Dipole in a Uniform External Field

Maximum torque occurs when sin θ = 1, so τmax = pE. Rearranging gives p = τmax/E = 99/11 = 9 C m. Therefore option A is correct. Option B simply repeats the field value, option C results from an incorrect multiplication idea, and option D is 99 × 11 rather than the required division.

At maximum torque, sin θ = 1, so τmax = pE. Solving for the field gives E = τmax/p = 108/12 = 9 N/C. Hence option B is correct. Option A confuses the given dipole moment with the field, option C subtracts or combines values incorrectly, and option D multiplies 108 by 12 instead of dividing.

For a dipole in a uniform field, U = −pE cos θ. Substituting pE = 28 J and U = −14 J gives −14 = −28 cos θ, so cos θ = 1/2. In the listed angular range, this corresponds to θ = 60°. At 120° the energy would be positive, at 90° it would be zero, and at 180° it would be +28 J. Therefore option A is correct.

Use the dipole-energy equation U = −pE cos θ. With U = 20 J and pE = 40 J, 20 = −40 cos θ, giving cos θ = −1/2. The angle among the options with this cosine is 120°. At 60° or 30°, cosine is positive and energy would be negative; at 0° the energy would be −40 J. Thus option B is correct.

The maximum torque is pE = 24, so τ/τmax = 12/24 = 1/2. Therefore sin θ = 1/2, giving θ = 30° or 150°. Since U = −pE cos θ is negative, cos θ must be positive. This selects 30°, because cos 30° is positive while cos 150° is negative. Option C is therefore correct.

Here τmax = pE = 48, so τ/τmax = 24/48 = 1/2 and hence sin θ = 1/2. The possible listed angles are 30° and 150°. Positive potential energy requires U = −pE cos θ > 0, so cos θ must be negative. Only 150° has a negative cosine; therefore option D is correct. At 30°, the energy is negative.

For a dipole, τ = pE sin θ, so torque is zero at θ = 0° and 180°. Its potential energy is U = −pE cos θ. At 0° the energy is minimum, U = −pE, whereas at 180° it is maximum, U = +pE. Therefore the dipole moment is antiparallel to the field, making option C correct. Perpendicular orientation gives maximum torque instead.

The torque relation τ = pE sin θ gives zero torque at θ = 0° or 180°. The potential energy relation U = −pE cos θ shows that U is minimum when cos θ = 1, which occurs at θ = 0°. Hence the dipole moment is parallel to the electric field, so D is correct. The antiparallel position also has zero torque but represents maximum, not minimum, energy.

For a dipole in a uniform field, U = −pE cos θ. Setting U = 0 gives cos θ = 0, so θ = 90° within the usual 0°–180° range. At this angle, τ = pE sin 90° = pE, which is the maximum torque. Therefore A is correct. At 0° or 180° the energy is not zero and the torque is zero.

In a uniform electric field, the two charges of a dipole experience equal and opposite forces, so their vector sum is zero and there is no translational acceleration. Because the forces act along different lines when the dipole is inclined, they form a couple with torque τ = pE sin θ. This torque causes rotation despite zero net force, so B is correct.

The dipole energy is U = −pE cos θ. Zero energy occurs at θ = 90°. A small rotation toward the parallel direction makes θ slightly less than 90°, so cos θ becomes positive. Therefore U becomes negative. Since the dipole is moving toward θ = 0°, the minimum-energy orientation, its energy decreases from zero. Thus option A is correct.

For a dipole, U = −pE cos θ, and U is zero at θ = 90°. Rotating slightly toward the opposite or antiparallel direction makes θ greater than 90°. In that range cos θ is negative, so −pE cos θ is positive. The energy therefore rises above zero, although it is not yet maximum unless θ reaches 180°. Hence B is correct.

A freely released dipole experiences torque τ = pE sin θ and tends to rotate toward lower potential energy. Its energy is U = −pE cos θ, which is minimum at θ = 0°, when the dipole moment is parallel to the field. Although the initial angle 135° has a nonzero torque, the dipole does not settle at 90°; it moves toward the parallel stable orientation. Thus A is correct.

