Dipole in a Uniform External Field

Correct answer: A. 1/2

Explanation: For an electric dipole, τ = pE sin θ and τmax = pE because the sine reaches its maximum value of one at 90°. Dividing gives τ/τmax = sin θ. At θ = 30°, sin 30° = 1/2, so the torque is half the maximum torque. Therefore option A is correct. √3/2 belongs to 60°, unity to 90°, and zero to 0° or 180°.

Correct answer: A. Dipole parallel to the field

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

Correct answer: C. Dipole opposite to the field

Explanation: For a dipole in a uniform electric field, torque is τ = pE sin θ. At θ = 180°, sin θ = 0, so the torque vanishes. The potential energy is U = −pE cos θ, which is maximum at 180°. A small displacement lowers the energy and makes the dipole move farther from this orientation; therefore it is unstable equilibrium. The parallel position also has zero torque, but it is stable because its energy is minimum.

Correct answer: C. Zero

Explanation: The torque on a dipole is τ = pE sin θ, so its magnitude is maximum when θ = 90°. The potential energy is U = −pE cos θ. Since cos 90° = 0, U becomes zero when the usual zero-energy reference is used. Thus option C is correct. Minimum energy occurs at θ = 0°, and maximum energy occurs at θ = 180°, so options A and B describe the two aligned positions rather than the maximum-torque position.

Correct answer: B. Zero

Explanation: For a dipole in a uniform field, U = −pE cos θ. The minimum value, −pE, occurs when θ = 0°, meaning that the dipole moment is parallel to the electric field. Torque is τ = pE sin θ, and sin 0° = 0. Therefore the torque is zero, so option B is correct. This is stable equilibrium because a small angular displacement increases the potential energy rather than lowering it.

Correct answer: B. Change in potential energy

Explanation: During a slow, controlled rotation, the dipole is treated as moving quasistatically, so its kinetic energy does not acquire a lasting change. The external agent supplies work against or along the electric torque. Consequently, the work done by the external agent changes the dipole’s potential energy: W_external = ΔU. For example, rotating from θ₁ to θ₂ gives ΔU = −pE(cos θ₂ − cos θ₁). Hence option B is correct.

Correct answer: A. The potential energy and angle identification may become wrong

Explanation: By definition, the electric dipole moment points from the negative charge to the positive charge. The angle θ used in τ = pE sin θ and U = −pE cos θ is measured between this directed dipole moment and the external field. Reversing the direction changes the interpreted angle, often replacing θ with 180° − θ, and can therefore reverse or alter the energy and torque conclusion. It cannot change charge units, erase the field, or automatically double a distance.

Correct answer: A. The charge magnitudes are equal and the field is the same at both positions

Explanation: The electric force on a charge is F = qE. An ideal dipole contains charges +q and −q, whose magnitudes are equal. In a uniform field, E has the same magnitude at the locations of both charges. Therefore the force magnitudes are |F₊| = qE and |F₋| = qE, although their directions are opposite. This equality produces zero net force but can still produce a torque. Equal signs or zero separation are not properties of a dipole.

Correct answer: B. The net force may be nonzero

Explanation: For charges +q and −q, the force magnitudes are qE₊ and qE₋. In a uniform field E₊ = E₋, so the forces cancel and the net force is zero. In a nonuniform field, the field values at the two charge positions can differ, giving a resultant force approximately related to the field gradient. Thus the dipole may translate while also rotating. Nonuniformity does not remove the dipole moment or guarantee zero torque.

Correct answer: A. There is no translational acceleration because the net force is zero

Explanation: In a uniform electric field, the +q and −q charges of a dipole experience equal forces in opposite directions. Their vector sum is therefore zero, even though the separated forces can produce a torque. Newton’s second law for the centre of mass is F_net = M a_cm. Since F_net = 0, the translational acceleration a_cm is zero. The centre of mass may remain at rest or move with constant velocity, but it does not acquire acceleration from this electric force pair.

Correct answer: B. It increases

Explanation: The torque magnitude for a dipole is |τ| = pE sin θ. Assuming p and E remain constant, at 30° the factor is sin 30° = 1/2, whereas at 90° it is sin 90° = 1. Thus the torque changes from pE/2 to pE, so it increases and reaches its maximum value. Option D is not appropriate because the question asks for magnitude, which is nonnegative; the vector direction may change separately.

Correct answer: B. It decreases

Explanation: For a dipole in a uniform field, the torque magnitude is |τ| = pE sin θ. At 90°, sin 90° = 1, so the torque is pE, its maximum value. At 180°, sin 180° = 0, so the torque is zero. As the angle increases continuously from 90° to 180°, the sine value decreases from one to zero; therefore the torque magnitude decreases. It does not become negative or infinite because the question concerns magnitude.

