Question

The orbital magnetic moment \((m_{orb})\) of a revolving electron around the nucleus varies with the principal quantum number (\(n\)) as

• A. \(m_{orb}\propto n^2\)
• B. \(m_{orb}\propto n\)
• C. \(m_{orb}\propto \frac{1}{n^2}\)
• D. \(m_{orb}\propto \frac{1}{n}\)

Hint:

\(m_{orb}=\dfrac{e}{2m}L\) with \(L=\dfrac{nh}{2\pi}\).

Answer 1

The Correct Option is B

Step 1: Use current times area
$m=IA=\dfrac{ev}{2\pi r}\cdot\pi r^2=\dfrac{evr}{2}$.

Step 2: Use Bohr's rule
$m_evr=\dfrac{nh}{2\pi}$, so $m=\dfrac{e}{2m_e}\dfrac{nh}{2\pi}\propto n$, option (B).

Final Answer:
$m_{orb}\propto n$, option (B). \[ \boxed{m_{orb}\propto n} \]