Dipole in a Uniform External Field

For a dipole in a uniform electric field, the torque magnitude is τ = pE sin θ. At θ = 180°, the dipole is antiparallel to the field and sin 180° = 0, so τ = 0. The dipole may be in an equilibrium orientation, although this orientation is unstable for small angular disturbances. Maximum torque occurs at 90°, not at 180°; the other numerical choices are unsupported.

The potential energy of an electric dipole in a uniform 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 small displacement then produces a restoring torque, so the equilibrium is stable. At θ = 180°, energy is maximum and equilibrium is unstable; perpendicular orientation is not an equilibrium position for small rotations.

The potential energy of a 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 U = +pE, the maximum value. A small displacement then lowers the energy and produces a turning tendency away from this position, so it is unstable equilibrium. Parallel orientation has minimum energy and is stable.

A dipole in a uniform electric field experiences a torque τ = pE sin θ. This torque tends to reduce the angle θ between the dipole moment and the field. Equivalently, the potential energy U = −pE cos θ is minimum at θ = 0°. Therefore the dipole naturally turns until its dipole moment points along the electric field. The perpendicular position gives torque, not final alignment.

The magnitude of torque on an electric dipole is τ = pE sin θ. Thus it depends on the dipole moment magnitude p, the external electric-field magnitude E, and the sine of the angle θ between them. It is zero for parallel or antiparallel orientations and maximum at 90°. Mass, colour, temperature, and unrelated distance do not appear in this ideal uniform-field formula.

The torque magnitude on a dipole is τ = pE sin θ. If the dipole moment p and angle θ are fixed, then τ is directly proportional to the field magnitude E. Increasing E therefore increases the torque by the same factor; for example, doubling E doubles τ. The result is zero only when sin θ = 0, so the claim that it always remains zero is incorrect.

For a dipole in a uniform electric field, the torque magnitude is τ = pE sin θ. With E and θ unchanged, τ is directly proportional to the dipole moment p. Consequently, increasing p increases the torque by the same proportion; doubling p, for instance, doubles τ. Torque is zero only for angles 0° or 180°, so the unchanged-angle statement does not automatically imply zero torque.

The torque magnitude on an electric dipole in a uniform field is τ = pE sin θ. When the electric-field magnitude E is zero, the product pE sin θ is zero for every orientation θ. Hence no electric torque acts on the dipole, regardless of its dipole moment. The answer is not “negative” or “equal to p,” because torque and dipole moment are different physical quantities with different units.

The torque on an electric dipole in a uniform field is given by τ = pE sin θ, where p is the dipole moment, E is the field magnitude, and θ is the angle between them. If p = 0, the entire product becomes zero for every value of E and θ. Therefore the torque is zero. Maximum torque is possible only when p and E are nonzero and perpendicular.

A dipole consists of charges +q and −q separated by a fixed distance. In a uniform electric field, the field has the same magnitude and direction at both charge positions. Thus the forces have magnitudes qE and qE, so they are equal. Because the charges have opposite signs, the force directions are opposite. Their resultant is zero, although they can produce torque.

For a charge in an electric field, the force is F = qE. In a uniform field, the field vector is identical at the locations of the positive and negative charges. The +q charge experiences a force qE in the field direction, while the −q charge experiences an equal force in the opposite direction. These forces cancel vectorially, so the net translational force is zero, though a separated pair may still have torque.

A couple is a pair of equal, opposite, and non-collinear forces. In a uniform electric field, the +q and −q charges of a dipole experience equal forces in opposite directions. Since the charges are separated, the two lines of action are generally different and parallel. Their resultant force is zero, but their moment is nonzero when the dipole is not parallel or antiparallel to the field. Hence such forces form a couple.

The moment of a couple is a turning effect. For a dipole in a uniform electric field, the equal and opposite forces on the two charges have zero resultant translation but may have a nonzero torque τ = pE sin θ. This torque tends to rotate the dipole so that its dipole moment aligns with the field. A couple therefore produces rotation, not a change in mass or charge and not simple translation.

