Dipole in a Uniform External Field

For a dipole in a uniform electric field, U = −pE cos θ. In the opposite orientation, θ = 180° and U = +pE, the maximum possible potential energy. A slight rotation reduces θ below 180°, so cos θ becomes greater than −1 and U decreases. This is unstable equilibrium: any small displacement lowers the energy, rather than increasing or preserving it.

The moment of a couple is defined as the product of either force and the perpendicular separation between the two lines of action: τ = Fd. For an electric dipole in a uniform field, this becomes τ = pE sin θ, where p is the dipole moment, E is the field strength, and θ is the angle between them. Therefore the basic dependence is on force magnitude and perpendicular distance; the other choices are irrelevant.

A couple produces torque τ = Fd, where d is the perpendicular distance between the lines of action of its equal and opposite forces. If both lines of action coincide, d = 0, so τ = 0. In a uniform electric field the net force on a dipole is also zero, but the question specifically asks about the rotational effect. The dipole moment itself remains p = qℓ and does not vanish.

The torque on a dipole is τ = pE sin θ. At θ = 90°, its magnitude is maximum, but its direction is such that the angle decreases and the potential energy U = −pE cos θ falls. Therefore the torque rotates the dipole toward alignment with the field, θ = 0°. It does not translate the dipole, keep it fixed, or alter its intrinsic dipole moment.

The governing concept is the action of a uniform electric field on a dipole. The field exerts equal and opposite forces, +qE and −qE, on the two charges, so the resultant force is zero. Because these forces act along separated lines, they can produce torque τ = pE sin θ and rotate the dipole toward alignment with the field. Thus A is correct; the other choices confuse translation, energy dependence, or the force on one charge.

For a dipole in a uniform electric field, the torque magnitude is τ = pE sin θ, while the maximum torque is τmax = pE. Hence τ/τmax = sin θ = √3/2. In the usual range from 0° to 180°, θ can be 60° or 120°. Since the question asks for one possible value, 60° is correct. Thirty degrees gives 1/2, while 90° and 180° give 1 and 0 respectively.

The torque magnitude on an electric dipole is τ = pE sin θ. At 30°, τ1 = pE sin 30° = pE/2. At 150°, τ2 = pE sin 150° = pE/2, because sin(180° − θ) = sin θ. Thus τ1 = τ2, so option A is correct. The torque directions may differ because the rotational tendency reverses, but the question asks only about magnitude.

Equilibrium requires zero net torque, and stability requires that a small displacement produce a restoring tendency. In energy terms, this means the potential energy must be at a minimum. For a dipole, U = −pE cos θ has its minimum at θ = 0°, when the dipole aligns with the field; the torque pE sin θ is then zero. Therefore option A combines both necessary conditions. Option C describes unstable equilibrium at θ = 180°.

For a dipole, τ = pE sin θ. At θ = 180°, the torque is zero, so the dipole is in equilibrium. Its potential energy is U = −pE cos 180° = +pE, which is the maximum value. A small rotation lowers the energy and does not produce a restoring tendency toward 180°; the equilibrium is therefore unstable. Thus option B is correct. At 0°, energy is minimum and the equilibrium is stable, while 90° and 60° are not equilibrium positions because the torque is nonzero.

The work done by the electric field is Wfield = −ΔU. For a dipole, U = −pE cos θ. At 60°, Uinitial = −pE/2, while at 0°, Ufinal = −pE. Thus ΔU = −pE − (−pE/2) = −pE/2, and Wfield = +pE/2. The field helps the dipole move toward alignment, so its work is positive. Therefore option A is correct.

The governing relation for a dipole in a uniform electric field is τ = p × E, where p is the dipole-moment vector and E is the electric-field vector. The right-hand rule gives the direction of this cross product, while its magnitude is pE sin θ. Reversing the order to E × p reverses the direction, so option A is correct. A dot product gives scalar energy-related information, not torque direction.

For a dipole in a uniform electric field, the potential energy is U = −p · E = −pE cos θ. The dot product supplies the cosine dependence, and the negative sign means that the lowest energy occurs when the dipole moment is parallel to the field. A cross product would describe a vector perpendicular to both vectors and is used in torque, not energy. Therefore option A is correct.

The torque on a dipole is τ = pE sin θ. At θ = 90°, sin θ = 1, so the torque has its maximum magnitude and immediately produces angular acceleration. In a uniform electric field, the forces on +q and −q are equal and opposite, giving zero net translational force. The charge and dipole moment do not change merely because the dipole is released. Hence rotation starts first, so A is correct.

