Torque on an Electric Dipole

Figure 15.1: Definition sketch for an electric dipole. Two charges, $q$ and $-q$ are connected by an uncharged bar of length $d$. The vectors $\mbox{\bf d}/2$ and $-\mbox{\bf d}/2$ give the positions of the two charges relative to the central point between them. The two forces are due to the electric field $\mbox{\bf E}$.

Let us now imagine a ``dumbbell'' consisting of positive and negative charges of equal magnitude $q$ separated by a distance $d$, as shown in figure 15.1. If there is a uniform electric field $\mbox{\bf E}$, the positive charge experiences a force $q \mbox{\bf E}$, while the negative charge experiences a force $-q \mbox{\bf E}$. The net force on the dumbbell is thus zero.

The torque acting on the dumbbell is not zero. The total torque acting about the origin in figure 15.1 is the sum of the torques acting on the two charges:

$$\mbox{\boldmath$\tau$} = (-q) (-\mbox{\bf d}/2) \times \mbo... ... \times (\mbox{\bf E}) = q \mbox{\bf d} \times \mbox{\bf E} .$$ (16.8)

The vector $\mbox{\bf d}$ can be thought of as having a length equal to the distance between the two charges and a direction going from the negative to the positive charge.

The quantity $\mbox{\bf p} = q \mbox{\bf d}$ is called the electric dipole moment. (Don't confuse it with the momentum!) The torque is just

$$\mbox{\boldmath$\tau$} = \mbox{\bf p} \times \mbox{\bf E} .$$ (16.9)

This shows that the torque depends on the dipole moment, or the product of the charge and the separation, but not either quantity individually. Thus, halving the separation and doubling the charge results in the same dipole moment.

The tendency of the torque is to rotate the dipole so that the dipole moment $\mbox{\bf p}$ is parallel to the electric field $\mbox{\bf E}$. The magnitude of the torque is given by

$$\tau = p E \sin (\theta ) ,$$ (16.10)

where the angle $\theta$ is defined in figure 15.1 and $p = \vert \mbox{\bf p} \vert$ is the magnitude of the electric dipole moment.

The potential energy of the dipole is computed as follows: The electrostatic potential associated with the electric field is $\phi = -E z$ where $E$ is the magnitude of the field, assumed to point in the $+z$ direction. Thus, the potential energy of a single particle with charge $q$ is $U = q \phi = -qEz$. The total potential energy of the dipole is the sum of the potential energies of the individual charges:

$$U = (+q) (-E z_+ ) + (-q) (-E z_- ) = -qE (z_+ - z_- )$$

$$= -qEd \cos (\theta ) = -p E \cos (\theta ) = -\mbox{\bf p} \cdot \mbox{\bf E} ,$$ (16.11)

where $z_+$ and $z_-$ are the $z$ positions of the positive and negative charges. The equating of $z_+ - z_-$ to $d \cos (\theta )$ may be verified by examining the geometry of figure 15.1.

The tendency of the electric field to align the dipole moment with itself is confirmed by the potential energy formula. The potential energy is lowest when the dipole moment is aligned with the field and highest when the two are anti-aligned.