A current loop of magnetic dipole moment `vec(M)` is placed in uniform magnetic field B. Initially the angle between `vec(M) and vec(B) " is " theta_(1)`. Find the work done by magnetic field as the angle is increased from `theta_(1) " to " theta_(2)`

Explanation

When the angular between `vec(M) and vec(B) " is " theta`, the torque exerted by magnetic field is `vec(tau) = vec(M) xx vec(B) = - MB sin theta hat(k) (vec(M) xx vec(B) " is through " -z-"axis")` Now, the dipole is rotated anti -clockwise by `dtheta`, then work done is
`dW = vec(tau).d vec(theta)`
`= - MB sin theta hat(k).(d theta hat(k))`
`dW = - MB sin theta d theta`
`W = - int_(0_(1))^(theta_(2)) MB sin theta d theta`
`W = MB [cos theta_(2) - cos theta_(1)]`

NOTE : As `DeltaU = -W`
`rArr Delta U = -MB [cos theta_(2) - cos theta_(1)]`
If `theta_(1) = 90^(@)` is taken as reference i.e. zero of potential energy, then `U = MB cos theta`