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Question

Deduce the expression for the magnetic dipole moment of an electron orbiting around central nucleus .

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Solution

Consider an electron revolving around the nucleus of an atom. Electron is in uniform circular motion around the nucleus of charge + Ze. This constitutes a current.
I=eT...(i)
If 'r' is the orbital radius of the electron and 'v' is the orbital speed,
then thetime period is,

T=2πrv.........(ii)
Putting the value of T in (i), we get
I=ev2πr
magnetic moment is μ=Iπr2
μ=evr2
Multiplying and dividing by m, we get
μ=evrm2m=e2ml
where l is angular momentum
According to Bohr's Hypothesis,
l=nh2π=e2ml
Therefore
μ=enh2π(2m)=neh4πm


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