The governing idea is the separate treatment of translation and rotation. In a uniform electric field, the +q and −q charges experience equal and opposite forces, so the resultant force is zero and F_net = M a gives zero acceleration of the centre of mass. Because these forces act along different lines, they can form a couple with torque τ = pE sin θ, allowing rotation. Thus A is correct; the other choices confuse force, torque, mass, or field energy.

A nonuniform field has different magnitudes or directions at different positions. The two charges of a dipole are separated, so the forces qE and −qE need not have equal magnitudes when evaluated at their respective locations. Their vector sum can therefore be nonzero, producing translational acceleration as well as possibly torque. Hence A is correct; a dipole contains opposite charges, torque is directional, and its separation is not required to vanish.

Torque on an electric dipole is represented by the vector relation τ = p × E, where p is the dipole moment and E is the external field. The direction of a cross product is perpendicular to both vectors and is determined by the right-hand rule. Therefore B is correct. Energy conservation can analyze work and stability, but it does not directly specify the cross-product direction; Ohm’s law and the distractor about mass are unrelated.

For a dipole in a uniform field, torque is τ = p × E, whose magnitude is τ = pE sin θ. In the parallel position θ = 0°, so sin 0° = 0 and the vector product vanishes. Consequently the torque is zero even though both p and E may be nonzero. Thus A is correct. The field, charges, and separation do not disappear or become infinite; those claims are physically irrelevant.

By definition, the electric dipole moment points from the negative charge toward the positive charge. The stated electric-field direction is from positive toward negative, which is exactly opposite to the dipole-moment direction. The smaller angle between two opposite vectors is 180°. Therefore D is correct. Zero degrees would mean parallel directions, 90° would mean perpendicular directions, and 30° does not describe the stated opposite orientation.

The governing idea is that a dipole has a directed dipole moment, pointing from the negative charge to the positive charge. First identify this direction and measure the angle with the external field. Then apply τ = pE sinθ for torque and U = −pE cosθ for potential energy. Therefore, option A is safest; the other choices make unjustified assumptions.

For a dipole of moment p in a uniform electric field E, the torque magnitude is τ = pE sin θ, where θ is the angle between p and E. Since sin θ has its maximum value, 1, at 90°, the torque is greatest when the dipole is perpendicular to the field. At 0° or 180° it is zero, and if E is zero there is no torque.

A dipole contains equal charges, +q and −q, separated by a small distance. In a uniform field, the forces on them have equal magnitude qE and opposite directions, so their vector sum is zero. Therefore option A is correct for net translational force. The dipole can still experience torque, τ = pE sin θ, so options involving pE or qE do not describe the net force.

The torque on an electric dipole in a uniform field is τ = pE sin θ. It becomes zero when sin θ = 0, which occurs at θ = 0° and θ = 180°. Thus the dipole must be parallel or antiparallel to the field, making option C correct. At 90° the torque is maximum, and changing p changes the magnitude but does not by itself make torque zero.

The potential energy of a dipole in a uniform electric field is U = −pE cos θ. Stable equilibrium occurs at minimum potential energy, which is obtained when cos θ = 1, or θ = 0°. Hence the dipole moment points along the field and option A is correct. The antiparallel position has maximum energy and is unstable; other angles have nonzero torque.

For a dipole in a uniform field, U = −pE cos θ. At θ = 180°, cos θ = −1, so U = +pE, its maximum value. A small angular displacement then produces a torque that moves the dipole away from this orientation and toward alignment with the field. Therefore option B is the unstable equilibrium position; alignment at 0° is stable.

The potential energy of an electric dipole in a uniform field is U = −pE cos θ, where θ is the angle between the dipole moment and the field. At θ = 0°, cos θ = 1, so U = −pE, the minimum possible value for fixed p and E. Thus option A is correct. At 90° the energy is zero, while at 180° it is maximum, +pE.