Dipole in a Uniform External Field

For a dipole in a uniform electric field, the torque is τ = pE sin θ and the potential energy is U = −pE cos θ. When the dipole moment and field point in the same direction, θ = 0°, so τ = 0 and U is minimum. A small angular displacement produces a restoring torque, so the orientation is stable equilibrium. Maximum torque occurs at 90°, not at 0°.

Opposite directions give θ = 180°. Using τ = pE sin θ, the torque is zero at this orientation. However, U = −pE cos θ becomes +pE, its maximum value. A small displacement creates a torque that increases the displacement rather than restoring the original position. Therefore the antiparallel orientation is unstable equilibrium. Maximum torque occurs at 90°.

For a dipole in a uniform field, τ = pE sin θ and U = −pE cos θ. Perpendicular orientation means θ = 90°, so sin 90° = 1 and the torque becomes τmax = pE. Also cos 90° = 0, giving U = 0 when the zero of energy is chosen in the usual way. The net force remains zero in a uniform field, so option A is correct.

Zero net force concerns translational motion only: the vector sum of all forces is zero. It does not automatically make the sum of moments zero. For a dipole in a uniform electric field, equal and opposite forces act on its two charges, so the net force is zero. If their lines of action are separated, they form a couple with τ = pE sin θ, which can be nonzero. Thus option B is correct.

The potential energy of a dipole in a uniform electric field is U = −pE cos θ. The electric torque τ = pE sin θ acts so that the dipole tends toward θ = 0°, where the energy is minimum and the dipole is aligned with the field. Consequently, the torque generally drives the system toward decreasing potential energy. The other choices do not describe the physical action of electric torque.

The angle θ controls two important quantities for a dipole in a uniform electric field. Torque is τ = pE sin θ, while potential energy is U = −pE cos θ. Thus an incorrect angle can change whether torque is zero or maximum and whether the energy is minimum, zero, or maximum. The angle does not alter the unit of charge, remove the field, or determine mass, so option A is correct.

A dipole in a uniform electric field experiences equal and opposite forces on its two charges, so its net force is zero for every orientation. Its torque is τ = pE sin θ, which is zero when θ = 0° or 180°. Among the given choices, a dipole parallel to the field has θ = 0° and therefore satisfies both conditions. At 90°, torque is maximum, so the perpendicular option is incorrect.

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. The torque is τ = pE sin θ. Its maximum value, pE, occurs when sin θ = 1, that is, θ = 90°. Therefore the dipole moment is perpendicular to the electric field. Parallel and antiparallel positions have zero torque, not maximum torque.

The governing relations for a dipole in a uniform electric field are τ = pE sin θ for torque and U = −pE cos θ for potential energy. Therefore, first determine the dipole-moment direction and the angle θ between p and E. Next choose the appropriate relation and substitute the quantities with units. The other choices are invalid because direct addition, always assuming zero torque, or ignoring charge separation does not follow the physics of a dipole.

For a dipole in a uniform electric field, torque is τ = pE sin θ. Substituting p = 6 C m, E = 2 N/C, and sin 30° = 1/2 gives τ = 6 × 2 × 1/2 = 6 N m. The value 12 N m would result from forgetting the sine factor. A zero value applies at 0° or 180°, not at 30°. Hence option C is the only correct answer.

For a dipole, τ = pE sin θ, and the maximum occurs at 90°, where sin θ = 1. Thus τmax = pE. Rearranging gives p = τmax/E = 42 N m ÷ 7 N/C = 6 C m. Option B confuses the field with the moment, while the larger values result from multiplication rather than the required division.

At maximum torque, θ = 90° and sin θ = 1, so the dipole relation becomes τmax = pE. Solving for the field gives E = τmax/p = 54 N m ÷ 9 C m = 6 N/C. The metre units cancel correctly, leaving newtons per coulomb. The other choices come from using an input value directly or performing addition or subtraction instead of division.

The potential energy of a dipole in a uniform electric field is U = −pE cos θ. In the parallel position, θ = 0° and cos 0° = 1. Therefore U = −5 × 6 × 1 = −30 J. The negative sign indicates the aligned orientation is a lower-energy, stable orientation. Zero energy belongs to 90°, while a positive value occurs in the opposite orientation.

For a dipole, U = −pE cos θ. In the opposite or antiparallel position, the dipole moment makes θ = 180° with the electric field, so cos 180° = −1. Hence U = −(8)(4)(−1) = +32 J. The negative value belongs to the parallel orientation, and zero belongs to the perpendicular orientation; neither describes the stated opposite position.