The potential energy of an electric dipole in a uniform field is U = −pE cos θ. When the dipole is parallel to the field, θ = 0°, so U = −pE, its minimum value for fixed p and E. The torque is zero at this position, and a small angular displacement produces a restoring tendency, so the equilibrium is stable. Therefore option A is correct; maximum energy occurs in the antiparallel position.

For an electric dipole in a uniform field, U = −pE cos θ. In the antiparallel orientation, θ = 180° and cos 180° = −1, giving U = +pE, the maximum value for fixed p and E. The torque is zero there, but a small displacement produces a torque that moves the dipole farther from this orientation. Thus the state is unstable equilibrium and has maximum energy, not zero or infinite energy.

Unstable equilibrium is a state in which a small displacement produces a tendency to move farther from the original position rather than return to it. The potential energy is locally maximum, not minimum. For an electric dipole in a uniform field, the antiparallel orientation is unstable because a slight rotation creates torque away from that orientation. Hence option A is correct; option B describes stable equilibrium.

For a complete electric dipole in a uniform electric field, the forces on its positive and negative charges have equal magnitude and opposite directions, so their vector sum is zero. However, because their lines of action are separated, they form a couple. Its torque is τ = pE sin θ, which is nonzero for a general angle other than 0° or 180°. Therefore option A is correct.

Electric dipole moment is defined as p = qd, where d is the displacement vector directed from the negative charge to the positive charge. Therefore p has both magnitude and a definite direction, so it is a vector quantity. It is not merely a number or a unit. Option B is incorrect because a scalar has magnitude only; hence option A is correct.

Torque is defined by the vector product τ = r × F. Its direction is perpendicular to the plane containing the position vector r and force F, and it is determined by the right-hand rule: curl the fingers from r toward F while the thumb gives the torque direction. The other listed rules do not determine vector-product direction. Therefore option A is correct.

The torque on an electric dipole in a uniform field is τ = pE sin θ. It rotates the dipole so that its dipole moment tends to become parallel to the electric field, where θ = 0 and the potential energy U = −pE is minimum. It does not destroy the dipole, remove the field, or change the charges to zero. Thus option A is correct.

The potential energy of an electric dipole in a uniform electric field is U = −pE cos θ. When the dipole is parallel to the field, θ = 0°, so cos θ = 1 and U = −pE, which is negative for nonzero p and E. This is also the minimum-energy orientation. Therefore option A is correct; positive energy occurs for the antiparallel orientation.

For a dipole in a uniform electric field, U = −pE cos θ. In the antiparallel orientation, θ = 180° and cos 180° = −1. Hence U = −pE(−1) = +pE, which is positive for nonzero p and E. This is the maximum-energy and unstable orientation. Therefore option A is correct; the negative value belongs to the parallel case.

The torque on an electric dipole in a uniform field is τ = pE sin θ, where p is dipole moment, E is field strength, and θ is the angle between them. At θ = 90°, sin 90° = 1, so τ = pE. This is the maximum possible torque for fixed p and E. Thus option A is correct; it is not zero at 90°.

The torque on an electric dipole in a uniform electric field is given by τ = pE sin θ, where p is the dipole moment, E is the field magnitude, and θ is the angle between them. Here, p = 4 C m, E = 5 N/C, and θ = 0°. Since sin 0° = 0, τ = 4 × 5 × 0 = 0 N m. Thus option A is correct; multiplying p and E without the sine factor would incorrectly give option B.

For a dipole in a uniform electric field, torque is τ = pE sin θ. In the perpendicular position, the angle between the dipole moment and the field is 90°, so sin 90° = 1. Substituting p = 3 C m and E = 2 N/C gives τ = 3 × 2 × 1 = 6 N m. Therefore option A is correct. Options B and C use only one given quantity, while option D would apply at 0° or 180°.