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Question
Use Bohr’s model of hydrogen atom to obtain the relationship between the angular momentum and the magnetic moment of the revolving electron.
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Solution
In the Bohr model of the hydrogen atom, the electron is modeled as a point negative charge rotating in a circular orbit about a fixed axis about a nucleus.
Let us consider,
r = radius of the orbit
v = velocity
e = charge of electron
m = mass of electron
Time period (T) = `"circumference"/"velocity"`
= `(2 pi r)/v`
Current (I) = `(-e)/T`
= `(-e)/((2 pi r)/v)`
= `(-e v)/(2 pi r)`
The magnetic moment due to a current loop enclosing an area A is given by:
ML = IA
= `(-e V)/(2 pi r) xx A`
= `(-e v)/(2 pi r) xx pi r^2`
= `(-e r v)/2`
= `(-m e r v)/(2 m)`
L = Angular momentum = mvr
So, ML = `(-e)/(2 m) L`
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