Dipole in a Uniform External Field

Correct answer: A. 60°

Explanation: For a dipole in a uniform field, potential energy is U = −pE cos θ. Substituting U = −10 J and pE = 20 J gives −10 = −20 cos θ, so cos θ = 1/2. In the usual interval 0°–180°, this corresponds to θ = 60°. Therefore option A is correct. At 120° the energy would be +10 J, at 90° it would be zero, and at 180° it would be +20 J.

Correct answer: B. 120°

Explanation: The dipole potential-energy formula is U = −pE cos θ. With U = +14 J and pE = 28 J, we obtain 14 = −28 cos θ, hence cos θ = −1/2. In the interval 0°–180°, this gives θ = 120°, so option B is correct. At 60° the energy would be −14 J, while 30° and 0° would both give negative energy.

Correct answer: C. 30°

Explanation: The torque magnitude is τ = pE sin θ, so 8 = 16 sin θ and sin θ = 1/2. Thus θ may be 30° or 150° between 0° and 180°. Negative potential energy requires U = −pE cos θ < 0, which means cos θ > 0. Only 30° has a positive cosine, so option C is correct. At 150°, the energy would be positive.

Correct answer: D. 150°

Explanation: Since τ = pE sin θ, the given values yield 16 = 32 sin θ, or sin θ = 1/2. Therefore the possible angles are 30° and 150°. Positive potential energy means U = −pE cos θ > 0, so cos θ must be negative. This occurs at 150°, making option D correct. At 30° the energy is negative, and at 90° the torque would be maximum rather than 16.

Correct answer: A. 6√3 times

Explanation: Torque on a dipole is τ = pE sin θ. Hence the ratio is τ₂/τ₁ = (p₂/p₁)(E₂/E₁)(sin 60°/sin 30°). The first two factors are 2 and 3, giving 6; the angular factor is (√3/2)/(1/2) = √3. Therefore τ₂/τ₁ = 6√3, so option A is correct. Ignoring the angle gives only 6, while ignoring one scaling factor gives smaller results.

Correct answer: B. 2 times

Explanation: For a dipole, τ = pE sin θ. The ratio of new to old torque is (p₂/p₁)(E₂/E₁)(sin 30°/sin 90°). The parameter changes give (1/2)×8 = 4, while the angular factor is (1/2)/1 = 1/2. Thus τ₂/τ₁ = 4×1/2 = 2. The new torque is therefore twice the old torque, so option B is correct.

Correct answer: C. It becomes one and a half times

Explanation: The torque formula is τ = pE sin θ. Thus τ₂/τ₁ = (3)(1/2)(sin 120°/sin 60°). Since sin 120° = sin 60° = √3/2, the angular ratio is 1. The remaining factor is 3/2, so the new torque is one and a half times the original torque. Therefore option C is correct; the angle change does not alter the magnitude because the two sine values are equal.

Correct answer: A. The torque magnitude remains the same, while the sign of potential energy changes.

Explanation: For a dipole in a uniform electric field, the torque magnitude is τ = pE sin θ and the potential energy is U = −pE cos θ. Since sin 30° = sin 150° = 1/2, the torque magnitudes are equal. However, cos 30° is positive whereas cos 150° is negative, so U changes sign. Thus option A is correct; the other options confuse the sine and cosine dependences.

Correct answer: B. The torque remains the same and the energy changes from negative to positive.

Explanation: For a dipole, τ = pE sin θ and U = −pE cos θ. The values sin 45° and sin 135° are both √2/2, so the torque magnitude is unchanged. At 45°, cos θ is positive and U is negative; at 135°, cos θ is negative and U is positive. Therefore option B is correct. Neither quantity becomes zero at these angles.

Correct answer: C. Opposite or antiparallel to the field

Explanation: The torque magnitude is τ = pE sin θ, so it is zero at θ = 0° or 180°. The potential energy is U = −pE cos θ. At 0°, U = −pE, its minimum value; at 180°, U = +pE, its maximum value. Therefore the dipole is antiparallel to the field, making option C correct. A identifies the minimum-energy position instead.

Correct answer: D. Parallel to the field

Explanation: For a dipole in a uniform field, τ = pE sin θ and U = −pE cos θ. Zero torque occurs at 0° and 180°, but these positions do not have the same energy. At θ = 0°, U = −pE, which is the minimum possible value; at 180°, U = +pE, the maximum. Hence the dipole is parallel to the field, so D is correct.

Correct answer: A. The angle is 90° and the torque is maximum.

Explanation: The dipole potential energy is U = −pE cos θ. For U = 0, cos θ must be zero, which gives θ = 90° in the usual range from 0° to 180°. At this angle, τ = pE sin 90° = pE, the maximum torque. Therefore A is correct; at 0° and 180° the torque is zero, while 30° does not make the energy zero.

Correct answer: B. A couple can produce rotation even when the net force is zero.

Explanation: The electric forces on the positive and negative charges of a dipole in a uniform field are equal and opposite, so their vector sum, the net force, is zero. Because they act along different parallel lines, however, they form a couple with torque τ = pE sin θ. Net force governs translation, whereas torque governs rotation. Thus B correctly explains the motion.

