Dipole in a Uniform External Field

For a dipole in a uniform electric field, the potential energy is U = −pE cos θ. During a slow rotation from 0° to 90°, external work increases the potential energy by pE, so that work is positive. During the return from 90° to 0°, the potential energy decreases by the same pE and the external work is negative. Because the path ends at the initial orientation, the net change in potential energy and total external work are both zero. Therefore, option A is correct.

The torque magnitude on an electric dipole is τ = pE sin θ, where θ is the angle between the dipole moment and the electric field. Its maximum value pE occurs at θ = 90°. A small rotation toward the field reduces θ below 90°. In this range, sin θ becomes smaller than one, so the torque magnitude decreases from its maximum value. It does not remain constant or become zero after only a small rotation. Hence option A is correct.

For a dipole, electric potential energy is U = −pE cos θ. At 90°, U is zero, and as the dipole turns toward the field, θ decreases toward 0°, so U becomes more negative and decreases. If the field is conservative and no dissipative force is present, conservation of mechanical energy requires this decrease in potential energy to appear as an increase in rotational kinetic energy. Thus option A is correct; potential energy decreases rather than increases.

For a dipole, U = −pE cos θ. The parallel position at θ = 0° is stable because it has minimum potential energy, while the antiparallel position at θ = 180° is unstable because it has maximum potential energy. Near the unstable position, a small angular displacement produces a torque that drives the dipole farther from θ = 180°, thereby increasing the displacement and lowering the potential energy. Therefore option A is correct; the torque is not restoring there.

The potential energy of a dipole is U = −pE cos θ. At θ = 0°, the dipole is aligned with the field and its potential energy is minimum, so this is stable equilibrium. A small angular displacement raises the energy, and the torque τ = pE sin θ acts so as to reduce the displacement and return the dipole toward alignment. Thus the torque is restoring and points toward the original equilibrium position. Therefore option A is correct.

The potential energy of a dipole is U = −pE cos θ. Its minimum value is Umin = −pE at θ = 0°. The statement that U equals half its minimum value means U = −pE/2. Substituting into the formula gives −pE cos θ = −pE/2, so cos θ = 1/2. For the usual angle range from 0° to 180°, this gives θ = 60°. Therefore option A is correct.

For a dipole, U = −pE cos θ and the minimum energy is Umin = −pE, whose magnitude is pE. The given condition says |U| = pE/2, while U is positive, so U = +pE/2. Hence −pE cos θ = pE/2, giving cos θ = −1/2. In the range 0° to 180°, this occurs at θ = 120°. Thus option B is correct; 60° would give negative energy, not positive energy.

For a dipole, torque is τ = pE sin θ, while potential energy is U = −pE cos θ; its magnitude contains the cosine factor. At θ = 45°, sin 45° = cos 45° = 1/√2. Thus the orientation factors multiplying pE are equal in magnitude, even though the energy itself carries a negative sign according to the chosen orientation. Therefore option A is correct.

The dipole energy is U = −pE cos θ. Relative to the stable parallel position, the increase is ΔU = pE(1 − cos θ). For a small angle measured in radians, cos θ ≈ 1 − θ²/2, so ΔU ≈ pEθ²/2. Hence the increase is proportional to the square of the small angle. It is not inverse, angle-independent, or always zero, so option A is correct.

The dipole moment p is defined from the negative charge toward the positive charge. Consider a point on the axial line beyond the positive charge. The field due to the positive charge points outward, along p, while the field due to the negative charge points toward the negative charge and therefore opposes it. Because the point is closer to the positive charge, its contribution is larger, so the net field remains along p. Thus A is correct.

For a point on the equatorial line, the distances from the positive and negative charges are equal, so the two field magnitudes are equal. Their components perpendicular to the dipole axis cancel, while the components along the axis combine toward the negative charge. Since the dipole moment points from negative to positive, the resultant field is opposite to p. It is not zero except at an infinitely distant limit.

For a uniform field, the positive and negative charges experience forces of equal magnitude, qE, in opposite directions. Their vector sum is therefore zero, so the dipole has no net translational force. If the dipole moment makes an angle with the field, these equal opposite forces form a couple and produce torque τ = pE sinθ; thus option A includes the important rotational effect.

