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An electron in the ground state of hydro...

An electron in the ground state of hydrogen atom is revolving in anticlock-wise direction in a circular orbit of radius `R`.
(i) Obtain an experssion for the orbital magnetic dipole moment of the electron.
(ii) The atom is placed in a uniform magnetic induction `vec(B)` such that the plane - normal of the electron - orbit makes an angle of ` 30^(@)` with the magnetic induction . Find the torque experienced by the orbiting electron.

Text Solution

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The correct Answer is:
A, B, C, D
(i) KEY CONCEPT : Orbital magnetic dipole moment ` M = IA` where `I` is the current due to orbital motion of electron and `A` is the area of loop made by electron.
rArr `M = (e)/(T) xx piR^(2) rArr M = (e omega)/( 2 pi) xx pi R^(2)`
rArr `M = (1)/(2) eomegaR^(2)`
But according to Bohr's postulate
`mRomega^(2) = ( nh)/( 2pi) rArr R omega^(2) = ( nh)/(2pim)`
rArr `M = (e)/(2)xx(nh)/( 2pim) = (nhe)/( 4 pim) = (eh)/( 4 pim)` (because `n=1` for ground state) ltbr. (ii) KEY CONCEPT : We know that torque
` vec(tau) = vec(M)xxvec(B) rArr tau = MB sin theta `
where ` theta` is the angle between `M and B`
rArr `tau = (he)/( 4 pim) xx B sin 30^(@) = ( heB)/( 8 pim)`

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