Question

Question 5.25: The magnetic moment vectors μsand μlassociated with the intrinsic spin angular momentum S and orbital angular momentum l, respectively, of an electron are predicted by quantum theory (and verified experimentally to a high accuracy) to be given by: μs= –(e/m) S, μl = –(e/2m)l Which of these relations is in accordance with the result expected classically? Outline the derivation of the classical result.

Class 12 - Physics - Magnetism And Matter Magnetism And Matter Page-203

Answer 1

out of the two relations given,only one is in accordance with classical physics.
this is not other than $\vec{\mu_l}=-(e/2m)\vec{l}$

it follows from the definitions of $\mu_l$ and l.
$\mu_l=iA=(-e/T)\pi r^2$---------(1)
[ we know, current = charge/ time . so, i = (-e/T)]

$l=mvr=m(2\pi r/T)r$--------(2)
where r is the radius of circular path and T is the time period of electron evolves arround the circular path.

dividing equation (1) by (2),
$\frac{\mu_l}{l}=\frac{-e}{T}\pi r^2\times\frac{T}{m2\pi r^2}=\frac{-e}{2m}\\\therefore\:\vec{\mu_l}=\frac{e}{2m}\vec{l}$

clearly $\vec{\mu_l}$ and $\vec{l}$ will be antiparallel (both being normal to the plane of the orbit).
in constrast , $\frac{\mu_s}{S}=\frac{e}{m}$.it is obtained on the basis of quantum mechanics.