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

step 1 So for Part A, we're going to have a dipole in an electric field, and we're looking for the orie

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 $E$ . (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.

Key Concepts

Electric Dipole

An electric dipole consists of two equal and opposite charges separated by a fixed distance. It is characterized by its dipole moment, a vector quantity whose magnitude is the product of the charge magnitude and the separation distance, and which points from the negative charge to the positive charge. Dipoles are fundamental in understanding interactions with electric fields, as well as in molecular chemistry and many physical phenomena.

Torque on an Electric Dipole

When an electric dipole is placed in a uniform electric field, it experiences a torque given by the vector cross product of the dipole moment and the electric field. This torque tends to rotate the dipole so that its dipole moment aligns with the field, thereby reducing the system's potential energy. The magnitude of the torque is proportional to the sine of the angle between the dipole moment and the field direction.

Equilibrium Orientation

Equilibrium occurs when the net torque on the dipole is zero. In the context of an electric dipole in a uniform electric field, this happens when the dipole moment is either parallel or anti-parallel to the field. This corresponds to two possible orientations: one where the dipole aligns with the field and one where it is aligned opposite to the field.

Stability Analysis of Equilibrium Orientations

Stability of an equilibrium orientation is determined by how the system responds to small perturbations. A stable equilibrium is one where a small displacement results in forces or torques that restore the system to its original configuration. For a dipole in a uniform electric field, the orientation where the dipole moment aligns with the electric field is stable, since small deviations produce a restoring torque, while the anti-parallel configuration is unstable, as small deviations further drive the dipole away from equilibrium.

Effect of the Dipole's Own Electric Field

In the stable orientation, not only is the dipole aligned with the external field, but the electric field produced by the dipole itself tends to oppose the external electric field at regions away from the dipole. This phenomenon is an example of how internal field contributions can modulate the net field distribution; it highlights the interplay between an external applied field and the induced or self-generated fields, which is an important concept in understanding screening and self-interaction effects in electromagnetic systems.

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