Correct answer: A. Because the cosine of the angle can change sign.

Explanation: By definition, the electric dipole moment points from the negative charge to the positive charge. Its potential energy in a uniform field is U = −pE cos θ. Reversing the chosen direction changes θ to its supplementary angle, 180° − θ, and cos(180° − θ) = −cos θ. This reverses the calculated energy sign, so A is correct; the charge, field, and force units do not change.

Correct answer: A. It becomes negative and decreases.

Explanation: For a dipole, U = −pE cos θ. Zero energy corresponds to θ = 90°. Rotating slightly toward the parallel direction makes θ less than 90°, so cos θ becomes positive. Consequently U becomes negative. It is also lower than its initial zero value, because the field does positive work while the dipole moves toward the stable parallel orientation. Hence A is correct.

Correct answer: B. It becomes positive.

Explanation: The relation U = −pE cos θ gives zero potential energy at θ = 90°. A small rotation toward the opposite direction makes the angle greater than 90°, so cos θ becomes negative. Therefore −pE cos θ becomes positive, and the energy rises above zero. The antiparallel orientation at 180° is the maximum-energy position, not the minimum. Thus option B is correct.

Correct answer: A. Toward the parallel stable position

Explanation: A freely released dipole experiences torque τ = pE sin θ and tends to move toward lower potential energy. Its energy is U = −pE cos θ, which is reduced as the angle decreases from 135° toward 0°. The parallel orientation has minimum energy and is stable, whereas the antiparallel orientation has maximum energy and is unstable. Therefore the dipole tends toward the parallel position, so A is correct.

Correct answer: A. Net force is zero, but torque can be nonzero

Explanation: The governing idea is the separate treatment of translation and rotation. In a uniform electric field, the positive and negative charges experience equal and opposite forces, so their vector sum is zero; hence F_net = 0 and the centre of mass has no translational acceleration. However, the forces act along different lines and form a couple. Therefore τ = pE sinθ may be nonzero, producing angular acceleration and rotation. Option B wrongly makes torque zero in every orientation.

Correct answer: A. Field values can differ at the two charges

Explanation: A dipole contains equal and opposite charges at two distinct positions. In a uniform field, both charges receive forces of equal magnitude, giving zero resultant force. In a nonuniform field, the field values at the two positions can differ, so the forces qE and -qE need not cancel. Their vector sum can therefore be nonzero, while a torque may also act. Option A states this positional field difference; the other options contradict the definition of a dipole or confuse force with torque.

Correct answer: B. Right-hand rule

Explanation: For an electric dipole in a uniform field, the torque is represented by the vector relation τ = p × E, where p is the dipole moment and E is the electric field. The magnitude is pE sinθ, while the direction is perpendicular to the plane containing p and E. Applying the right-hand rule to this cross product gives the torque direction, so option B is correct. Energy conservation can describe changes in energy but does not directly give the vector direction; Ohm’s law concerns circuits.

Correct answer: A. The cross product of parallel vectors is zero

Explanation: The torque on a dipole is τ = p × E, and its magnitude is τ = pE sinθ. In the parallel position, the angle between the dipole moment p and field E is 0° (or 180° for antiparallel alignment), so sinθ = 0. Consequently, the cross product and torque vanish even though p, E, and the charge separation remain finite. Thus option A is correct; none of the physical quantities disappears or becomes infinite.

Correct answer: A. Energy is maximum, and a small displacement leads toward lower energy

Explanation: For a dipole in a uniform field, potential energy is U = -pE cosθ. When p and E are opposite, θ = 180° and U = +pE, the maximum value. The torque τ = pE sin180° is zero only at the exact orientation, so it is an equilibrium position. A small angular displacement makes the torque act farther away from the aligned direction and lowers the energy, rather than restoring the dipole; therefore the equilibrium is unstable. Option A is correct.

Correct answer: A. The parallel position has minimum energy

Explanation: The potential energy of a dipole in a uniform electric field is U = -pE cosθ. For the parallel position, θ = 0° and U = -pE, which is the minimum possible energy. After a small displacement, the field produces a torque τ = pE sinθ directed toward decreasing θ, so it tends to restore the dipole to θ = 0°. Thus option A is correct. The torque is zero exactly at equilibrium but becomes restoring after displacement; it is not always maximum.

Correct answer: A. Torque is maximum and energy is zero

Explanation: The conditions sinθ > 0 and cosθ = 0 identify θ = 90° for the usual angle range from 0° to 180°. The dipole torque has magnitude τ = pE sinθ, so τ = pE, its maximum value. Its potential energy is U = -pE cosθ = 0. Therefore option A is correct. Options B and C describe zero torque at parallel or antiparallel orientations, while option D does not follow from the angle alone.

Correct answer: B. Unstable equilibrium

Explanation: The conditions sinθ = 0 and cosθ = -1 correspond to θ = 180°. Thus the dipole moment is antiparallel to the electric field. At this orientation, τ = pE sin180° = 0, but the potential energy U = -pE cos180° = +pE, its maximum value. A small displacement lowers the energy and produces a torque away from the original orientation, so the equilibrium is unstable. Hence option B is correct, not stable equilibrium or maximum torque.