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.

Explanation

`(i)` Orbital magnetic dipole moment `M=(e)/(T)xxpi R^(2)`
`impliesM=(1)/(2)eomegaR^(2)`
According to orhr's postualtes
`mRomega^(2)=(nh)/(2pi)impliesM=(nhe)/(4pim)`
`(ii) tau=MBsin theta=(nhe)/(4pim)xxBsin30^(@)=(nheB)/(8pim)`
filed due to other semicircle `KNM`.
Therefore,
`vec(B)=-(mu_(0)I)/(4R)(-hat(i))+(mu_(0)I)/(4R)`
`impliesvec(B)=-(mu_(0))/(4R)hat(i)+(mu_(0)I)/(4R)(-hat(i)+hat(j))`
`|vec(B)` due to a circular current carrying loop is `(mu_(0)I)/(2R)` For semicircle it is half `|`
Therefore magnetic force acting on the particle
`vec(F)q{(-v_(0)hat(i))xx(-hat(i)+hat(j))xx(mu_(0)I)/(4R)}=(-mu_(0)qv_(0)I)/(4R)hat(k)`
`(B) vec(F)_(KLM)=vec(F)_(KNM)=vec(F)_(KM)`
and Total force on the loop `=4BIRhat(i)`