Question

An electric dipole of dipole moment $\overrightarrow{P}$ is placed in a uniform electric field $\overrightarrow{E}$. Write the expression for the torque $\overrightarrow{r}$ experienced by the dipole. Identify two pairs of perpendicular vectors in the expression. Show diagrammatically the orientation of the dipole in the field for which the torque is (i) Maximum (ii) Half the maximum value (iii) Zero.

Answer 1

$\overrightarrow{\tau }=\overrightarrow{p}\times \overrightarrow{E}$
Pairs of perpendicular vector (a) $(\overrightarrow{\tau },\overrightarrow{p})$ (b) $(\overrightarrow{\tau },\overrightarrow{E})$
(i) Magnitude of torque $\tau =pE\sin \theta$
For maximum torque $(\sin \theta {)}_{max}=1\Rightarrow \theta ={90}^{\circ }$
Orientation is shown in figure (i)
(ii) For $\tau =\frac{1}{2}{\tau }_{max}$
$pE\sin \theta =\frac{1}{2}pE\Rightarrow \sin \theta =\frac{1}{2}$ or $\theta ={30}^{\circ }$
Orientation is shown in figure (ii)
(iii) For zero torque, $\sin \theta =0\Rightarrow \theta =0$
The orientations is shown in the figure (iii).