In Fig. 4.6, p1 and p2 are (perfect) dipoles a distance r...

In Fig. $4.6, p_{1}$ and $p_{2}$ are (perfect) dipoles a distance $r$ apart. What is the torque on $\mathbf{p}_{1}$ due to $\mathbf{p}_{2} ?$ What is the torque on $\mathbf{p}_{2}$ due to $\mathbf{p}_{1}$ ? [In each case I want the torque on the dipole about its own center. If it bothers you that the answers are not equal and opposite, see Prob. 4.29.]

Key Concepts

Evaluation of Torque About the Dipole's Center

When calculating the torque on a dipole, it is important to specify the point about which the torque is measured, often the dipole’s own center. This consideration explains why torques calculated on different dipoles, even if interacting with each other, might not appear as equal and opposite when taken about their respective centers.

Dipole-Dipole Interaction

When two dipoles are separated by a distance, each dipole creates an electric field that interacts with the other dipole, leading to forces and torques. The interaction depends on both the magnitudes and orientations of the dipoles, as well as the separation distance, and it highlights the complex nature of forces in distributed charge systems.

Electric Dipole

An electric dipole is a pair of equal and opposite charges separated by a distance, characterized by a dipole moment, a vector pointing from the negative charge to the positive charge. This concept is central because it defines how dipoles interact with external fields and with each other.

Electric Field of a Dipole

A dipole produces an electric field that has a characteristic spatial distribution, decaying with distance and varying in direction. Understanding this field is crucial because the field from one dipole exerts forces (and thus torques) on another dipole.

Torque on a Dipole in an External Electric Field

In an electric field, a dipole experiences a torque given by the cross product of its dipole moment and the electric field (? = p × E). This fundamental relationship is used to calculate how one dipole is rotated by the field produced by another dipole.

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