For a dipole in a uniform electric field, the potential energy is U = −pE cos theta. In the parallel position, theta = 0 degrees, so U1 = −(3)(5) = −15 J. In the opposite position, theta = 180 degrees, so U2 = +(3)(5) = 15 J. For slow rotation, external work equals the increase in potential energy: Wext = U2 − U1 = 15 − (−15) = 30 J. Thus option B is correct; 15 J is only the magnitude of one energy value, while the negative answer represents work done by the field.

The interaction energy of a dipole is U = −pE cos theta. Initially the dipole is parallel to the field, so theta1 = 0 degrees and U1 = −(6)(2) = −12 J. Finally it is perpendicular, so theta2 = 90 degrees and U2 = 0. Therefore the external work required for a controlled rotation is the increase in potential energy, Wext = U2 − U1 = 0 − (−12) = 12 J. Option A is correct. The value 24 J would correspond to a parallel-to-antiparallel rotation, not this motion.

For a dipole in a uniform field, U = −pE cos theta. At the opposite position, theta = 180 degrees and U = +pE, which is the maximum potential energy. At the parallel position, theta = 0 degrees and U = −pE, the minimum value. The electric torque naturally drives the dipole toward alignment, so its potential energy decreases as it rotates from opposite to parallel. Hence option A is correct; it does not remain constant because the angle and therefore U change.

The potential energy of an electric dipole in a uniform field is U = −pE cos theta, where p is the dipole moment and E is the field magnitude. For theta = 60 degrees, cos 60 degrees = 1/2. Substitution gives U = −pE(1/2) = −pE/2. Therefore option A is correct. The positive sign in option B ignores the negative sign in the interaction-energy formula, while pE and zero apply to different angles.

Use the dipole-energy relation U = −pE cos theta. Here p = 8 C m, E = 2 N/C, and theta = 60 degrees, for which cos theta = 1/2. Thus U = −(8)(2)(1/2) J = −8 J. Option B is correct. A positive 8 J results from dropping the negative sign, 16 J results from forgetting the cosine factor, and zero would be correct only at 90 degrees.

The torque on a dipole is tau = pE sin theta. Its maximum value is tau_max = pE, reached when theta = 90 degrees. Therefore the fraction is tau/tau_max = sin theta. At theta = 30 degrees, sin 30 degrees = 1/2, so the torque is half the maximum torque. Option A is correct. √3/2 corresponds to 60 degrees, unity to maximum torque, and zero to parallel or antiparallel alignment.

For a dipole, torque is tau = pE sin theta, while the maximum torque is tau_max = pE. Dividing gives tau/tau_max = sin theta. At 60 degrees, sin 60 degrees = √3/2. Therefore the torque is √3/2 of its maximum value, making option B correct. One half is the value for 30 degrees, one fourth is not the relevant sine value here, and zero occurs only when the dipole is parallel or antiparallel to the field.

The torque on a dipole is tau = pE sin theta, so it is zero at theta = 0 degrees and 180 degrees. Stability is decided by potential energy U = −pE cos theta: at theta = 0 degrees, U is minimum, so a small displacement produces a restoring torque and the equilibrium is stable. At 180 degrees, energy is maximum and equilibrium is unstable. Thus option A is correct; perpendicular and 45-degree positions do not have zero torque.

Torque is given by tau = pE sin theta, so it vanishes when theta is 0 degrees or 180 degrees. The potential energy is U = −pE cos theta. At the opposite, or antiparallel, position theta = 180 degrees and U = +pE, its maximum value. A small angular displacement lowers the energy and causes the dipole to move farther from that position, so the equilibrium is unstable. Therefore option C is correct.

The torque on a dipole is tau = pE sin theta, so it is maximum when theta = 90 degrees. The potential energy is U = −pE cos theta. Since cos 90 degrees = 0, the potential energy at the maximum-torque orientation is U = 0, taking the standard zero reference used for dipole-field interaction energy. Therefore option C is correct. Minimum energy occurs at parallel alignment, maximum energy at opposite alignment, and no finite dipole-field energy becomes infinite here.

For a dipole in a uniform electric field, potential energy is U = −pE cos θ, while the torque magnitude is τ = pE sin θ. Minimum energy occurs when θ = 0°, because the dipole moment is parallel to the field. Therefore τ = pE sin 0° = 0. Option B is correct. Maximum torque occurs at 90°, not at the minimum-energy orientation; the other choices therefore do not apply.