Torque on a Dipole. An electric dipole with dipole moment p is in a uniform external electric field E. (b) Which of the orientations in part (a) is stable, and which is unstable? (Hint: Consider a small rotation away from the equilibrium position and see what happens.)

Verified step by step guidance

1. Understand the concept of an electric dipole: An electric dipole consists of two equal and opposite charges separated by a distance. The dipole moment \( \mathbf{p} \) is a vector pointing from the negative to the positive charge, with magnitude \( p = qd \), where \( q \) is the charge and \( d \) is the separation distance.
2. Recognize the effect of an external electric field \( \mathbf{E} \) on a dipole: The torque \( \mathbf{\tau} \) experienced by a dipole in an electric field is given by \( \mathbf{\tau} = \mathbf{p} \times \mathbf{E} \). This torque tends to rotate the dipole to align \( \mathbf{p} \) with \( \mathbf{E} \).
3. Identify stable and unstable orientations: A stable orientation occurs when the dipole moment \( \mathbf{p} \) is aligned with the electric field \( \mathbf{E} \). In this position, any small rotation will result in a torque that restores the dipole to its original position. Conversely, an unstable orientation occurs when \( \mathbf{p} \) is anti-aligned with \( \mathbf{E} \), as any small rotation will increase the torque, causing further rotation away from equilibrium.
4. Consider the potential energy of the dipole: The potential energy \( U \) of a dipole in an electric field is given by \( U = -\mathbf{p} \cdot \mathbf{E} \). The stable orientation corresponds to the minimum potential energy, while the unstable orientation corresponds to the maximum potential energy.
5. Apply the hint: For a small rotation away from the equilibrium position, analyze the direction of the torque. If the torque acts to return the dipole to its equilibrium position, the orientation is stable. If the torque acts to further rotate the dipole away from equilibrium, the orientation is unstable.

Key Concepts

Electric Dipole Moment

The electric dipole moment is a vector quantity that represents the separation of positive and negative charges within a system. It is defined as the product of the charge magnitude and the distance between the charges, pointing from the negative to the positive charge. In an external electric field, the dipole moment interacts with the field, influencing the dipole's orientation and stability.

Torque on a Dipole

Torque on a dipole in an electric field is the rotational force experienced by the dipole due to the field. It is given by the cross product of the dipole moment and the electric field vector, τ = p × E. This torque tends to align the dipole with the field, and its direction and magnitude depend on the angle between the dipole moment and the field.

Stable and Unstable Equilibrium

Stable equilibrium occurs when a system returns to its equilibrium position after a small disturbance, while unstable equilibrium results in the system moving further away. For a dipole in an electric field, stability is determined by the orientation of the dipole moment relative to the field. If a small rotation leads to a restoring torque, the orientation is stable; if it leads to further rotation away, it is unstable.