Dipole in a Uniform External Field

In a uniform electric field, the positive and negative charges of a dipole experience forces of equal magnitude and opposite direction, so their vector sum is zero. However, these forces act at different points and form a couple. Its torque has magnitude τ = pE sin θ and is nonzero for a general angle θ. It becomes zero only when the dipole is parallel or antiparallel to the field.

The two charges of a dipole experience equal and opposite electric forces in a uniform field. Because these forces act along parallel lines at different points, they form a couple: the net force is zero but a turning effect remains. The couple produces torque τ = pE sin θ, which can rotate the dipole. Friction, weight and pressure are not the characteristic mechanical effects described here.

For a dipole in a uniform field, U = −pE cos θ. At θ = 0°, the dipole is parallel to the field and has minimum potential energy, so this is a stable equilibrium. If it is turned slightly, the torque τ = pE sin θ acts in a restoring sense and tends to decrease the displacement, bringing the dipole back toward the parallel direction. The charges and field do not disappear.

The dipole’s potential energy in a uniform field is U = −pE cos θ. At θ = 180°, the dipole is opposite to the field and U = +pE, the maximum value. A small angular displacement makes the torque act away from this orientation and toward smaller θ, ultimately favoring the parallel, lower-energy state. Thus the antiparallel position is unstable; the net force need not become large.

The dipole potential energy is U = −pE cos θ. In the antiparallel position, θ = 180° and U = +pE; in the parallel position, θ = 0° and U = −pE. Therefore the change is ΔU = (−pE) − (+pE) = −2pE. The negative change means that the energy decreases as the dipole moves from opposite to parallel orientation. Option B would describe the reverse motion.

The dipole energy is U = −pE cos θ. It changes from −pE at θ = 0° to +pE at θ = 180°, so ΔU = +2pE. If the dipole is moved slowly and its kinetic energy does not change, the external agent must supply work equal to this positive increase in potential energy. The field itself does negative work during this forced motion. Hence the required external work is positive.

Stable equilibrium means that a small displacement produces a restoring effect, causing the system to tend toward its original position. For a dipole in a uniform field, the parallel orientation has U = −pE, the minimum energy. If the dipole is slightly rotated, the torque acts to reduce the angular displacement and restore alignment with the field. Thus stability is defined by a restoring tendency, not by zero energy or the permanent absence of force.

The torque on an electric dipole in a uniform electric field is τ = pE sin θ. Here p = 2 C m, E = 3 N/C, and θ = 90°, for which sin 90° = 1. Thus τ = (2)(3)(1) = 6 N m. The units reduce to N m because C in the dipole moment cancels C in the field unit. Therefore option A is correct; zero would occur only for a parallel or antiparallel dipole.

The potential energy of an electric dipole in a uniform electric field is U = −pE cos θ. It is minimum when cos θ = 1, which occurs at θ = 0°, meaning the dipole moment is parallel to the field. A minimum-energy orientation is stable equilibrium. At 180° the energy is maximum and the equilibrium is unstable, while other angles are not equilibrium positions. Therefore option A is correct.

The potential energy of a dipole in a uniform electric field is U = −pE cos θ. At the opposite orientation, θ = 180° and U = +pE, the maximum value. A small displacement allows the dipole to rotate toward smaller potential energy, ultimately reaching θ = 0°, where it is parallel to the field and energy is minimum. The rotation is therefore caused by the electric torque and energy reduction, not by charge loss or a changing mass. Option A is correct.

The direction of the dipole moment and the angle θ with the electric field determine the main results. Torque is τ = pE sin θ, while potential energy is U = −pE cos θ. Therefore, the first useful step is to identify the direction of p and measure its angle with E. Diagram colour, page number, and question length have no role in the physics calculation. Hence option D is correct.

In a uniform electric field, the positive and negative charges of a dipole experience forces of equal magnitude because the field is the same at both locations. The forces point in opposite directions, so their vector sum and net force are zero. Since they act at separate points, however, they form a couple with torque τ = pE sinθ. Thus option A correctly explains the possible rotation.