A charge in an electric field experiences force F = qE at its location. In a non-uniform field, the positive and negative charges of the dipole occupy different positions, so the field magnitudes, and sometimes directions, at those positions can differ. Their forces then fail to cancel completely, leaving a net force as well as possibly a torque.

For a short dipole at a far axial point, the field magnitude is E_axial = 2kp/r³, whereas at a far equatorial point it is E_equatorial = kp/r³. At the same distance, p, k, and r³ are common, so their ratio is E_axial:E_equatorial = 2:1. The directions are different, but the question asks for magnitudes; hence option A is correct.

Consider equal positive and negative charges and a point on the perpendicular bisector. The distances from the point to both charges are equal, so the field components perpendicular to the dipole axis cancel by symmetry. The components along the axis point toward the negative charge and add. Since the dipole moment points from negative to positive charge, the resultant equatorial field is opposite to p.

For a dipole at a far point, the axial field magnitude is E_axial = 2kp/r³, whereas the equatorial field magnitude is E_equatorial = kp/r³. At equal distance r, the common factors k, p, and 1/r³ cancel. Therefore E_axial/E_equatorial = 2, giving the ratio 2:1. Option B reverses the order, while C ignores the axial coefficient and D uses an unsupported factor of four.

In a uniform electric field, the forces on the positive and negative charges are equal and opposite, so the net translational force is zero. They nevertheless form a couple whose torque magnitude is τ = pE sin θ, where θ is the angle between dipole moment and field. Since sin θ has its maximum value, 1, at θ = 90°, the torque is maximum when the dipole is perpendicular to the field. Thus A is correct.

The potential energy of a dipole in a uniform field is U = −pE cos θ, and the torque is τ = pE sin θ. At θ = 180°, the torque is momentarily zero, but this is unstable equilibrium because a small displacement makes the torque act in the direction that increases the displacement and rotates p toward the field. The dipole therefore does not return to the opposite orientation. Hence A is correct.

A dipole contains equal and opposite charges, and each charge experiences F = qE. In a uniform field, the field magnitudes at both charge positions are equal, so the two forces cancel and the net force is zero, although a torque may act. In a non-uniform field, the two charges occupy different positions where E can have different magnitudes, so the forces need not be equal. A nonzero net force can therefore result; option A is correct.

For a dipole in a uniform electric field, the torque is tau = pE sin theta, and its potential energy is U = -pE cos theta. At theta = 180 degrees, the dipole is in an unstable equilibrium: an exact, perfectly undisturbed arrangement could remain momentarily, but any slight disturbance produces a torque that increases rotation toward theta = 0. The parallel orientation is stable and has minimum potential energy. Hence A is correct.

The torque on an electric dipole in a uniform field is τ = pE sin θ. At θ = 180°, the torque is zero, but this alone does not establish stability. If the dipole is displaced slightly, the restoring condition is absent; the torque acts so that the dipole turns farther away from the antiparallel orientation and toward alignment with the field. Therefore this is unstable equilibrium, making option A correct; option B describes stable behavior.

The potential energy of a dipole in a uniform electric field is U = −pE cos θ. For the parallel position, θ = 0° and U = −pE, the minimum value. For the perpendicular position, θ = 90° and U = 0. Thus the energy increases by pE, so an external agent must supply positive work, assuming the rotation is controlled slowly. Hence option A is correct; the dipole moment and net charge do not disappear.

For an ideal dipole in a uniform electric field, the forces on its two charges are equal and opposite, so the net translational force is zero. However, if the dipole makes angle θ with the field, these forces produce torque τ = pE sin θ. The potential energy is U = −pE cos θ, which changes with orientation even though the net force remains zero. Thus torque, not net force, explains the angular dependence.

When the dipole is parallel to the electric field, θ = 0 and its potential energy U = −pE cos θ is minimum, equal to −pE. If it is displaced through a small angle, the torque τ = pE sin θ acts in the direction that reduces the displacement and restores alignment. Hence the parallel orientation is stable equilibrium. The effect is caused by torque and energy curvature, not by a change in charge or flux.

For the antiparallel orientation, θ = π and the dipole energy U = −pE cos θ is maximum, equal to +pE. At exactly θ = π the torque is zero, but a small angular displacement makes τ = pE sin θ act in the direction that increases the displacement and turns the dipole toward parallel alignment. Therefore the antiparallel position is unstable equilibrium, not a restoring situation.