**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*/2*m*)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 *i *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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