An electron in the ground state of hydrogen atom is removing in unanticlockwise direction in a circle orbit of radius `R`

a. Obtain an expression for the orbital magnetic dipole moment of the electron .
b. The atom is placed in a uniform magnetic indution `bar B` such that the plane normal to the electron orbitmakes an angle of `30^(@)` with the magnetic induction. Find the tarque experienced by the orbiting electron .

Explanation

(a) In ground state `(n=1)` according to Bohr's theory:
`mvR=h/(2pi) or v=h/(2pimR)`
Now time peroid, `T=(2piR)/v=(2piR)/(h//2pimR)=(4pi^2mR^2)/h`
Magnetic moment `M=iA`
where `I=("charge")/("time period")=e/((4pi^2mR^2)/h)=(eh)/(4pi^2mR^2)`
and `A=piR^2`
`:. M=(piR^2)((eh)/(4pi^2mR^2)) or M=(eh)/(4pim)`
Direction of magnetic moment `vecM` is perpendicular to the
plane of orbit.
(b) `vectau=vecMxxvecB`
`|vectau|=MB sin theta`
where `theta` is the angle between `vecM and vecB theta=30^@`
`:. tau=((eh)/(4pim))(B) sin 30^@=(ehB)/(8pim)`
The direction of `vectau` is perpendicular to both `vecM and vecB`.