The torque on an electric dipole in a uniform field is τ = pE sinθ. Here p = 2 C m, E = 5 N/C, and θ = 90°, so sin90° = 1. Substitution gives τ = 2 × 5 × 1 = 10 N m. The product has the correct torque unit, so option B is correct; option A would apply at zero angle.

The governing relation is τ = pE sinθ. With p = 5 C m, E = 2 N/C, and θ = 30°, use sin30° = 1/2. Thus τ = 5 × 2 × 1/2 = 5 N m. Ten N m ignores the angular factor, while zero N m would correspond to a parallel or antiparallel orientation. Hence option B is correct.

For a dipole in a uniform electric field, the torque magnitude is τ = pE sinθ. Substituting p = 6 C m, E = 4 N/C, and sin30° = 1/2 gives τ = 6 × 4 × 1/2 = 12 N m. The value 24 N m would omit the sine factor, and zero would be valid only at θ = 0° or 180°. Therefore option A is correct.

For a dipole, τ = pE sinθ. At maximum torque, θ = 90° and sinθ = 1, so τmax = pE. Rearranging gives p = τmax/E = 18 N m ÷ 3 N/C = 6 C m. The answer is not obtained by adding or multiplying the given values; the defining maximum-torque relation gives option A.

Maximum dipole torque occurs at 90°, so τmax = pE because sin90° = 1. Solving for the field gives E = τmax/p = 24 N m ÷ 6 C m. The metre units cancel, leaving E = 4 N/C. Thus option B is correct; options A, C, and D result from incorrect division or combination of the given quantities.

The potential energy of an electric dipole in a uniform electric field is U = −pE cos θ, where θ is the angle between the dipole moment and the field. For the opposite orientation, θ = 180° and cos 180° = −1. Therefore U = −(2)(7)(−1) = +14 J. Hence option D is correct. A negative value would correspond to the parallel orientation, while zero would occur at 90°.

For a dipole in a uniform electric field, the potential energy is U = −pE cos θ. In the perpendicular position, the angle is θ = 90°, so cos 90° = 0. Substitution gives U = −(5)(2)(0) = 0 J. Thus option C is correct. The values 10 J and −10 J would apply to parallel and opposite orientations, respectively, not to the perpendicular orientation.

For a slow rotation with no change in kinetic energy, external work equals the increase in dipole potential energy. Initially, θ = 0°, so Ui = −pE = −(3)(4) = −12 J. Finally, θ = 180°, so Uf = +pE = +12 J. Thus Wext = Uf − Ui = 12 − (−12) = 24 J. Therefore option B is correct; 12 J is only the magnitude of one endpoint energy.

The dipole potential energy is U = −pE cos θ. At the opposite orientation, θ = 180° and U = +pE, which is the maximum value. At the parallel orientation, θ = 0° and U = −pE, the minimum value. When released, the dipole tends toward lower potential energy, although its kinetic energy increases. Therefore its potential energy decreases, making option A correct.

The required external work for a quasistatic rotation is the change in potential energy, Wext = Uf − Ui. In the parallel position, Ui = −pE = −(2)(5) = −10 J. In the perpendicular position, cos 90° = 0, so Uf = 0. Consequently Wext = 0 − (−10) = +10 J. Option A is correct; the positive sign means external energy must be supplied.

The governing relation is U = −pE cos θ for a dipole in a uniform electric field. With θ = 60°, cos 60° = 1/2. Therefore U = −pE(1/2) = −pE/2. Option A is correct. The positive expression has the wrong sign, pE would correspond to the opposite orientation in magnitude, and zero applies only when the dipole is perpendicular to the field.

For a dipole in a uniform electric field, U = −pE cos θ. Here p = 4 C m, E = 3 N/C, and cos 60° = 1/2. Substitution gives U = −(4)(3)(1/2) = −6 J. Hence option B is correct. A positive 6 J neglects the negative sign in the formula, 12 J neglects the cosine factor, and zero would require a 90° angle.

The torque on an electric dipole is τ = pE sin θ, while the maximum torque is τmax = pE, reached at 90°. Therefore τ/τmax = sin θ. For θ = 60°, sin 60° = √3/2. Thus the torque is √3/2 of its maximum value, so option B is correct. One-half would correspond to 30°, and zero would occur at 0° or 180°.