Dipole in a Uniform External Field

The potential energy of a dipole is U = −pE cos θ. The torque tends to rotate the dipole toward θ = 0°, where U is minimum and the dipole is aligned with the field. This is a stable orientation, so the aligning torque acts in the direction that decreases potential energy, not always in the direction of increasing energy. Therefore A is correct.

The angle θ controls two important dipole quantities: torque has magnitude τ = pE sin θ, while potential energy is U = −pE cos θ. Therefore changing or misidentifying the angle can change both the predicted torque and the energy, including whether equilibrium is stable or unstable. It does not alter charge units, mass, or the existence of the field. Thus A is correct.

For a dipole in a uniform electric field, the forces on its two charges are equal and opposite, so the net force is zero for every orientation. The torque is τ = pE sin θ, which is zero only when θ = 0° or 180°. Among the listed choices, parallel orientation represents θ = 0°, so option A is possible; perpendicular and oblique positions have nonzero torque.

The governing relations for a dipole in a uniform electric field are τ = pE sin θ for torque and U = −pE cos θ for potential energy. First determine the direction of the dipole moment, which is from negative to positive charge, and then measure θ from the field direction. This makes option A correct. Direct addition, assuming zero torque, or ignoring angular and field information cannot give a reliable result.

For a dipole in a uniform electric field, potential energy is U = −pE cos θ. In the parallel position, θ = 0°, so U₁ = −pE = −(2)(7) = −14 J. In the antiparallel position, θ = 180°, so U₂ = +pE = +14 J. The required slow external work equals the increase in potential energy: W = U₂ − U₁ = 14 − (−14) = 28 J. Thus option B is correct; 14 J is only one pE, while zero and negative work are incorrect here.

The governing relation is U = −pE cos θ. Initially the dipole is parallel to the field, so θ₁ = 0° and U₁ = −(5)(3) = −15 J. Finally it is perpendicular, so θ₂ = 90° and U₂ = 0. The external work required for a controlled rotation is the increase in potential energy, W_ext = U₂ − U₁ = 0 − (−15) = 15 J. Therefore option A is correct; 30 J would incorrectly use the full parallel-to-antiparallel change.

Torque is τ = pE sin θ, so it is zero at θ = 0° and 180°. Stability is determined by U = −pE cos θ: at θ = 0°, the energy is minimum, so a small displacement produces a restoring torque and the equilibrium is stable. At 180° the energy is maximum and equilibrium is unstable. Thus option A is correct; the perpendicular and 45° cases have nonzero torque.

The torque τ = pE sin θ vanishes at both 0° and 180°. To determine stability, use U = −pE cos θ. At the antiparallel orientation, θ = 180° and U = +pE, the maximum potential energy; a small displacement lowers the energy and drives the dipole farther from that orientation. Hence the equilibrium is unstable, making option C correct. Parallel alignment is stable.

For a dipole in a uniform electric field, potential energy is U = −pE cos θ, while the torque magnitude is τ = pE sin θ. The minimum energy occurs at θ = 0°, when the dipole moment is parallel to the field. Therefore sin 0° = 0 and τ = 0. Maximum torque occurs at 90°, not at the minimum-energy orientation, so option B is correct.

The potential energy of a dipole is U = −pE cos θ, and its torque magnitude is τ = pE sin θ. Maximum potential energy occurs at θ = 180°, when the dipole moment is antiparallel to the field. Since sin 180° = 0, the torque is zero. The torque is maximum at 90°, so option A is not correct; option B is the unique answer.

For a slow rotation, the dipole is moved quasistatically, so its kinetic energy does not undergo a lasting change. The external agent supplies or removes energy against the electric torque. Consequently, the external work is W_ext = ΔU, the change in electrostatic potential energy. It is not a change in charge, separation, or force; those quantities need not change during ideal rotation. Hence option B is correct.

By convention, the electric dipole moment points from the negative charge toward the positive charge. The angle θ in U = −pE cos θ and τ = pE sin θ is measured from this direction to the electric field. Reversing the dipole-moment direction changes the angle to its supplementary value and can reverse the energy interpretation or torque direction. It does not alter charge units, the metre definition, or mass, so option A is correct.

The electric force on a charge is given by F = qE in magnitude. A dipole contains charges +q and −q, whose magnitudes are equal. In a uniform field, E has the same magnitude at both charge locations, so each charge experiences a force of magnitude qE. The force directions are opposite because the charge signs differ, but the magnitudes are equal. Therefore option A is correct.

