Question

Deduce the expression for the magnetic dipole moment of an electron orbiting around central nucleus .

Answer 1

Consider an electron revolving around the nucleus of an atom. Electron is in uniform circular motion around the nucleus of charge + Ze. This constitutes a current.

$I=\frac{e}{T}$...(i)

If 'r' is the orbital radius of the electron and 'v' is the orbital speed,

then thetime period is,

$T=\frac{2\pi r}{v}$.........(ii)

Putting the value of T in (i), we get

$I=\frac{ev}{2\pi r}$

$magneticmomentis\mu =I\pi r^2$

$\mu =\frac{evr}{2}$

Multiplying and dividing by m, we get

$\mu =\frac{evrm}{2m}=\frac{e}{2m}l$

where l is angular momentum

According to Bohr's Hypothesis,

$l=\frac{nh}{2\pi }=\frac{e}{2m}l$

Therefore

$\mu =\frac{enh}{2\pi \left(2m\right)}=\frac{neh}{4\pi m}$