question_answer

Two small identical electrical dipoles AB and CD, each of dipole moment\[\mathbf{'p'}\] are kept at an angle of 120° as shown in Fig. What is the resultant dipole moment of this combination? If this system is subjected to electric field (\[\mathbf{\vec{E}}\]) directed along \[\mathbf{+X}\] direction, what will be the magnitude and direction of the torque acting on this?   

Answer:

                The direction of the two dipole moments and their resultant are shown in Fig. Given \[{{p}_{A}}={{p}_{C}}=p\] Resultant dipole moment, \[{{p}_{R}}=\sqrt{{{p}^{2}}+{{p}^{2}}+2\times p\times p\cos {{120}^{\circ }}}\] \[=\sqrt{2{{p}^{2}}+2{{p}^{2}}\left( -\frac{1}{2} \right)}=p.\] This dipole moment acts along the bisector of \[\angle AOC\]i.e., at an angle of \[{{30}^{\circ }}\]with \[\text{+X}\]direction. \[\therefore \,\,Torque,\,\,\tau =pE\,\sin 30{}^\circ =\frac{1}{2}pE\] By right hand rule, the torque \[\tau \] acts into the plane of paper along Z-direction.