In a uniform field, the two equal and opposite forces on a dipole cancel, giving zero net force, although they may produce torque. In a nonuniform field, the field magnitudes at the positive and negative charges can differ. The forces qE₊ and qE₋ then need not have equal magnitudes, so their vector sum may be nonzero. The dipole moment does not disappear, and torque is not necessarily zero. Thus option B is correct.

The translational acceleration of a system’s centre of mass is determined by F_net = M a_cm. In a uniform electric field, the forces on +q and −q have equal magnitudes and opposite directions, so their vector sum is zero. Hence F_net = 0 and a_cm = 0, although the pair may experience a torque and rotate. Therefore option A is correct; the other choices incorrectly predict a linear acceleration or deny the centre of mass.

The torque magnitude on a dipole in a uniform field is τ = pE sin θ. At 30°, sin 30° = 1/2, so τ₃₀ = pE/2. At 90°, sin 90° = 1, so τ₉₀ = pE, which is twice the initial magnitude and is the maximum possible value. Thus the torque magnitude increases. The sign of torque direction is a separate issue from its magnitude, so option B is correct.

For an electric dipole, the torque magnitude is τ = pE sin θ. At 90°, sin 90° = 1, so the torque has its maximum value pE. At 180°, sin 180° = 0, so the torque becomes zero. Therefore, as the angle increases through this interval, the torque magnitude decreases from maximum to zero. It does not become infinite or remain constant, making option B correct.

For a dipole in a uniform electric field, U = −pE cos θ and τ = pE sin θ. At θ = 90°, cos 90° = 0, so the potential energy is zero. At the same angle, sin 90° = 1, giving τ = pE, the maximum torque magnitude. At 0° or 180°, torque is zero, while at 30° it is only half of pE. Therefore option B is correct.

For a dipole in a uniform electric field, the torque is τ = pE sin θ, and it tends to reduce the angle θ between the dipole moment and the field. Stable equilibrium occurs when the dipole is parallel to the field, where potential energy U = −pE cos θ is minimum. A small displacement therefore produces a restoring torque directed back toward the stable position. Hence option A is correct; the other choices contradict the restoring nature of stable equilibrium.

The unstable equilibrium of an electric dipole occurs when its dipole moment is antiparallel to the electric field, at θ = 180°. Here U = −pE cos θ is maximum. A small displacement makes the torque τ = pE sin θ act so that the angle moves away from 180° and toward 0°, the parallel stable orientation. Thus option B is correct. It does not return to the unstable position, and the torque is not always zero.

The magnitude of torque on an electric dipole is τ = pE sin θ. Since the angle remains unchanged, sin θ is unchanged. If p becomes 2p and E becomes E/2, the new torque is τ′ = (2p)(E/2)sin θ = pE sin θ = τ. Therefore the torque remains unchanged, making option C correct. Options A, B, and D ignore the compensating changes in p and E.

For an electric dipole in a uniform field, torque is τ = pE sin θ. At the same angle, sin θ remains constant, so torque is directly proportional to both p and E. With p′ = 3p and E′ = 4E, τ′ = (3p)(4E)sin θ = 12pE sin θ = 12τ. Thus option D is correct; adding the factors would incorrectly give seven instead of multiplying them.

The torque on a dipole is τ = pE sin θ. Because the angle is unchanged, the sine factor is the same before and after the change. The new factors give p′ = p/2 and E′ = 6E, so τ′ = (p/2)(6E)sin θ = 3pE sin θ = 3τ. Therefore option A is correct. The field change alone would suggest six times, but the halved dipole moment reduces that factor by two.

When the dipole is parallel to the field, θ = 0° and U = −pE cos θ = −pE, the minimum possible potential energy. A small rotation makes θ nonzero and decreases cos θ from 1, so U becomes less negative and therefore increases. This is the signature of stable equilibrium: displacement raises the energy and the torque acts to restore alignment. Hence option B is correct.

For the antiparallel orientation, θ = 180°, so U = −pE cos 180° = +pE, which is the maximum potential energy. A slight rotation reduces the angle below 180°; cos θ becomes greater than −1, causing U = −pE cos θ to decrease from +pE. This is an unstable equilibrium, because the dipole moves away from the antiparallel orientation. Therefore option A is correct.