Dipole in a Uniform External Field

The electric force on a charge is F = qE. In a uniform field, E has the same magnitude and direction at both charge positions. The dipole charges are +q and −q, so their force magnitudes are both qE, while their directions are opposite because their signs are opposite. Hence option A is correct; the forces are not generally zero.

For a dipole with charges +q and −q in a uniform field E, the forces are +qE and −qE. Their vector sum is F_net = qE − qE = 0 because the field is identical at both positions. These forces can still produce a torque if their lines of action differ, but they cannot produce net translation. Therefore B is correct.

A couple consists of two equal, opposite, and parallel forces whose lines of action are separated by a perpendicular distance. In a uniform electric field, the opposite charges of a dipole experience equal and opposite forces. Because the charges are at different positions, the force lines are generally distinct, producing torque without net force. Thus C gives the required condition.

A couple has zero resultant force but a nonzero moment when its equal and opposite forces act along separate lines. For a dipole in a uniform field, this moment is τ = pE sin θ. It tends to turn the dipole so that its dipole moment aligns with the field. Therefore the mechanical effect is rotation, not translation, mass change, or charge change.

The potential energy of an electric dipole in a uniform field is U = −pE cos θ. When the dipole is parallel to the field, θ = 0°, so U = −pE, its minimum value for fixed p and E. This is stable equilibrium because a small angular displacement produces a restoring torque. Hence A is correct; the energy is not necessarily zero.

For a dipole in a uniform field, U = −pE cos θ. In the antiparallel orientation, θ = 180° and cos 180° = −1, so U = +pE, the maximum value for fixed p and E. This orientation is unstable equilibrium: a small displacement causes torque away from it. Therefore B is correct; the energy is not zero or necessarily negative.

The dipole potential energy is U = −pE cos θ. At θ = 180° (opposite), U = +pE, while at θ = 0° (parallel), U = −pE. Thus the change is ΔU = −pE − (+pE) = −2pE, which is a decrease. Consequently, the field can do positive work during a slow natural rotation, while external work would be negative.

For a slow, controlled rotation, the external work equals the increase in the dipole’s potential energy. Initially, at θ = 0°, U_i = −pE; finally, at θ = 180°, U_f = +pE. Therefore W_ext = ΔU = U_f − U_i = 2pE, which is positive. The field tends to rotate the dipole back, so an external agent must supply energy. Hence D is correct.

Stable equilibrium means that a small displacement produces a restoring tendency toward the original position. For a dipole, U = −pE cos θ has a minimum at θ = 0°, when the dipole is parallel to the field. The torque τ = pE sin θ opposes a small displacement from this orientation. Thus A is correct; moving farther and maximum energy describe instability, not stability.

Unstable equilibrium is a position in which a small displacement produces a tendency to move farther from the original position. For a dipole in a uniform electric field, the antiparallel position has maximum potential energy, U = −pE cos θ; at θ = 180°, U = +pE. Thus even a slight disturbance makes the dipole rotate away, so option B is correct. Option A describes stable equilibrium, while C describes its opposite energy condition.

In a uniform electric field, the two equal and opposite forces acting on the positive and negative charges of a dipole cancel vectorially, so the net force is zero. Their lines of action generally differ, forming a couple. The torque is τ = pE sin θ, which is nonzero for an oblique angle except at 0° or 180°. Therefore option C is correct; the other choices incorrectly assign a maximum, zero, or infinite force or torque.

Electric dipole moment is defined as p = qd, where d is the displacement vector directed from the negative charge to the positive charge. Hence p has both a magnitude, qd, and a definite direction. It is therefore a vector quantity. Calling it merely a number or a unit is incorrect, and it is not scalar even though its magnitude may be quoted separately. Thus option A is correct.

Torque is defined by the vector product τ = r × F. The direction of a cross product is perpendicular to the plane containing r and F and is determined by the right-hand rule: curl the fingers from r toward F, while the thumb gives the direction of τ. Therefore option B is correct. Heat, pressure, and mass rules are unrelated to the directional convention for a torque vector.

