Dipole in a Uniform External Field

For slow rotation without energy loss, external work equals the increase in potential energy. Initially, at 0°, Ui = −pE = −2 × 6 = −12 J. Finally, at 120°, Uf = −pE cos 120° = +6 J. Thus Wext = Uf − Ui = 6 − (−12) = 18 J, so option A is correct; the other choices omit one energy contribution or assume no work.

The torque magnitude is τ = pE sin θ, so τ/τmax = sin θ. Half maximum torque requires sin θ = 1/2. The energy is U = −pE cos θ, and U/(pE) = −√3/2 requires cos θ = √3/2. Both conditions are satisfied at θ = 30°. Therefore option A is correct; 60° reverses the sine and cosine fractions, while 90° gives maximum torque but zero energy.

For a dipole, U = −pE cos θ. Initially, at 30°, Uᵢ = −20 cos 30° = −20(√3/2) = −10√3 J. Finally, at 150°, U_f = −20 cos 150° = −20(−√3/2) = +10√3 J. Hence ΔU = U_f − Uᵢ = 10√3 − (−10√3) = 20√3 J. Option B is correct; the positive value means energy increases during rotation.

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. Opposite directions mean θ = 180°, so sin 180° = 0 and cos 180° = −1. Hence τ = 0, while U = +pE, its maximum value. Parallel alignment gives minimum energy, whereas perpendicular alignment gives maximum torque, so option C is correct.

A dipole placed in a uniform electric field experiences equal and opposite forces on its two charges, so its net force is zero. Its torque is τ = pE sin θ, which is maximum when sin θ = 1, or θ = 90°. Therefore the dipole must be perpendicular to a nonzero uniform field. In a parallel position the torque is zero, and a nonuniform field can produce a net force, so option B is correct.

In a uniform electric field, the forces on the positive and negative charges are equal and opposite, so the net force on the dipole is zero. Its potential energy is U = −pE cos θ. This is minimum when cos θ = 1, or θ = 0°, meaning that the dipole moment is parallel to the field. Perpendicular orientation gives zero energy, and opposite orientation gives maximum energy; therefore option A is correct.

For a dipole in a uniform field, U = −pE cos θ. Stable equilibrium is at θ = 0°, where Ustable = −pE, while unstable equilibrium is at θ = 180°, where Uunstable = +pE. If the dipole is moved slowly, the external work equals the increase in potential energy: Wext = Uunstable − Ustable = pE − (−pE) = 2pE. Thus option B is correct.

The dipole potential energy is U = −pE cos θ. At unstable equilibrium, θ = 180° and U = +pE; at stable equilibrium, θ = 0° and U = −pE. The electric field does work equal to the decrease in potential energy: Wfield = Ui − Uf = pE − (−pE) = 2pE. This work is positive because the field drives the dipole toward stable alignment, so option C is correct.

For a dipole in a uniform field, τ = pE sin θ. Zero torque can result when θ = 0° or θ = 180°, so the dipole may be parallel or antiparallel to the field. The parallel orientation is stable, whereas the antiparallel orientation is unstable. Therefore zero torque alone does not prove stable equilibrium. Option B is the conclusion that is not certain; the other statements follow directly.

The torque magnitude on a dipole is τ = pE sin θ, so it reaches its maximum value pE at θ = 90°. At this angle, the potential energy U = −pE cos θ becomes zero because cos 90° = 0. Maximum torque does not mean maximum or minimum potential energy; nor does it imply maximum net force, since the net force in a uniform field is zero. Hence option A is correct.

The torque magnitude is τ = pE sin θ. Initially, τi = pE sin 30° = pE/2. Finally, the dipole moment is 2p and the angle is 90°, so τf = (2p)E sin 90° = 2pE. Therefore the ratio is τf/τi = 2pE ÷ (pE/2) = 4. The torque becomes four times its initial value, so option B is correct.

Use τ = pE sin θ. Initially, τi = pE sin 90° = pE. After the change, the field is 2E and the angle is 30°, so τf = p(2E)sin 30° = 2pE × 1/2 = pE. Thus τf/τi = 1, meaning the torque remains unchanged. The increase in field exactly compensates for the decrease in the sine factor, so option A is correct.

