Dipole in a Uniform External Field

The potential energy of an electric dipole in a uniform field is U = −pE cos θ. For a dipole parallel to the field, θ = 0° and U = −pE, the lowest possible value for fixed p and E. A small rotation raises the energy and produces a restoring torque, so this orientation is stable equilibrium. The antiparallel orientation has maximum energy, while zero energy occurs only at 90°, not always.

A dipole contains equal and opposite charges separated by a finite distance. In a non-uniform electric field, the field magnitude or direction can differ at the locations of the two charges. Consequently, the forces qE on the positive and negative charges need not be equal and opposite, leaving a non-zero resultant force. Thus A is correct; B and C violate charge conservation, while D is false for a uniform field.

The torque on an electric dipole in a uniform field is τ = pE sinθ. In the perpendicular position, θ = 90°, so sinθ = 1 and τ = pE. If p changes to 3p and E changes to E/2, the new torque is τ′ = (3p)(E/2) = 3τ/2. Therefore, the torque becomes one and a half times its original value. The factors must be multiplied, not added.

The torque on a dipole is τ = pE sinθ. Because the angle remains unchanged, sinθ has the same value before and after the change. Replacing p by 2p and E by 2E gives τ′ = (2p)(2E)sinθ = 4pE sinθ = 4τ. Thus the torque becomes four times its original value. Doubling only one quantity would give two times, but both independent factors are doubled here.

A dipole in a uniform electric field experiences torque τ = p × E, whose magnitude is τ = pE sin θ. At θ = 90°, the torque is maximum and rotates the dipole toward θ = 0°, where its dipole moment is parallel to the field and the potential energy U = −p·E is minimum. Therefore option A is correct. The antiparallel orientation is an unstable maximum-energy position, and the field cannot remove the dipole charges or make p zero.

The governing relation for a dipole in a uniform electric field is τ = pE sinθ, where θ is the angle between the dipole moment and the field. Initially, τ₁ = pE sin45° = pE/√2. Finally, τ₂ = pE sin90° = pE. Since pE is unchanged and pE is greater than pE/√2, the torque increases and becomes maximum at 90°. Therefore A is correct; C applies only at 0° or 180°.

For a dipole in a uniform electric field, the torque magnitude is τ = pE|sinθ|. At 120°, τ₁ = pE sin120° = (√3/2)pE. At 180°, τ₂ = pE sin180° = 0. Thus the magnitude decreases continuously to zero as the angle changes from 120° to 180°. Option A is correct. It does not increase or remain constant, and the final torque is zero rather than maximum.

The torque on an electric dipole in a uniform field is τ = pE sinθ. Let the original torque be τ = pE sinθ. When the dipole moment becomes 2p and the field becomes 2E, while θ remains unchanged, the new torque is τ′ = (2p)(2E)sinθ = 4pE sinθ = 4τ. Hence the torque becomes four times its original value. A is correct; doubling only one factor would produce two times, not four times.

The torque on an electric dipole in a uniform electric field is τ = pE sin θ, where p is dipole moment, E is field strength, and θ is the unchanged angle between them. After the changes, p′ = 3p and E′ = E/3. Hence τ′ = (3p)(E/3)sin θ = pE sin θ = τ. Therefore the torque remains unchanged. Options B and C consider only one change and ignore the product relationship.

In a uniform electric field, the positive charge experiences a force qE along the field, while the negative charge experiences an equal force qE opposite to it. Thus the vector sum of forces is zero. However, the charges are separated, so the forces act along different parallel lines and form a couple. This couple produces torque τ = pE sin θ, allowing rotation. Therefore option A is correct; the other statements contradict the force law.

A dipole consists of equal and opposite charges located at different positions. In a uniform field, both charges experience forces of equal magnitude, so the net force is zero. In a non-uniform field, the field value can differ from one charge to the other; consequently, the forces qE+ and qE− need not have equal magnitudes. Their vector sum can therefore be nonzero, causing translation as well as possible rotation. Hence A is correct.

For a dipole in a uniform electric field, the torque is τ = pE sin θ and acts to reduce the angle θ between the dipole moment and the field. At θ = 0, the dipole is in stable equilibrium. A small displacement makes the torque act in the direction opposite to the displacement, so it tends to restore alignment. The dipole moment, charge, and field do not vanish. Thus option A correctly identifies the restoring torque.

