An electric dipole with dipole moment p p is in a uniform external electric field EE. Find the orientations of the dipole for which the torque on the dipole is zero.

Ch 21: Electric Charge and Electric Field

Young & Freedman Calc - University Physics 14th Edition

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Young & Freedman Calc 14th Edition

Ch 21: Electric Charge and Electric Field

Problem 55a

Chapter 21, Problem 55a

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

Verified step by step guidance

Understand the concept of torque on a dipole: Torque (τ) on an electric dipole in an external electric field is given by the cross product of the dipole moment (p) and the electric field (E). The formula is τ = p × E.

Recall the properties of the cross product: The magnitude of the torque is given by |τ| = |p| |E| sin(θ), where θ is the angle between the dipole moment vector and the electric field vector.

Identify when the torque is zero: The torque will be zero when the angle θ is such that sin(θ) = 0. This occurs when θ = 0° or θ = 180°, meaning the dipole moment is either parallel or antiparallel to the electric field.

Visualize the orientations: When the dipole is aligned parallel (θ = 0°) or antiparallel (θ = 180°) to the electric field, the vectors are either pointing in the same direction or directly opposite, resulting in no rotational effect.

Conclude the orientations: Therefore, the orientations of the dipole for which the torque is zero are when the dipole moment is aligned with the electric field, either in the same direction or opposite direction.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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 Electric Fields

Torque in electric fields is the rotational force experienced by a dipole when placed in an external electric field. It is calculated as the cross product of the dipole moment and the electric field vector, τ = p × E. The torque tends to align the dipole with the field, and its magnitude depends on the angle between the dipole moment and the electric field.

Equilibrium Orientation

Equilibrium orientation refers to the position where the net torque on the dipole is zero, meaning the dipole is either aligned or anti-aligned with the electric field. At these orientations, the dipole experiences no rotational force, as the angle between the dipole moment and the electric field is either 0° or 180°, resulting in a zero cross product.

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Textbook Question

An electric dipole with dipole moment $p$ is in a uniform external electric field $E$ . Show that for the stable orientation in part (b), the dipole's own electric field tends to oppose the external field. Note: Part (b) asked 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.) Also, part (a) asked to find the orientations of the dipole for which the torque on the dipole is zero.

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Textbook Question

The dipole moment of the water molecule (H₂O) is $6.17 \times 10^{- 30}$ Cm. Consider a water molecule located at the origin whose dipole moment $p$ points in the $+ x$ -direction. A chlorine ion (Cl^-), of charge $- 1.60 \times 10^{- 19}$ C, is located at $x = 3.00 \times 10^{- 9}$ m. Find the magnitude and direction of the electric force that the water molecule exerts on the chlorine ion. Is this force attractive or repulsive? Assume that $x$ is much larger than the separation $d$ between the charges in the dipole, so that the approximate expression for the electric field along the dipole axis derived in Example $21.14$ can be used.

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Textbook Question

Point charges $q_{1} = - 4.5$ nC and $q sub 2 equals positive 4.5$ nC are separated by $3.1$ mm, forming an electric dipole. The charges are in a uniform electric field whose direction makes an angle of $36.9$ ° with the line connecting the charges. What is the magnitude of this field if the torque exerted on the dipole has magnitude $7.2 \times 10^{- 9}$ Nm?

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Textbook Question

An electric dipole with dipole moment $p$ is in a uniform external electric field $E$ . 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.) Note: Part (a) asked to find the orientations of the dipole for which the torque on the dipole is zero.

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Textbook Question

A very long, straight wire has charge per unit length $3.20 \times 10^{- 10}$ C/m. At what distance from the wire is the electric field magnitude equal to $2.50$ N/C?

Textbook Question

A charge of $negative 6.50$ nC is spread uniformly over the surface of one face of a nonconducting disk of radius $1.25$ cm. Why is the field in part (a) stronger than the field in part (b)? Why is the field in part (c) the strongest of the three fields? Note: Part (a) asked to find the magnitude and direction of the electric field this disk produces at a point $P$ on the axis of the disk a distance of $2.00$ cm from its center. Part (b) asked to find the magnitude and direction of the electric field at point $P$ , supposing that the charge were all pushed away from the center and distributed uniformly on the outer rim of the disk. Part (c) asked to find the magnitude and direction of the electric field at point $P$ if the charge is all brought to the center of the disk.