Torque on a Dipole. An electric dipole with dipole moment p is in a uniform external electric field E. (c) Show that for the stable orientation in part (b), the dipole's own electric field tends to oppose the external field.

Verified step by step guidance

1. Identify the stable orientation of the dipole in the external electric field. In this case, the stable orientation is when the dipole moment vector \( \vec{p} \) is aligned with the electric field vector \( \vec{E} \).
2. Recall that the electric dipole consists of two equal and opposite charges separated by a distance. The dipole creates its own electric field, which affects the surrounding space.
3. Analyze the direction of the electric field created by the dipole. The field lines emerge from the positive charge and terminate at the negative charge, forming a pattern that opposes the direction of the external electric field at certain points around the dipole.
4. Consider the effect of the dipole's electric field on the external field. At points along the axis of the dipole, the dipole's field adds to the external field on one side (where the fields are aligned) and subtracts from it on the other side (where the fields are anti-aligned).
5. Conclude that the dipole's own electric field tends to oppose the external electric field in the region where the fields are anti-aligned, which is consistent with the behavior of fields from opposite charges.

Key Concepts

Electric Dipole Moment

The electric dipole moment (p) is a vector quantity that represents the separation of positive and negative charges in a system. It is defined as the product of the charge magnitude and the distance between the charges. In the context of a dipole in an electric field, the dipole moment determines how the dipole interacts with the field, influencing the torque experienced by the dipole.

Torque on a Dipole

Torque (τ) on an electric dipole in an electric field is the measure of the rotational force acting on the dipole due to the field. It is given by the equation τ = p × E, where p is the dipole moment and E is the electric field. The direction of the torque tends to align the dipole with the external field, and understanding this relationship is crucial for analyzing the dipole's behavior in the field.

Stable Orientation

Stable orientation refers to the position of the dipole where it experiences a minimum potential energy in the presence of an external electric field. For an electric dipole, this occurs when the dipole moment is aligned with the field direction. In this orientation, the dipole's own electric field opposes the external field, leading to a stable equilibrium where any small displacement results in a restoring torque that returns the dipole to its aligned position.