Question

The magnetic moment vectors µ_s and µ_l associated 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:

µ_s = –(e/m) S,

µ_l = –(e/2m) l

Which of these relations is in accordance with the result expected classically? Outline the derivation of the classical result.

Answer 1

The magnetic moment associated with the orbital angular momentum is valid with classical mechanics.

The magnetic moment associated with the orbital angular momentum is given as

µ_l = `-("e"/(2"m")) "l"`

For current i and area of cross-section A, we have the relation:

Magnetic moment

µ_l = iA ……….(1)

Where,

e = Charge of the electron

r = Radius of the circular orbit

T = Time taken to complete one rotation around the circular orbit of radius r

Orbital angular momentum, l = mvr

L = `"m" xx (2π"r")/"T" xx "r"` ……….(2)

Where,

m = Mass of the electron

v = Velocity of the electron

r = Radius of the circular orbit

Dividing equation (1) by equation (2), we get:

`(µ_"l")/"T" = -("e"/(2"m"))`

µ_l = `-("e"/(2"m")) "l"`