The potential energy of a dipole in a uniform electric field is U = −pE cos θ. Given pE = 16 J and U = −8 J, substitute these values: −8 = −16 cos θ. Dividing by −16 gives cos θ = 1/2. The standard angle between 0° and 180° having cosine 1/2 is θ = 60°. Therefore option B is correct; 30°, 90°, and 120° give different energy values.

For a dipole in a uniform electric field, the potential energy is U = −pE cos θ. Here pE = 18 and U = 9 J, so 9 = −18 cos θ, giving cos θ = −1/2. In the usual range 0° to 180°, this occurs at θ = 120°. The other listed angles give different cosine values or the wrong sign of energy, so option C is correct.

The torque on a dipole is τ = pE sin θ, while its maximum value is τmax = pE. Therefore τ/τmax = sin θ = √3/2. In the range 0°–180°, sine has this value at 60° and 120°. Thus both angles are possible; 30° and 150° give 1/2, while 90° gives the maximum ratio 1. Option B is correct.

Since τ = pE sin θ and τmax = pE, the condition τ = τmax/2 gives sin θ = 1/2. The possible angles between 0° and 180° are 30° and 150°. Potential energy is U = −pE cos θ; for U to be negative, cos θ must be positive. That selects 30°, whereas cos 150° is negative. Therefore option A is correct.

Using τ = pE sin θ and τmax = pE, half maximum torque means sin θ = 1/2. Hence the possible angles in the standard range are 30° and 150°. The dipole energy is U = −pE cos θ. Positive energy requires cos θ < 0, which occurs at 150° among these choices. Therefore option C is the unique correct answer.

For a dipole in a uniform electric field, U = −pE cos θ. If U = 0 and pE is nonzero, then cos θ = 0, so θ = 90°. The torque is τ = pE sin θ; at 90°, sin θ = 1, giving τ = pE = τmax. Thus the torque is maximum. Zero torque occurs at 0° or 180°, not at 90°, so option B is correct.

The dipole energy is U = −pE cos θ. Stable equilibrium occurs at θ = 0°, where U has its minimum value, −pE. For a small displacement δ, cos δ is slightly less than 1, so U = −pE cos δ becomes greater than −pE. Thus the energy increases on either side of the stable position, providing a restoring tendency. Option B is correct.

For a dipole, U = −pE cos θ. Unstable equilibrium occurs at θ = 180°, where U = +pE, the maximum possible value. If the dipole is displaced slightly from 180°, the cosine becomes slightly greater than −1, so −pE cos θ becomes less than +pE. Therefore the potential energy decreases, and the dipole tends to move farther from the unstable position. Option A is correct.

The torque magnitude on a dipole is τ = pE sin θ. The dipole moment p is fixed. If the new sine factor is half the original value, then τnew = pEnew(sin θold/2). To make this equal to τold = pEold sin θold, Enew/2 must equal Eold. Hence Enew = 2Eold: the field must double. Therefore option A is correct.

Newton's second law for the centre of mass is Fnet = M acom. Therefore, when the net external force on the dipole is zero, its centre of mass has no net translational acceleration. This does not imply zero torque: the two equal and opposite forces can form a couple, producing τ = pE sin θ when the dipole is not parallel or antiparallel to the field. Option A is correct.

In a uniform field, the electric field at the positive and negative charges is the same, so the forces are equal and opposite and the net force is zero. In a nonuniform field, the field values at the two charge locations can differ. Consequently, the two forces need not cancel, and a net force can act on the dipole, in addition to possible torque. Therefore option B is correct.

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. Zero torque requires θ = 0° or 180°. At 0°, U is negative, whereas at 180°, cos θ = −1 and U = +pE, which is positive. Hence the dipole is antiparallel, or opposite to the field. The perpendicular position gives maximum torque, not zero torque.

The torque on a dipole is τ = pE sin θ, so it vanishes at θ = 0° and 180°. Its potential energy is U = −pE cos θ. For θ = 0°, U = −pE, which is negative and corresponds to the stable parallel orientation. For θ = 180°, U = +pE, which is positive. Therefore the dipole must be parallel to the electric field. A perpendicular orientation would produce maximum torque.