Dipole in a Uniform External Field

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°.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.