question_answer

An electric dipole of dipole moment \[\mathbf{\vec{p}}\] is placed in a uniform electric field\[\mathbf{\vec{E}}\]. Write the expression for the torque \[\mathbf{\vec{\tau }}\] 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:

                Torque experienced by the electric dipole of dipole moment \[\vec{p}\] in a uniform electric field \[\vec{E}\] is given by                                                                 \[\vec{\tau }=\vec{p}\times \vec{E}\] The pairs of perpendicular vectors are: (i) When \[\theta ={{90}^{\circ }}\], torque is maximum [Fig. (a)]. \[{{\tau }_{\max }}{{=}_{p}}E\sin {{90}^{\circ }}=pE\]  (ii) When\[\theta ={{30}^{\circ }}\]\[{{150}^{\circ }}\] torque is half the maximum value [Fig. (b)]. \[\tau {{=}_{p}}E\sin ({{30}^{\circ }}\text{or 15}{{\text{0}}^{\circ }})\] \[=\frac{1}{2}pE=\frac{1}{2}{{\tau }_{\max }}\] (iii) When \[\theta ={{0}^{\circ }}\]or\[~\text{18}0{}^\circ \], torque is minimum [Fig. (c)] \[{{\tau }_{\min }}=pE\sin ({{0}^{\circ }}or{{180}^{\circ }})=0\]