A hypothetical model for the quantized motion of an electron in a uniform magnetic field $B$ assumes

Question

A hypothetical model for the quantized motion of an electron in a uniform magnetic field $B$ assumes that the magnetic flux passing through the circular orbit of the electron is quantized as $n(h/2e)$, where $n$ is an integer. According to this model, what will be the magnetic moment of the electron in its lowest energy state? ($m$ is the mass of the electron)

Professor PC Thomas Classes · Answer

(a) : Given flux $\phi = B 	imes \pi r^2 = n(h/2e)$. This implies $r^2 = \frac{nh}{2\pi Be}$. The centripetal force is provided by the magnetic force: $\frac{mv^2}{r} = Bve \implies v = \frac{Ber}{m}$. The magnetic moment is $\mu = IA = \left(\frac{e}{T}ight) \pi r^2 = \left(\frac{ev}{2\pi r}ight) \pi r^2 = \frac{evr}{2}$. Substituting $v$, we get $\mu = \frac{e(Ber/m)r}{2} = \frac{eB r^2}{2m}$. Now substitute $r^2$: $\mu = \frac{eB}{2m} \left(\frac{nh}{2\pi Be}ight) = \frac{nh}{4\pi m}$. For the lowest state $n=1$, the magnetic moment is $\frac{he}{4\pi m}.$