Magnetic moment due to the motion of the electron in `n^(th)` energy state of hydrogen atom is proportional to :

Options

A. `n`
B. `n_0`
C. `n^5`
D. `n^3`

Answer

To solve the problem of determining the proportionality of the magnetic moment due to the motion of an electron in the \( n^{th} \) energy state of a hydrogen atom, we can follow these steps:

### Step 1: Understand the Formula for Magnetic Moment
The magnetic moment (\( \mu \)) due to the motion of an electron can be expressed as:
\[ \mu = -\frac{e}{2m} L \]
where:
- \( e \) is the charge of the electron,
- \( m \) is the mass of the electron,
- \( L \) is the angular momentum of the electron.

**Hint:** Remember that the magnetic moment is related to the angular momentum of the electron.

### Step 2: Express Angular Momentum in Terms of Quantum Numbers
The angular momentum (\( L \)) of an electron in the \( n^{th} \) energy state is given by:
\[ L = n \frac{h}{2\pi} \]
where:
- \( h \) is Planck's constant,
- \( n \) is the principal quantum number.

**Hint:** Recall that angular momentum in quantum mechanics is quantized and depends on the principal quantum number \( n \).

### Step 3: Substitute Angular Momentum into the Magnetic Moment Formula
Substituting the expression for \( L \) into the magnetic moment formula, we get:
\[ \mu = -\frac{e}{2m} \left(n \frac{h}{2\pi}\right) \]
This simplifies to:
\[ \mu = -\frac{e h n}{4\pi m} \]

**Hint:** Simplify the expression carefully and note the constants involved.

### Step 4: Identify the Proportionality
From the final expression for the magnetic moment, we can see that:
\[ \mu \propto n \]
This indicates that the magnetic moment is directly proportional to the principal quantum number \( n \).

**Hint:** Look for the variable that changes in the final expression to determine proportionality.

### Conclusion
Thus, the magnetic moment due to the motion of the electron in the \( n^{th} \) energy state of a hydrogen atom is proportional to \( n \).

**Final Answer:** The magnetic moment is directly proportional to \( n \).