of a revolving electron around the mucleaus varies with principle quantum number `n` as

Options

A. `mupropn`
B. `muprop1//n`
C. `mupropn^(2)`
D. `muprop1//n^(2)`

Explanation

To determine how the magnetic moment \((\mu)\) of a revolving electron around the nucleus varies with the principal quantum number \(n\), 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: Calculate the Current
For an electron revolving around the nucleus, the current \(I\) can be expressed as the charge of the electron divided by the time period of revolution. The time period \(T\) can be calculated as:
\[ T = \frac{\text{Circumference}}{\text{Velocity}} = \frac{2\pi r}{v} \]
Thus, the current \(I\) is:
\[ I = \frac{e}{T} = \frac{e}{\frac{2\pi r}{v}} = \frac{ev}{2\pi r} \]

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

### Step 4: Substitute into the Magnetic Moment Formula
Substituting the expressions for current and area into the magnetic moment formula:
\[ \mu = I \cdot A = \left(\frac{ev}{2\pi r}\right) \cdot (\pi r^2) = \frac{evr}{2} \]

### Step 5: Relate Velocity and Angular Momentum
The angular momentum \(L\) of the electron is given by:
\[ L = mvr \]
From this, we can express \(vr\) as:
\[ vr = \frac{L}{m} \]

### Step 6: Substitute Angular Momentum into Magnetic Moment
Substituting \(vr\) into the magnetic moment equation:
\[ \mu = \frac{e}{2} \cdot \frac{L}{m} \]

### Step 7: Use Bohr's Quantization Condition
According to Bohr's model, the angular momentum is quantized and given by:
\[ L = n\frac{h}{2\pi} \]
Substituting this into the magnetic moment equation:
\[ \mu = \frac{e}{2} \cdot \frac{n\frac{h}{2\pi}}{m} \]
This simplifies to:
\[ \mu = \frac{n \cdot e \cdot h}{4\pi m} \]

### Step 8: Conclusion
From the final expression, we can see that the magnetic moment \(\mu\) is directly proportional to the principal quantum number \(n\):
\[ \mu \propto n \]

### Final Answer
Thus, the magnetic moment \((\mu)\) of a revolving electron around the nucleus varies with the principal quantum number \(n\) as:
\[ \mu \propto n \]