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(B) The magnetic moment $M$ is given by $M = I A$,where $I$ is the current and $A$ is the area of the orbit.$I = \frac{e}{T} = \frac{ev}{2 \pi r}$ and

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$\frac{h e}{2 \pi m}$

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(B) The magnetic moment $M$ is given by $M = I A$,where $I$ is the current and $A$ is the area of the orbit.$I = \frac{e}{T} = \frac{ev}{2 \pi r}$ and $A = \pi r^2$.Thus,$M = \left( \frac{ev}{2 \pi r} \right) (\pi r^2) = \frac{evr}{2}$.Given the flux condition: $B(\pi r^2) = n(h/e)$. For the lowest energy state,$n = 1$,so $B \pi r^2 = h/e$,which implies $r^2 = \frac{h}{B \pi e}$.Also,for an electron in a magnetic field,the centripetal force is provided by the Lorentz force: $\frac{mv^2}{r} = evB$,which gives $\frac{v}{r} = \frac{eB}{m}$.Substituting $v = \frac{eBr}{m}$ into the expression for $M$:$M = \frac{e}{2} \left( \frac{eBr}{m} \right) r = \frac{e^2 B r^2}{2m}$.Substituting $r^2 = \frac{h}{B \pi e}$ into the equation for $M$:$M = \frac{e^2 B}{2m} \left( \frac{h}{B \pi e} \right) = \frac{eh}{2 \pi m}$.

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