We have to show (a) that the magnetic moments of the electrons in the various Bohr orbit are given by

in which is the Bohr magneton and (b) We have to compare this expression with the actual values.

Solution:             Click For PDF Version

(a)

The Bohr quantization condition is

Let the speed of the electron in the shell with quantum number and orbit radius r be v. This motion will result in current

,

where T is the period of the orbital motion of the electron and is given by

, and so . We know that magnetic moment in a current carrying planar loop of area A is given by

.

Thus the magnetic moment of the orbit of radius r will be

(b)

We expect from this result that the magnetic moment of the shell be , but experimentally it is found to be zero. Also, the state with principal quantum number n contains subshells of angular momentum quantum number , and the angular momentum of the state with quantum number is .