Torque on a Dipole. An electric dipole with dipole moment p⃗ is in a uniform electric field k .

(a) The torque on a dipole in a uniform electric field is given by $vec{tau} = vec{p} times vec{

Torque on a Dipole. An electric dipole with dipole moment p⃗...

Torque on a Dipole. An electric dipole with dipole moment $\vec{p}$ is in a uniform electric field $\overrightarrow{\boldsymbol{k}} .$ (a) Find the orientations of the dipole for which the torque on the dipole is zero. (b) Which of the orientations in part (a) is stable, and which is unstable? (Hint: Consider a small displacement away from the equilibrium position and see what happens.) (c) Show that for the stable orientation in part $(b),$ the dipole's own electric field tends to oppose the external field.

Transcript

00:01 In this problem, we have been given the electric field, which is in the right direction, as represented by the right arrow.

00:08 And there is a dipole.

00:10 That's the electric dipole, which is placed in the region of uniform electric field.

00:14 So the dipole here, we know the dipole moment vector that's from negative to the positive.

00:20 So this dipole indicates that a negative charge and an equally positive charge placed here.

00:27 So this is a dipole.

00:28 Let's represent it with the letter p.

00:30 So we need to determine the orientation of this dipole if the torque on this dipole is zero.

00:37 So we know the expression of the torque, that's p cross -e.

00:41 It's the cross -product of dipole moment vector and the electric field vector.

00:46 And the cross -product results into p -e -sign theta, where theta is the angle between the dipole moment vector and the electric field vector.

00:54 And for the torque to be zero, we see that theta should be zero or it should be pi.

01:03 That means we can see if the dipole moment is in the direction of the electric field vector or exactly opposite, that in this case the torque will be zero.

01:12 And now we have to again determine the orientation for the stable and unstable equilibrium configuration.

01:20 So we know the expression of the potential energy that's minus p .e.

01:25 So expanding this, we get potential energy as minus p .e.

01:28 Cost theta and the condition for stable equilibrium is that the potential energy should be minimum and the minimum potential energy will be when theta is zero because in this case the value of cost zero will be one and you will come out to be minus be so we can say the condition for stable equilibrium is that the dipole moment should be parallel to the electric field factor...

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