Question

If a magnetic dipole of dipole moment $ M $ rotated through an angle $ \theta $ with respect to the direction of the field $ H $ , then the work done is :

• A. $ MH\, sin\, θ $
• B. $ MH \,(1 − sin\, θ) $
• C. $ MH\, cos\, θ $
• D. $ MH\, (1 − cos\,θ) $

Answer 1

Answer: (D)

When a magnetic dipole of dipole moment $M$ is rotated through an angle $\theta$ with respect to the direction of the field $H$, then work done is done. This work is stored as magnetic potential energy in the dipole. The potential energy of a magnetic dipole of magnetic moment $M$ placed in magnetic field $H$ is
$U_{\theta}=-\vec{ M } \cdot \vec{ H }=-M H\, \cos \,\theta$
where $\theta$ is angle between the vector $\overrightarrow{ M }$ and $\overrightarrow{ H }$. Initially the dipole possesses minimum potential energy $U_{0}$,therefore work required to turn through angle $\theta$ is
$W=U_{\theta}-U_{0}=-M H \cos \theta-(-M H \,\cos \,\theta)$
$=-M H \,\cos\, \theta+M H$
$\therefore \, W=M H(1-\cos \,\theta)$