An orbital electron is the ground state of hydrogen has the magnetic moment `mu_1`. This orbital electron is excited to 3rd excited state by some energy transfer to the hydrogen atom. The new magnetic moment fo the electron is `mu_2` then

Options

A. `mu_1 = 4mu_2`
B. `2mu_1 = mu_2`
C. `16mu_1 = mu_2`
D. `4mu_1 = mu_2`

Explanation

To solve the problem of finding the new magnetic moment \( \mu_2 \) of an electron in the third excited state of hydrogen, we can follow these steps:

### Step 1: Understand the Magnetic Moment of an Electron
The magnetic moment \( \mu \) of an electron in an orbit can be expressed in terms of the current \( I \) and the area \( A \) of the orbit:
\[ \mu = I \cdot A \]

### Step 2: Determine the Current \( I \)
The current \( I \) due to the revolving electron can be defined as:
\[ I = \frac{e}{T} \]
where \( e \) is the charge of the electron and \( T \) is the time period of revolution. The time period \( T \) can be expressed as:
\[ T = \frac{2\pi r_n}{v_n} \]
where \( r_n \) is the radius of the orbit and \( v_n \) is the velocity of the electron in the \( n \)-th orbit.

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

### Step 4: Substitute \( I \) and \( A \) into the Magnetic Moment Formula
Substituting the expressions for current and area into the magnetic moment formula gives:
\[ \mu = I \cdot A = \left(\frac{e}{T}\right) \cdot \left(\pi r_n^2\right) \]
Substituting \( T \) into this equation:
\[ \mu = \left(\frac{e v_n}{2\pi r_n}\right) \cdot \left(\pi r_n^2\right) = \frac{e v_n r_n}{2} \]

### Step 5: Analyze the Dependence on Quantum Number \( n \)
From quantum mechanics, we know:
- The velocity \( v_n \) is inversely proportional to \( n \): \( v_n \propto \frac{1}{n} \)
- The radius \( r_n \) is proportional to \( n^2 \): \( r_n \propto n^2 \)

Thus, combining these relationships:
\[ v_n r_n \propto n^2 \cdot \frac{1}{n} = n \]
This implies that the magnetic moment \( \mu \) is directly proportional to \( n \):
\[ \mu \propto n \]

### Step 6: Relate the Magnetic Moments of Different States
For the ground state (where \( n = 1 \)):
\[ \mu_1 \propto 1 \]
For the third excited state (where \( n = 4 \)):
\[ \mu_2 \propto 4 \]

Thus, we can express the relationship between the two magnetic moments:
\[ \mu_2 = 4 \mu_1 \]

### Conclusion
The new magnetic moment \( \mu_2 \) when the electron is in the third excited state is four times the magnetic moment \( \mu_1 \) in the ground state:
\[ \mu_2 = 4 \mu_1 \]