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.

Answer

(i) `(he)/(4pi m)` (ii) `(heB)/(8pi m)`

Explanation

(i) In ground state `(n=1)`, according to Bohr's theory
`mvR=(h)/(2pi) or v=(h)/(2pimR)`
Now time period `T=(2piR)/(v)=(2piR)/(h//2pimR)=(4pi^(2)mR^(2))/(h)`
Magnetic moment `M=iA`
where `i=("charge")/("Time period")=((e )/(4pi^(2)mR^(2)))/(h)=(eh)/(4pi^(2)mR^(2))`
and `A= piR^(2)`
`:. M=(piR^(2))((eh)/(4pi^(2)mR^(2)))`
or `M=((eh)/(4pim))` Ans. (i)
Direction of magnetic moment `vec(M)` is perpendicular to the plane of orbit
(ii) `vec(tau)=vec(M)xxvec(B)`
or `tau=MB sin theta`
where `theta` is the angle between `vec(M)` and `vec(B)`
`theta=30^(@)`
`:. tau=((eh)/(4pim))Bsin 30^(@) :. tau=(ehB)/(8pim)`
The direaction of `vec(tau)` is perpendicular to both `vec(M)` and `vec(B)`