Correct answer: B. Ninety degrees

Explanation: For a dipole in a uniform electric field, potential energy is U = −pE cos θ and the magnitude of torque is τ = pE sin θ. At θ = 90°, cos 90° = 0, so U = 0, while sin 90° = 1, giving the maximum possible torque pE. At 0° or 180°, torque is zero, so option B is correct.

Correct answer: A. Toward the stable position

Explanation: For a dipole in a uniform field, stable equilibrium occurs when the dipole moment is parallel to the field and U = −pE is minimum. A small angular displacement produces a torque that opposes the displacement, just as a restoring torque acts on a pendulum. Thus the dipole tends to return to its parallel, stable orientation, making option A correct; the other choices do not describe this restoring action.

Correct answer: B. It starts rotating toward the parallel stable position

Explanation: The antiparallel orientation, θ = 180°, is unstable because U = −pE cos θ has its maximum value there. If the dipole is displaced slightly, the torque acts so that the angle decreases toward 0°, where the dipole is parallel to the field and energy is minimum. Therefore it begins rotating toward the parallel stable position, so option B is correct.

Correct answer: C. It remains unchanged

Explanation: The magnitude of torque on an electric dipole is τ = pE sin θ. Since the angle is unchanged, sin θ is the same before and after the changes. With p′ = 2p and E′ = E/2, the new torque is τ′ = (2p)(E/2)sin θ = pE sin θ = τ. The increase in p exactly cancels the decrease in E, so option C is correct. The other choices ignore this cancellation.

Correct answer: C. Six times

Explanation: For an electric dipole in a uniform field, τ = pE sin θ. Since the angle is unchanged, the sine factor is constant. Replacing p by 3p and E by 2E gives τ′ = (3p)(2E)sin θ = 6pE sin θ = 6τ. Therefore the torque becomes six times its original value, so option C is correct.

Correct answer: B. It becomes double

Explanation: The dipole torque is τ = pE sin θ. With the angle fixed, only the product pE changes. The new product is (p/2)(4E) = 2pE, so τ′ = 2pE sin θ = 2τ. Thus the torque becomes double its original value. Option A would apply if only p were halved, while option C ignores the simultaneous change in p.

Correct answer: B. Energy increases

Explanation: When the dipole is parallel to the field, θ = 0° and U = −pE, the minimum possible potential energy. After a small rotation, θ becomes nonzero and cos θ becomes slightly less than 1, so U = −pE cos θ becomes less negative and therefore increases. The change is approximately ΔU ≈ pEθ²/2 for a small angle. Hence option B is correct.

Correct answer: A. Energy decreases

Explanation: The antiparallel orientation has θ = 180° and U = −pE cos 180° = +pE, which is the maximum potential energy. A small rotation makes the angle 180° − δ, and cos(180° − δ) = −cos δ, whose magnitude is slightly less than 1. Consequently U decreases from +pE. Therefore option A is correct; the energy does not become infinite or remain fixed.

Correct answer: A. Dipole moment and electric field

Explanation: The governing relation for a dipole in a uniform electric field is τ = p × E, where p is the dipole-moment vector and E is the electric-field vector. Its magnitude is τ = pE sin θ, and its direction follows the right-hand rule. Therefore option A is correct. The other pairs do not define electric-dipole torque.

Correct answer: A. Stable equilibrium

Explanation: For a dipole in a uniform field, the potential energy is U = −pE cos θ and the torque magnitude is τ = pE sin θ. When p and E point in the same direction, θ = 0°, so torque is zero and energy is minimum. A small angular displacement produces a restoring torque, making this stable equilibrium. Thus A is correct.

Correct answer: B. Unstable equilibrium

Explanation: Opposite directions correspond to θ = 180°. Using τ = pE sin θ, the torque is zero, while U = −pE cos θ becomes +pE, its maximum value. A small displacement makes the torque increase the angular departure rather than restore alignment. Hence the antiparallel orientation is unstable equilibrium, so option B is correct.

Correct answer: A. Torque maximum and energy zero

Explanation: Perpendicular vectors make θ = 90°. The dipole torque is τ = pE sin θ, so τ = pE, its maximum possible value. The potential energy is U = −pE cos θ, which becomes zero because cos 90° = 0. In a uniform field the net force is zero, so option A is the only correct statement.

Correct answer: B. No, torque can exist when there is an angle

Explanation: Zero net force concerns translation, not rotation. In a uniform electric field, the forces on the positive and negative charges of a dipole are equal and opposite, so their vector sum is zero. However, if the dipole makes an angle θ with the field, these forces form a couple with τ = pE sin θ. Thus torque may be nonzero, making B correct.