Question

The least value of magnetic moment (where $m$ is the mass of the electron, $e$ is the charge of electron) is :

• A. $\frac{eh}{m}$
• B. $\frac{eh}{4\pi m}$
• C. $\frac{2eh}{\pi m}$
• D. $\frac{eh}{\pi m}$

Answer 1

The correct option is B $\frac{eh}{4\pi m}$

For an electron moving in an orbit around a nucleus produces an average current along its orbit.

$\therefore$ Period of motion, $T=\frac{distance}{velocity}=\frac{2\pi r}{V}$

Magnetic moment associated = current X area

$\mu =\frac{charge}{time}\times area$

$\mu =\frac{eV}{2\pi r}\times \pi r^2=\frac{eVr}{2}$

Angular momentum, $L=mvr=\frac{nh}{2\pi }$

$\therefore vr=\frac{nh}{2\pi m}$

where $n$ is quantum number associated.

$\therefore \mu =\frac{e}{2}\left(\frac{nh}{2\pi m}\right)$

$\therefore {\mu }_{min.}=\frac{eh}{4\pi m}$ (at $n=1$)