Dipole in a Uniform External Field

The potential energy of a dipole is U = −pE cos θ, so it is maximum when θ = 180°, with the dipole moment antiparallel to the electric field. The torque magnitude is τ = pE sin θ. Hence τ = pE sin 180° = 0. Option B is correct. Although this orientation is an unstable equilibrium, the instantaneous torque is still zero; maximum torque occurs at 90°.

For a slow rotation, the dipole is moved quasistatically, so its kinetic energy does not acquire a lasting change. The external agent supplies or removes energy according to the change in the dipole’s potential energy. Thus W_external = ΔU, provided the usual quasistatic convention is used. Option B is correct. Force, charge, and separation are not themselves equal to the work done in this rotational process.

By convention, the electric dipole moment vector points from the negative charge to the positive charge. The angle θ in U = −pE cos θ and τ = pE sin θ is measured from this vector to the electric field. Reversing the dipole-moment direction changes the angle interpretation and can change the calculated sign or value of energy and torque. Therefore option A is correct; the units of charge, metre definition, and mass are unrelated.

The two charges of an ideal electric dipole are +q and −q, so their charge magnitudes are equal. In a uniform electric field, the field vector has the same magnitude at both charge locations. Since the force magnitude on a charge is F = |q|E, each charge experiences the same force magnitude, although the directions are opposite because the charge signs differ. Thus option A is correct.

In a uniform field, the two equal and opposite forces on a dipole cancel in translation, giving zero net force. In a nonuniform field, however, the field magnitudes at the positive and negative charges can differ. Since F = qE, the two force magnitudes need not be equal, so their vector sum can be nonzero. Therefore option B is correct. The dipole moment does not disappear, and torque is not necessarily zero.

The translational motion of a system’s centre of mass is governed by F_net = M a_cm. In a uniform electric field, the forces on the +q and −q charges of a dipole have equal magnitudes and opposite directions, so their vector sum is zero. Consequently, a_cm = F_net/M = 0. The dipole may still rotate because a torque can act, but its centre of mass has no translational acceleration. Option A is correct.

The torque magnitude on a dipole in a uniform field is τ = pE sin θ. Assuming p and E remain constant, at 30° the value is pE sin 30° = 0.5pE, whereas at 90° it is pE sin 90° = pE. Since pE is twice 0.5pE, the torque magnitude increases as the angle changes from 30° to 90°. Thus option B is correct.

For a dipole in a uniform electric field, τ = pE sin θ. At 90°, sin 90° = 1, so the torque magnitude is maximum and equals pE. At 180°, sin 180° = 0, so the torque magnitude becomes zero. Therefore, over this angular interval the torque magnitude decreases from its maximum value to zero. Option B is correct; the signed torque direction may also change near equilibrium, but the question asks for magnitude.

For a dipole in a uniform electric field, U = −pE cos θ and τ = pE sin θ. The potential energy is zero when cos θ = 0, which occurs at θ = 90° in the given range. At the same angle, sin 90° = 1, giving τ = pE, the maximum possible torque. Therefore option B is correct. At 0° and 180° the torque is zero, while at 30° it is not maximum and the energy is not zero.

For a dipole in a uniform electric field, the torque is τ = pE sin θ, and its direction tends to reduce the angular displacement from the field direction. Parallel alignment is stable because the potential energy U = −pE cos θ is minimum there. Therefore, after a small displacement, the torque opposes that displacement and restores the dipole toward equilibrium. Option B describes unstable behavior, while C and D are not generally true.

The antiparallel orientation of a dipole is unstable because U = −pE cos θ has its maximum value at θ = 180°. A small displacement produces a torque that moves the dipole away from this maximum-energy orientation and toward the lower-energy parallel orientation at θ = 0°. Thus the dipole rotates toward the stable parallel position. A zero torque occurs only at exact equilibrium, not after displacement.

The magnitude of torque on an electric dipole is τ = pE sin θ. Since the angle is unchanged, sin θ is constant. If p changes to 2p and E changes to E/2, the new torque is τ′ = (2p)(E/2) sin θ = pE sin θ = τ. Hence the torque remains the same. It is not doubled or halved because the two changes cancel, and it is not quadrupled.

