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NCERT Class XII Chapter
Magnetism and Matter
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Question : 25 of 25
Marks: +1, -0
The magnetic moment vectors μ⃗s\vec{\mu}_s and μ⃗I\vec{\mu}_I associated with the intrinsic spin angular momentum S⃗\vec{S} and orbital angular momentum I⃗\vec{I} respectively, of an electron are predicted by quantum theory (and verified experimentally to a high accuracy) to be given by μ⃗S\vec{\mu}_S = - (em)S⃗\left(\frac{e}{m}\right)\vec{S} and μ⃗I\vec{\mu}_I = - (e2m)I⃗\left(\frac{e}{2m}\right)\vec{I}
Which of these relations is in accordance with the result expected classically?
Outline the derivation of the classical result.
Solution:  
Out of the two relations given, only one is in accordance with classical physics. This is μ⃗1\vec{\mu}_1 = - (e2m)I⃗\left(\frac{e}{2m}\right)\vec{I}
It follows from the definitions of μl\mu_l and l.
μI\mu_I = iA = (−eT)πr2\left(-\frac{e}{T}\right)\pi r^2 ... (i)
I = mvr = m (2Ï€rT)r\left(\frac{2\pi r}{T}\right) r ... (ii)
where r is the radius of the circular orbit, which the electron of mass m and charge (– e) completes in time T.
Divide (i) by (ii),
μIl\frac{\mu_I}{l} = −eTπr2-\frac{e}{T}\pi r^2 × T2πmr2\frac{T}{2\pi m r^2} = −e2m-\frac{e}{2m}
∴ μ⃗1\vec{\mu}_1 = (−e2m)l⃗\left(-\frac{e}{2m}\right)\vec{l}
Clearly μ⃗l\vec{\mu}_l and l⃗\vec{l} will be antiparallel (both being normal to the plane of the orbit)
In contrast, μSS\frac{\mu_S}{S} = em\frac{e}{m}. It is obtained on the basis of quantum mechanics.
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