Question

Show that the magnetic moment of an atom is M = $\frac{1}{2}$eωr², where e the charge of an electron, ω – the angular speed of electron and r – the radius of electron orbit.

Answer 1

The electrons are revolving around the nucleus of an atom. If ‘e’ is the charge of an electron and T its period of revolution, then
Current I = $\frac{e}{T}$ = $\frac{e\omega }{2\pi }$.
Since T = $\frac{2\pi }{\omega }$, ω is the angular velocity.
If A = πr², is the area of the orbit of radius ‘r’, then a magnetic moment of the atom,
M = IA = $\frac{e\omega }{2\pi }$.πr² = $\frac{1}{2}$eωr².