Magnetic moment of an electron in nth orbit of hydrogen atom is

Options

A. `(neh)/(pim)`
B. `(neh)/(4pim)`
C. `meh)/(2pin)`
D. `(meh)/(4 pin)`

Answer

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 formula for magnetic moment
The magnetic moment (μ) of a charged particle moving in a circular path is given by the formula:
\[ \mu = \frac{q}{2m} L \]
where:
- \( q \) is the charge of the particle,
- \( m \) is the mass of the particle,
- \( L \) is the angular momentum of the particle.

### Step 2: Identify the charge and mass of the electron
For an electron:
- The charge \( q = e \) (where \( e \approx 1.6 \times 10^{-19} \, C \)),
- The mass \( m = m_e \) (where \( m_e \approx 9.11 \times 10^{-31} \, kg \)).

### Step 3: Find the expression for angular momentum
The angular momentum \( L \) of an electron in the nth orbit of a hydrogen atom is given by:
\[ L = n \frac{h}{2\pi} \]
where:
- \( n \) is the principal quantum number (orbit number),
- \( h \) is Planck’s constant (approximately \( 6.626 \times 10^{-34} \, Js \)).

### Step 4: Substitute the expression for angular momentum into the magnetic moment formula
Substituting \( L \) into the magnetic moment formula, we get:
\[ \mu = \frac{q}{2m} \left(n \frac{h}{2\pi}\right) \]

### Step 5: Substitute the values of charge and mass
Substituting \( q = e \) and \( m = m_e \):
\[ \mu = \frac{e}{2m_e} \left(n \frac{h}{2\pi}\right) \]

### Step 6: Simplify the expression
This simplifies to:
\[ \mu = \frac{n e h}{4 \pi m_e} \]

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