For an electric dipole in a uniform field, torque is τ = pE sin θ. At the same angle, sin θ remains unchanged, so torque is directly proportional to the product pE. The changes give p′ = 3p and E′ = 5E. Therefore τ′ = (3p)(5E) sin θ = 15τ. The correct multiplier is thus fifteen; adding the factors would incorrectly give eight.

The torque magnitude is τ = pE sin θ. Because the angle is fixed, the factor sin θ does not change. The new values are p′ = p/2 and E′ = 8E, so τ′ = (p/2)(8E) sin θ = 4pE sin θ = 4τ. Therefore the torque becomes four times its original value. The field increase dominates the halving of the dipole moment by a net factor of four.

The potential energy of a dipole in a uniform electric field is U = −pE cos θ, where θ is the angle between p and E. For 0° < θ < 90°, cos θ is positive; with p and E nonzero, the leading negative sign makes U negative. At 90° the energy is zero, while between 90° and 180° it is positive. Thus option A gives the stated condition.

For a dipole in a uniform field, U = −pE cos θ. In the interval 90° < θ < 180°, cos θ is negative. The minus sign in the energy expression therefore makes U positive, assuming p and E are nonzero. At θ = 0° the energy is minimum and negative, and at θ = 90° it is zero. Hence the correct condition is the interval in option A.

The potential energy is U = −pE cos θ. When the dipole is parallel to the field, θ = 0° and U = −pE, its minimum possible value. A small rotation makes θ nonzero, decreases cos θ below 1, and therefore makes −pE cos θ less negative. The potential energy consequently increases, showing that the parallel orientation is stable. It does not decrease or remain zero.

For a dipole, U = −pE cos θ. In the opposite orientation, θ = 180°, so U = +pE, the maximum value. A slight rotation reduces θ below 180°, making cos θ greater than −1; consequently U becomes smaller than +pE. Thus the energy decreases as the dipole moves away from the unstable antiparallel orientation toward a lower-energy orientation. Therefore option A is correct.

The torque on an electric dipole in a uniform electric field is given by the vector relation τ⃗ = p⃗ × E⃗. Its magnitude is τ = pE sin θ, where θ is the angle between the dipole moment and the electric field, and its direction follows the right-hand rule. Thus option A identifies both vectors correctly. The other pairs do not form the electrostatic torque relation.

For a dipole in a uniform electric field, the torque is τ = pE sin θ and the potential energy is U = −pE cos θ. If the dipole moment and field point in the same direction, θ = 0°, so torque is zero and energy is minimum. A small angular displacement produces a restoring torque; therefore this is stable equilibrium. Maximum torque occurs at 90°, while opposite directions give unstable equilibrium.

Opposite directions mean that the angle between the dipole moment p and the electric field E is θ = 180°. Thus τ = pE sin 180° = 0, so the dipole is in rotational equilibrium. However, U = −pE cos 180° = +pE, which is the maximum potential energy. A small displacement lowers the energy and moves the dipole away from this position, so it is unstable equilibrium.

For perpendicular vectors, θ = 90°. The torque on an electric dipole is τ = pE sin θ, so τ = pE × 1 = pE, its maximum possible value for fixed p and E. The potential energy is U = −pE cos θ = 0. In a uniform field the net translational force on an ideal dipole is zero, so option A correctly combines the torque and energy results.

Zero net force and zero net torque are separate conditions. In a uniform electric field, the forces on the positive and negative charges of an ideal dipole are equal and opposite, so their vector sum is zero. If the dipole is inclined, these forces act along parallel but different lines and form a couple. Consequently τ = pE sin θ can be nonzero for 0° < θ < 180°, so zero force does not guarantee zero torque.

The potential energy of a dipole in a uniform electric field is U = −pE cos θ. The torque τ = pE sin θ acts so that the dipole tends to rotate toward θ = 0°, where its potential energy is minimum. This is the stable aligned orientation. It does not always increase energy; in fact, an unrestrained dipole naturally moves in the direction of decreasing potential energy. Therefore option A is correct.

The angle θ between p and E appears in both key relations for a dipole. The torque is τ = pE sin θ, while the potential energy is U = −pE cos θ. Therefore identifying θ correctly determines whether torque is zero or maximum and whether the energy is minimum, zero, or maximum. The angle does not change the charge unit, remove the field, or determine mass, so option A is correct.