For a dipole in a uniform electric field, the torque is τ = pE sin θ. Its effect is to rotate the dipole so that its dipole moment becomes parallel to the field, reducing the angle θ. This orientation has minimum potential energy, U = −pE, and is stable. The torque does not break the dipole, remove the external field, or neutralize its charges. Hence option C is correct.

The potential energy of an electric dipole in a uniform field is U = −pE cos θ, where θ is the angle between p and E. For a parallel dipole, θ = 0° and cos 0° = 1, so U = −pE, a negative value when p and E are nonzero. This is the minimum-energy orientation. Zero energy occurs at 90° under this reference, while positive energy occurs at 180°. Therefore option D is correct.

For an electric dipole in a uniform field, the potential energy is U = −pE cos θ. In the antiparallel orientation, the angle is θ = 180° and cos 180° = −1. Therefore U = −pE(−1) = +pE, which is positive for nonzero p and E. This is the maximum-energy and unstable orientation. The negative value belongs to the parallel position, while zero occurs at 90°. Hence option A is correct.

The torque on an electric dipole in a uniform field is τ = pE sin θ, where p is the dipole-moment magnitude and E is the field magnitude. At θ = 90°, sin 90° = 1, so τ = pE. This is the maximum possible torque for fixed p and E. It is not zero, and neither E alone nor the charge alone has the units or complete expression for torque. Thus option B is correct.

Use the dipole-torque relation τ = pE sin θ. Here p = 2 C m, E = 4 N/C, and θ = 90°, so sin 90° = 1. Substitution gives τ = (2)(4)(1) = 8 N m. The units are correct because (C m)(N/C) = N m. Therefore option C is correct; options A and B result from incomplete multiplication, while D would apply only at 0° or 180°.

The torque on an electric dipole in a uniform electric field is given by τ = pE sin θ, where p is the dipole moment and θ is the angle between p and E. Here p = 5 C m, E = 2 N/C, and θ = 0°. Since sin 0° = 0, τ = 5 × 2 × 0 = 0 N m. Thus option D is correct. Options A, B, and C ignore the essential sine factor.

For a dipole in a uniform electric field, torque is τ = pE sin θ. In the perpendicular position, θ = 90° and sin 90° = 1. Substituting p = 3 C m and E = 5 N/C gives τ = 3 × 5 × 1 = 15 N m. Therefore option A is correct. Options B and C use only one given quantity, while D would apply only when the dipole is parallel or antiparallel to the field.

The torque on a dipole is τ = pE sin θ, so it is zero at θ = 0°, the parallel position. Equilibrium must also be classified by stability. The potential energy is U = −pE cos θ, which is minimum at θ = 0°. A small displacement therefore produces a restoring torque that brings the dipole back, making the parallel position stable. Hence option B is correct; zero torque alone does not imply instability.

At the antiparallel position, the angle between p and E is 180°. The torque τ = pE sin 180° is zero, so the dipole is in rotational equilibrium. However, its potential energy U = −pE cos 180° = +pE, which is maximum. A small displacement lowers the energy and produces a torque away from this position, so the equilibrium is unstable. Therefore option C is correct.

A torque that acts opposite to a small displacement and tends to restore the original position is called a restoring torque. For an electric dipole, the parallel orientation has θ = 0° and minimum potential energy U = −pE. A slight rotation increases the energy, so the torque acts to return the dipole to alignment with the field. This is the defining behavior of stable equilibrium, making option D correct.

The electric force on a charge is given by F = qE. For a positive charge, q is positive, so the force vector has the same direction as the electric field vector. Therefore the positive charge in a dipole experiences force along E. The negative charge experiences force opposite to E, but that does not change the answer for the positive charge. Hence option B is correct; the force is not always zero or necessarily upward.

The force on a charge in an electric field is F = qE. Since a negative charge has q < 0, multiplication by q reverses the direction of E. Thus the force on the negative charge of the dipole acts opposite to the electric field. Its direction is not necessarily rightward, and it is not always zero; its magnitude is |q|E. Therefore option C is correct.