For a dipole in a uniform electric field, the torque magnitude is τ = pE sinθ, where θ is the angle between p and E. For a nonzero dipole and field, τ becomes zero when sinθ = 0, namely at θ = 0° or 180°. Thus the dipole may be parallel or antiparallel, so option A is correct. Perpendicular orientation gives maximum torque, not zero torque.

The potential energy of an electric dipole in a uniform field is U = −pE cosθ. For fixed p and E, U is minimum when cosθ = 1, so θ = 0°. Hence the dipole moment points along the electric field and option A is correct. The antiparallel direction gives maximum energy, while the perpendicular direction gives zero energy rather than the minimum.

For a dipole in a uniform electric field, U = −pE cosθ. The maximum value of U occurs when cosθ = −1, which means θ = 180°. Thus the dipole moment is antiparallel to the electric field, so option A is correct. The parallel orientation gives minimum energy, 45° gives an intermediate value, and the dipole moment need not be zero.

The torque on an electric dipole in a uniform electric field is τ = pE sin θ, where p is the dipole moment and θ is the angle between p and E. If p becomes 2p and E becomes 3E while θ stays unchanged, then τ′ = (2p)(3E) sin θ = 6τ. Therefore the torque becomes six times its original value. The factors multiply because torque depends on the product pE.

For an electric dipole in a uniform field, torque is τ = pE sin θ. The angle remains the same, so sin θ is unchanged. The change factor is therefore (3) × (1/2) = 3/2. Hence τ′ = 1.5τ, meaning the new torque is one and a half times the original. The factors must be multiplied, not added, because p and E occur as a product in the governing equation.

The torque on a dipole is τ = pE sin θ. It is zero only when θ = 0° or 180°, when the dipole moment and electric field are collinear. At θ = 60°, sin 60° = √3/2, which is nonzero, so τ = pE√3/2 and a turning effect acts on the dipole. The net force in a uniform field can still be zero; torque does not require a nonzero net force.

A polar molecule possesses a permanent dipole moment p. In a uniform electric field E, the positive and negative charges experience equal and opposite forces. Their net force is zero, but the separated forces form a torque τ = pE sin θ. This torque tends to reduce θ, so the dipole aligns parallel to the field, where its potential energy U = −pE is minimum. The molecule does not need a net positive charge.

For a dipole in a uniform electric field, the forces on +q and −q are equal and opposite, so the net translational force is zero. Because the forces act at different positions, they produce torque τ = pE sin θ and can rotate the dipole. The work associated with changing orientation gives potential energy U = −pE cos θ. Thus energy depends on angle even when the net force is zero.

When the dipole is parallel to the field, θ = 0 and its potential energy U = −pE cos θ is minimum, so this is stable equilibrium. After a small angular displacement, the torque τ = pE sin θ acts in the restoring direction, approximately τ ≈ −pEθ for a small angle. It therefore tends to reduce the displacement and bring the dipole back toward the field direction. The net charge does not change.

For an antiparallel dipole, θ = π and U = −pE cos π = +pE, which is the maximum potential energy. Although the torque is zero exactly at θ = π, a small displacement produces a torque that increases the displacement and rotates the dipole toward the field direction. Therefore the antiparallel position is unstable equilibrium. The dipole moment does not become zero, and the absence of torque only applies at the exact position.

The potential energy of an electric dipole in a uniform field is U = −pE cos θ, where θ is the angle between the dipole moment and the field. In the parallel orientation, θ = 0° and U = −pE, the minimum possible value. A small rotation increases the energy and produces a restoring tendency, confirming stable equilibrium. The antiparallel orientation has maximum energy, not minimum.

In a uniform electric field, the positive and negative charges of a dipole experience equal forces in opposite directions. These forces cancel translationally, so the net force is zero. When the dipole moment makes an angle θ with the field, the forces form a couple with torque τ = pE sinθ. For an oblique angle, sinθ is non-zero, so torque exists. In parallel or antiparallel positions, sinθ = 0; zero separation also gives no dipole moment. Hence A is correct.

The torque on an electric dipole in a uniform field is τ = pE sinθ. At θ = 90°, sinθ = 1, so the initial torque is τ = pE. After the changes, p′ = p/2 and E′ = 2E; therefore τ′ = (p/2)(2E)(1) = pE = τ. The torque remains unchanged, so option A is correct. It does not double or halve because the two scaling factors cancel, and it is not zero because the angle is perpendicular, not parallel.