NEET · Physics · STD 12 - 12. atoms

A model for quantized motion of an electron in a uniform magnetic field \(B\) states that the flux passing through the orbit of the electron is \(n(h / e)\) where \(n\) is an integer, \(h\) is Planck's constant and \(e\) is the magnitude of electron's charge. According to the model, the magnetic moment of an electron in its lowest energy state will be ( \(m\) is the mass of the electron)

A \(\frac{h e}{\pi m}\)
B \(\frac{h e}{2 \pi m}\)
C \(\frac{h e B}{\pi m}\)
D \(\frac{h e B}{2 \pi m}\)

Verified Solution

Answer & Solution

Correct Answer

(B) \(\frac{h e}{2 \pi m}\)

Step-by-step Solution

Detailed explanation

Magnetic moment
\(M=I A=I\left(\pi r^2\right)\)
\(M=\left(\frac{e u }{2 \pi r}\right)\left(\pi r^2\right)\)
Given \(B\left(\pi r^2\right)=n\left(\frac{h}{e}\right)\)
\(\Rightarrow r ^2=\frac{ h }{ B \pi e } \quad \ldots(2) \quad(\because n =1)\)
And when charge moving in external magnetic field then \(r=\frac{m u}{q B}\)
\(\frac{ v }{ r }=\frac{e B}{ m } \ldots(3) \quad(\because q =e)\)
Put value from equation (2) and (3) in equation (1)
\(M =\left(\frac{e v}{2 \pi r}\right)\left(\pi r^2\right)\)
\(M =\frac{e}{2 \pi}\left(\frac{e B}{m}\right) \pi\left(\frac{h}{B \pi e}\right)\)
\(M =\frac{e h}{2 \pi m}\)

Apply the relevant identity from the chapter and substitute the values established in the previous step, keeping the original constraint in view.

Use the simplified expression to isolate the unknown, then plug the result back into the question setup to confirm the working is internally consistent.

Cross-check against a boundary or limiting case. Both the dimensional balance and the qualitative behaviour at the extreme should agree with the candidate answer.

Combine the steps above to identify the correct option. The remaining choices each fail at least one of the consistency, dimensional, or boundary checks.

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