A model for quantized motion of an electron in a uniform magnetic field states that the flux passing through the orbit of the electron is where is an integer, is Planck's constant and is the magnitude of electron’s charge. According to the model, the

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Solution

Q.
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) [2025]

1 $\frac{h e B}{π m}$

2 $\frac{h e B}{2 π m}$

3 $\frac{h e}{π m}$

4 $\frac{h e}{2 π m}$

Ans.
(4)

Flux, $Φ = B \times π r^{2}$

or   $\frac{n h}{e} = B \times π r^{2} or r^{2} = \frac{n h}{B e π}$ ...(i)

Also, $B v e = \frac{m v^{2}}{r} or m v = B e r$ ...(ii)

Let magnetic moment of electron in circular orbit of an atom be $μ$ .

$μ = I A = \frac{e}{T} A = \frac{e ω}{2 π} (π r^{2}) = \frac{e v r}{2}$ ...(iii)

where $m$ is the mass of electron and $L$ is angular momentum.

From eq. (iii),

$μ = \frac{e v r}{2} = \frac{e}{2 m} L (∵$ $L = m v r)$

or    $μ = \frac{e}{2 m} \times m v r = \frac{e}{2 m} \times B e r^{2} (Using eq (ii))$

or    $μ = \frac{e}{2 m} \times B e \times \frac{n h}{B e π} (Using eq (i))$

$∴$ $μ = \frac{h e}{2 π m}$

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