Dipole in a Uniform External Field

The governing idea is the behaviour of a dipole in a uniform electric field. The positive and negative charges experience equal forces in opposite directions, so their vector sum gives zero net force. However, these forces act along different parallel lines and form a couple. Its torque is τ = pE sin θ, so it is nonzero for most orientations, but becomes zero when the dipole is exactly parallel or antiparallel to the field. Therefore A is correct; the other choices use unjustified words such as “always.”

The torque on an electric dipole in a uniform electric field is τ = pE sin θ. For the initial angle, τ₁ = pE sin 30° = pE/2. For the final angle, τ₂ = pE sin 60° = √3pE/2. Therefore τ₁:τ₂ = (pE/2):(√3pE/2) = 1:√3, so option A is correct. The other ratios do not follow from the sine dependence.

For a dipole in a uniform electric field, potential energy is U = −pE cos θ. The stable position is θ = 0°, where U₁ = −pE, while the opposite position is θ = 180°, where U₂ = +pE. During slow rotation, the external work equals the increase in potential energy: W = U₂ − U₁ = pE − (−pE) = 2pE. Hence option A is correct.

The torque is τ = pE sin θ, and its maximum value is τmax = pE when θ = 90°. If τ = τmax/2, then pE sin θ = pE/2, so sin θ = 1/2. One possible angle is 30°, although 150° is another possible angle in the usual range. Since option A gives 30°, it is the correct choice; 60° does not give half the maximum torque.

The torque on a dipole is τ = p × E, and its rotational effect tends to align the dipole moment with the electric field. Equivalently, the potential energy U = −pE cos θ decreases as the dipole turns toward θ = 0°. Therefore, even when the initial angle is obtuse, the torque acts in the direction that reduces the angle between p and E. Hence option A is correct.

A dipole contains charges +q and −q of equal magnitude. In a uniform electric field, the field has the same magnitude and direction at both charge positions. Consequently, the forces have magnitudes qE and qE but point in opposite directions. Their vector sum is zero, so the dipole has no net translational force, although the separated forces can still produce torque. Therefore option A is correct.

In a uniform field, the forces on the positive and negative charges of a dipole are equal in magnitude and opposite in direction, so their vector sum is zero. However, the charges are separated, and the two forces act along parallel but distinct lines. This pair forms a couple whose turning effect is τ = pE sin θ. Thus zero net force removes translation but not necessarily rotation, making option A correct.

Stable equilibrium is identified by a local minimum of potential energy. If the dipole is displaced slightly from this position, the resulting torque tends to restore it. For a dipole in a uniform electric field, U = −pE cos θ, which is minimum at θ = 0°, when the dipole moment is parallel to the field. Therefore option A is correct; maximum energy corresponds to unstable equilibrium.

On a potential-energy curve, unstable equilibrium occurs at a local maximum. A small angular displacement from that point lowers the energy and produces a torque that carries the dipole farther from its original orientation. Since U = −pE cos θ, the dipole has U = +pE at θ = 180°, when its moment is opposite to the field; this is the maximum and unstable position. Thus option A is correct.

The torque on a dipole in a uniform electric field is τ = pE sin θ. If p becomes 2p and E becomes 2E while θ remains unchanged, the new torque is τ′ = (2p)(2E) sin θ = 4pE sin θ = 4τ. Therefore the torque becomes four times its original value, so option B is correct. The factor is not merely two because both independent factors are doubled.

The torque on an electric dipole in a uniform field is τ = pE sin θ. The angle remains unchanged, so sin θ is constant. If the dipole moment changes from p to p/2 and the field changes from E to 4E, the new torque is (p/2)(4E)sin θ = 2pE sin θ = 2τ. Therefore, the torque becomes double. It is not half or four times because both changes act together.

For a dipole in a uniform electric field, potential energy is U = −pE cos θ. At 60°, cos 60° = 1/2, so U is negative, equal to −pE/2. At 120°, cos 120° = −1/2, so U becomes +pE/2. Thus the energy changes from a negative value to an equal positive value and therefore increases. The distractor saying it decreases reverses this sign reasoning.

