The magnetic moment vectors
Question:

The magnetic moment vectors μsand μlassociated with the intrinsic spin angular momentum 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/mS,

μl = –(e/2m)l

 

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

Solution:

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

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

$\mu_{1}=-\left(\frac{e}{2 m}\right) \mid$

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

Magnetic moment 

$\mu=i \mathrm{~A}$

$=>\mu_{1}=\left(\frac{-e}{T}\right) \Pi r^{2}$   …….(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

$I=m * \frac{2 \pi r}{T} * r$    ….(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:

$\stackrel{\mu}{\mathrm{T}}=-\left(\frac{\mathrm{e}}{2 \mathrm{~m}}\right)$

$=>\mu=-\left(\frac{e}{2 m}\right) \mid$

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