Torque on a Dipole. An electric dipole with dipole moment p is in a uniform external electric field E. (a) Find the orientations of the dipole for which the torque on the dipole is zero.

Verified step by step guidance

1. Understand the concept of torque on a dipole: Torque (τ) on an electric dipole in an electric field is given by the cross product of the dipole moment (p) and the electric field (E), expressed as τ = p × E. The magnitude of the torque is τ = pE sin(θ), where θ is the angle between p and E.
2. Identify when the torque is zero: The torque will be zero when the angle θ between the dipole moment p and the electric field E is such that sin(θ) = 0. This occurs when θ is 0 degrees or 180 degrees.
3. Determine the orientations: When θ = 0 degrees, the dipole moment p is aligned with the electric field E, meaning they point in the same direction. When θ = 180 degrees, the dipole moment p is anti-aligned with the electric field E, meaning they point in opposite directions.
4. Visualize the orientations: Picture the dipole as an arrow representing the dipole moment p. In the zero torque orientations, this arrow is either pointing directly along the direction of the electric field E or directly opposite to it.
5. Summarize the conditions: The dipole experiences zero torque when it is either parallel or anti-parallel to the electric field. These orientations correspond to θ = 0 degrees and θ = 180 degrees, respectively.

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 the context of an electric field, it determines how the dipole interacts with the field.

Torque in an Electric Field

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 vector and the electric field vector, τ = p × E. The magnitude of the torque depends on the angle between the dipole moment and the electric field, being zero when they are aligned or anti-aligned.

Orientation of Dipole

The orientation of a dipole refers to the angle between its dipole moment vector and the external electric field vector. For the torque to be zero, the dipole must be oriented such that the dipole moment is either parallel or antiparallel to the electric field, meaning the angle between them is 0° or 180°. In these orientations, the cross product becomes zero, resulting in no torque.