Question

An electron is revolving in a circular orbit of radius 'r' in hydrogen atom. Using Bohr's theory, the angular momentum of the electron is
(M = magnetic dipole moment, m = mass of electron)

• A. \(\frac{mM}{e}\)
• B. \(\frac{e}{mM}\)
• C. \(\frac{2mM}{e}\)
• D. \(\frac{e}{2mM}\)

Hint:

Express the magnetic moment through the orbital current, then relate it to angular momentum m v r.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
A revolving electron acts like a current loop. Its magnetic dipole moment is \(M = IA\).

Step 2: Find the moment.
Current \(I = \dfrac{e}{T}\) with \(T = \dfrac{2\pi r}{v}\), so \(I = \dfrac{ev}{2\pi r}\). Area \(A = \pi r^2\).
\[ M = \frac{ev}{2\pi r}\times\pi r^2 = \frac{evr}{2} \]

Step 3: Link to angular momentum.
Angular momentum \(L = mvr\), so \(vr = \dfrac{L}{m}\).
\[ M = \frac{e}{2}\cdot\frac{L}{m} \Rightarrow L = \frac{2mM}{e} \]

Step 4: Check the options.
Option (A) misses the factor 2. Options (B) and (D) have \(e\) in the numerator, which is dimensionally wrong.

Final Answer:
The angular momentum is \(\dfrac{2mM}{e}\), option (C). \[ \boxed{\frac{2mM}{e}} \]