For a dipole, U = −pE cos θ, so |U| = pE|cos θ|. The minimum energy occurs at θ = 0°, giving Umin = −pE and |Umin| = pE. At 45°, |U| = pE cos 45° = pE/√2. Therefore |U|/|Umin| = 1/√2. It is smaller than the minimum-energy magnitude, not larger or zero.

The torque on a dipole is τ = pE sin θ. For nonzero p and E, zero torque requires sin θ = 0, so θ can be 0° or 180°. The potential energy is U = −pE cos θ. At 0°, U = −pE, the minimum value; at 180°, U = +pE, the maximum value. Since the energy is explicitly not minimum, the angle must be 180°. Ninety and sixty degrees give nonzero torque.

The torque magnitude is τ = pE sin θ, so it is maximum when sin θ = 1, that is, θ = 90°. The dipole potential energy is U = −pE cos θ. Since cos 90° = 0, U = 0 at this orientation. Minimum energy occurs at 0° and maximum energy at 180°, so neither of those choices applies. The result assumes a nonzero dipole moment and field.

The torque on an electric dipole is given by τ = p × E, with magnitude pE sin θ. Its direction is perpendicular to both p and E and follows the right-hand rule. If the dipole is placed at the corresponding angle on the opposite side, the angular orientation changes from θ to −θ, so the cross product reverses direction. Changing only E, p, or charge separation changes magnitude, not direction.

The dipole moment is p = qd, and the torque magnitude is τ = pE sin θ. After the change, q′ = 2q and d′ = d/2, so p′ = q′d′ = (2q)(d/2) = qd = p. Since the electric field and angle are unchanged, τ′ = p′E sin θ equals the original torque. Thus the torque remains the same.

The potential energy of a dipole in a uniform electric field is U = −p·E = −pE cos θ. When the dipole moment is along the field, θ = 0° and cos 0° = 1, giving U = −pE, the minimum possible value for fixed p and E. The opposite orientation gives maximum energy, while zero energy occurs at 90°, not at 0°.

The torque magnitude is |τ| = pE sin θ, where θ is the angle between p and E. Initially θ = 90°, so |τ| = pE. Reversing the field changes E to −E, but its magnitude remains E; the new angle is 90° or equivalently 270°, and the sine magnitude is still 1. Thus the torque magnitude remains pE, although its vector direction reverses.

For a dipole, U = −pE cos θ. Opposite alignment means θ = 180°, so U = −pE(−1) = +pE, the maximum potential energy. Although the torque is momentarily zero at exactly 180°, a small angular displacement produces a torque that increases the displacement rather than restoring the original orientation. Hence the antiparallel position is unstable.

The torque magnitude is |τ| = pE sin θ. For θ = 60°, |τ60| = pE sin 60° = (√3/2)pE. For θ = 120°, |τ120| = pE sin 120° = pE sin(180° − 60°) = (√3/2)pE. Therefore the two torque magnitudes are equal, although their vector directions may differ because the orientations are different.

The potential energy of a dipole in a uniform electric field is U = −pE cos θ. At θ = 60°, cos θ = 1/2, so U is negative, equal to −pE/2. At θ = 120°, cos θ = −1/2, so U is positive, equal to +pE/2. Therefore option B is correct; the signs are opposite, while neither value is zero.

In a uniform electric field, the positive and negative charges experience forces of equal magnitude, qE, in opposite directions. Their vector sum is therefore zero, so the translational equation M a_cm = F_net gives a_cm = 0. The forces can still form a couple and produce torque, so zero net force—not zero torque—is the reason. Option A is correct.

In a uniform electric field, the forces on the +q and −q charges have equal magnitude qE and opposite directions, so their vector sum gives F_net = 0 for every orientation. Because the forces act at separated points, they can form a couple with torque magnitude τ = pE sin θ. Thus torque varies with angle and is zero only at 0° and 180°. Option A is correct.

The potential energy of an electric dipole in a uniform field is U = −pE cos θ. When the dipole is aligned with the field, θ = 0° and U is minimum, equal to −pE. A small rotation makes θ nonzero, decreases cos θ, and therefore increases U. Thus the aligned position is stable equilibrium; the energy does not remain zero or become infinite.