For a dipole, the torque magnitude is |τ| = pE|sin θ|. At 90°, |sin 90°| = 1, and at 270°, |sin 270°| = 1 as well. Thus both orientations produce the maximum torque magnitude pE. The torque vectors have opposite senses, but that does not change their magnitudes. Therefore option A is correct; options B and C confuse direction or angle with magnitude.

The torque vector is τ = p × E. At 90° and 270°, the sine factor has equal magnitude, so the torque magnitudes are both pE. However, changing the dipole orientation by 180° reverses p relative to E, and therefore reverses the cross-product direction: p × E changes sign. The torque is not directionless, and mass is irrelevant to its electrostatic direction. Thus option A is correct.

A dipole at an angle θ in a uniform field experiences electric torque τe = pE sin θ, which tends to rotate it toward alignment with the field. If the dipole is held fixed, its angular acceleration must be zero, so the net torque must vanish. The applied external torque therefore has equal magnitude and opposite direction, τext = −τe. It need not increase net force or alter p or E, so option A is correct.

Slow, controlled rotation means the angular acceleration is kept very small. From τnet = Iα, the net torque is therefore approximately zero, so the external torque must nearly cancel the electric torque: τext ≈ −τelectric. Equal direction would reinforce the rotation rather than control it, while a fixed factor such as four or zero is not generally required. Hence option A is correct.

Initially p is parallel to E, but after the field reverses, p becomes antiparallel to the new field. For a dipole, U = −pE cos θ, so θ = 180° gives maximum potential energy, while τ = pE sin θ is zero. A small angular displacement lowers the energy and produces a torque away from this orientation. Therefore the new position is unstable equilibrium, making A correct.

Before reversal, the dipole moment is antiparallel to the old field. Reversing the field makes the new field parallel to p, so θ = 0°. The potential energy U = −pE cos θ is then minimum and the torque τ = pE sin θ is zero. A small displacement produces a restoring torque toward alignment, which is the defining feature of stable equilibrium. Therefore option A is correct.

For a dipole, U = −pE cos θ. Setting U = 0, with nonzero p and E, gives cos θ = 0 and therefore θ = 90° (or an equivalent perpendicular orientation). The torque magnitude is |τ| = pE|sin θ|, which becomes pE at 90°, its maximum value. Thus zero energy does not mean zero torque; option A is correct. Options B and C confuse energy with torque.

For a dipole in a uniform electric field, torque is τ = pE sin θ, so it becomes zero at θ = 0° or 180°. Its potential energy is U = −pE cos θ. At 0°, U = −pE, which is negative and minimum; at 180°, U = +pE, which is positive and maximum. Therefore, the dipole is opposite to the field. The ninety-degree position has maximum torque, while forty-five degrees has nonzero torque.

The torque on a dipole is τ = pE sin θ, so zero torque can occur at 0° or 180°. The potential energy is U = −pE cos θ. At θ = 0°, U = −pE, its minimum value; at θ = 180°, U = +pE, its maximum value. Hence the dipole moment must be parallel to the field and the angle is zero degrees. At 90°, torque is maximum, not zero.

Electrostatic force is conservative, so the work done by the electric field depends only on the initial and final orientations. During slow rotation, the external agent supplies work equal to the change in dipole potential energy, ΔU = Ufinal − Uinitial, with U = −pE cos θ. Thus the external work is independent of the path taken. It still depends on the initial and final angles and on pE.

The torque magnitude on a dipole is τ = pE sin θ. The dipole moment p and angle θ are specified to remain unchanged, so sin θ is also constant. Consequently, τ is directly proportional to the electric-field magnitude E. Replacing E by 2E gives τ′ = p(2E)sin θ = 2τ, which doubles the torque for any fixed angle. Halving the field would halve the torque, while making the angle zero would make it zero. Hence option A is correct.

For a dipole, torque magnitude is |τ| = pE|sin θ|, while potential-energy magnitude is |U| = pE|cos θ|. At 0° and 180°, sin θ is zero, so the torque vanishes. At 90°, cos θ is zero, so the energy magnitude vanishes. At 60°, both sin 60° and cos 60° are nonzero; therefore both torque magnitude and energy magnitude are nonzero. Hence option A is correct.