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Two small identical electric dipoles AB...

Two small identical electric dipoles AB and CD, each of dipole moment `vecp` are kept at an angle of `120^(@)` to each other in an external electric field `vecE` pointing along the x-axis as shown in the figure. Find the
(a) dipole moment of the arrangement, and
(b) magnitude and direction of the net torque acting on it.

Text Solution

Verified by Experts

As shown let two identical electric dipoles AB and CD are kept at an angle `theta=120^(@)`, where their dipole moments have magnitude
`|vecp_(A)|=|vecp_(C )|=p`
The resultant dipole moment of the combination
`|vecp_(R)|=2p "cos"(theta)/(2)=2p" cos"(120^(@))/(2)=2p" cos"60^(@)=2p xx (1)/(2)=p`.
and the resultant dipole moment subtends an angle `(theta)/(2)=60^(@)` from either of two dipoles `vecp_(A) or vecp_(B)`.
Therefore, `vecp_(R )` subtends an angle `30^(@)` from +X direction.

If the system is subjected to electric field `vecE` directed along +X direction, the torque acting on the system is
`vec tau=vecp_(R ) xx vecE`
Thus, the magnitude of torque is `|vectau|=pE sin 30^(@)=(1)/(2)pE` and the torque is directed into the plane of paper i.e., the torque tends to align the system along the direction of electric field `vecE`.
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