Dipole in a Uniform External Field

In a uniform electric field, the positive and negative charges of a dipole experience forces of equal magnitude, qE, but in opposite directions. Therefore their vector sum, or net force, is zero. However, the forces act at two separated charge locations, so they form a couple. This couple produces torque τ = pE sin θ, allowing rotation even when translation does not occur. Thus option A is correct; the other options incorrectly assume a common point of action or no force.

For a dipole in a uniform electric field, the torque is τ = pE sin θ. At θ = 0°, the torque is zero because sin 0° = 0, so the dipole is in equilibrium. Its potential energy is U = −pE cos θ, which is minimum at θ = 0°. A small displacement creates a restoring torque that tends to bring the dipole back. Therefore option A, stable equilibrium, is correct; maximum torque occurs at 90°.

When the dipole moment is opposite to the field, θ = 180°. The torque is τ = pE sin 180° = 0, so the dipole is momentarily in equilibrium. Its potential energy U = −pE cos θ is maximum at 180°. A small angular displacement produces a torque that moves it farther from the anti-parallel position and toward alignment with the field. Hence option A is correct. Maximum torque occurs at 90°, not 180°.

In a uniform electric field, the positive and negative charges of a dipole experience forces of equal magnitude in opposite directions, so the net force is zero, although a torque may act. In a non-uniform field, the field strengths at the two charge locations can differ. Consequently, the two electric forces need not cancel, and a net translational force can occur. The charge signs and dipole moment do not change merely because the external field is non-uniform.

The potential energy of a dipole in a uniform electric field is U = −pE cos θ. Opposite alignment corresponds to θ = 180° and is an unstable equilibrium, although the instantaneous torque is zero there because sin 180° = 0. After a small displacement, the torque τ = pE sin θ acts so that θ moves farther from 180° and toward 0°. Therefore the dipole is driven away from its original position, making option A correct.

For a dipole, U = −pE cosθ. Stable equilibrium is parallel to the field at θ = 0°, where U = −pE, while unstable equilibrium is antiparallel at θ = 180°, where U = +pE. The energy increase is ΔU = 2pE, so an external agent must supply positive work in a slow rotation. The charge, field, and dipole moment need not change.

Gauss’s law states that the net electric flux through a closed surface is Φ = Q_enclosed/ε₀. Only the algebraic sum of charges inside the surface determines the total flux. External charges may alter the electric field at individual points and may change positive and negative local contributions, but their overall contribution to closed-surface flux is zero. Since the enclosed net charge is positive, the total flux is positive; option A is correct.

For a dipole in a uniform field, the torque is τ = pE sin θ. At the antiparallel position, θ = 180°, so the instantaneous torque is zero. However, a small angular displacement makes sin θ nonzero, and the resulting torque increases the displacement and tends to turn the dipole toward the field direction. Thus this is unstable equilibrium; option B describes stable equilibrium.

The potential energy of a dipole in a uniform electric field is U = −pE cos θ. For the parallel position, θ = 0° and U = −pE, the minimum value. For the perpendicular position, θ = 90° and U = 0. Therefore the energy increases by pE, so an external agent must supply positive work. The charge and dipole moment do not disappear.

In a uniform electric field, the positive charge experiences force +qE and the negative charge experiences force −qE. These forces have equal magnitudes and opposite directions, so their vector sum is zero. However, each individual force can be nonzero and the pair may produce a torque on the dipole. Thus zero net force does not mean zero force on every charge.

For a dipole in a uniform electric field, the forces on its two charges are equal and opposite, so the net translational force is zero. The torque is τ = pE sin θ, where θ is the angle between the dipole moment and the field. Perpendicular placement gives θ = 90° and sin θ = 1, so τ = pE, its maximum value. Therefore option A is correct.

For a dipole in a uniform electric field, the torque is τ = pE sin θ. At the exactly anti-parallel position, θ = 180°, so the instantaneous torque is zero; however, the potential energy U = −pE cos θ is maximum there. A small displacement produces a restoring tendency away from this maximum-energy position, so the dipole rotates toward alignment with the field. Thus A is correct; the net charge and dipole moment do not change, and torque is not always maximum.

The torque on an electric dipole is τ = pE sin θ. When the dipole moment is parallel to the field, θ = 0°, so sin θ = 0 and the torque vanishes. Its potential energy is U = −pE cos θ, which is minimum at θ = 0°. A small angular displacement therefore raises the energy and produces a restoring torque that brings the dipole back. Hence A is correct; the charges and field have not disappeared, and p has not become negative.

Each charge experiences a force F = qE at its own position. In a uniform field, the positive and negative charges of an ideal dipole experience equal and opposite forces, so the net force is zero, although torque may act. In a non-uniform field, the magnitude or direction of E changes between the two charges. Their forces therefore need not cancel, producing a net force. Thus A is correct; the dipole remains electrically neutral.