Dipole in a Uniform External Field

An ideal electric dipole in a uniform electric field experiences zero net force because the forces on its two charges are equal and opposite. Its torque is τ = pE sin θ. For zero torque, sin θ must be zero, so θ = 0° or 180°. Among the listed choices, a dipole parallel to the field corresponds to θ = 0° and is therefore possible. Perpendicular, 30°, and 60° orientations produce nonzero torque.

In a uniform electric field, the net force on an ideal dipole is zero for every orientation because the forces on its charges cancel. The torque is τ = pE sin θ, which reaches its maximum value pE when sin θ = 1, that is, θ = 90°. Hence the dipole moment must be perpendicular to the field. Parallel and opposite orientations give zero torque, while a zero field cannot produce a maximum torque.

For a dipole in a uniform electric field, the torque magnitude is τ = pE sin θ. Substituting p = 4 C m, E = 6 N/C, and sin 30° = 1/2 gives τ = 4 × 6 × 1/2 = 12 N m. Therefore option A is correct. Option B omits the sine factor, option C uses an incorrect factor, and option D would apply only for a parallel or antiparallel orientation.

The torque on an electric dipole is τ = pE sin θ. With p = 8 C m, E = 3 N/C, and sin 60° = √3/2, τ = 8 × 3 × √3/2 = 12√3 N m. Hence option A is correct. Twenty-four N m results from incorrectly taking sine as one, while zero torque applies only when the dipole is parallel or antiparallel to the field.

The torque is τ = pE sin θ, and it becomes maximum when θ = 90°, so sin θ = 1. Therefore τmax = pE and p = τmax/E = 40/5 = 8 C m. Option B is correct. Option A comes from an incorrect division, while options C and D do not follow from the governing relation and have no physical calculation supporting them.

For a dipole, τ = pE sin θ. At maximum torque the angle is 90°, so τmax = pE. Rearranging gives E = τmax/p = 70/(10 C m) = 7 N/C. Thus option A is correct. The other values arise from adding, subtracting, or multiplying the given numbers instead of using the maximum-torque relation and its correct units.

The potential energy of a dipole in a uniform electric field is U = −pE cos θ. Since cos 120° = −1/2, U = −(6)(5)(−1/2) = +15 J. Therefore option C is correct. The negative sign in option A would correspond to an angle such as 60°, while zero energy occurs at 90°, not at 120°; 30 J has an incorrect magnitude.

For a dipole in a uniform field, U = −pE cos θ. Using p = 4 C m, E = 10 N/C, and cos 60° = 1/2 gives U = −4 × 10 × 1/2 = −20 J. Hence option D is correct. The positive 20 J misses the minus sign, zero is possible at 90°, and 40 J ignores the cosine factor.

For a dipole, U = −pE cos θ and τ = pE sin θ. Positive potential energy requires cos θ < 0, which occurs for an obtuse angle such as 120°. At 120°, sin 120° is nonzero, so the torque is also nonzero. Hence option C is correct. At 0° and 180° the torque is zero; at 90° the energy is zero rather than positive.

For an electric dipole in a uniform electric field, the torque magnitude is τ = pE sin θ, where p and E remain unchanged. Since sin 60° = √3/2 and sin 120° = sin(180° − 60°) = √3/2, the two torque magnitudes are equal. The torque direction may change, but its magnitude does not. Therefore option C is correct; it is neither halved, doubled, nor zero.

The potential energy of a dipole in a uniform electric field is U = −pE cos θ. At 60°, cos 60° = 1/2, so U = −pE/2, which is negative for positive p and E. At 120°, cos 120° = −1/2, so U = +pE/2, which is positive. Thus the sign changes from negative to positive. Option A is correct; the energy does not remain unchanged or become zero.

A dipole consists of equal and opposite charges separated by a distance. In a uniform field, the forces on the two charges have equal magnitude and opposite directions, so their resultant force is zero. When the dipole is inclined, these forces act along different parallel lines and form a couple. The couple produces torque τ = pE sin θ and rotates the dipole. Therefore option A is correct.

Stable equilibrium occurs when the dipole is aligned with the electric field, θ = 0, and its potential energy U = −pE cos θ is minimum. After a small angular displacement, the torque is τ = −pE sin θ. For a small positive θ, sin θ is positive, so the torque is negative and acts opposite to the displacement, tending to return the dipole to alignment. Hence option B is correct.

Unstable equilibrium occurs when the dipole is anti-parallel to the electric field, θ = 180°, where its potential energy U = +pE is maximum. A small displacement produces a torque τ = −pE sin θ that acts so as to move the dipole farther from the anti-parallel orientation and toward the lower-energy parallel orientation. Thus the torque increases the displacement rather than restoring it. Option B is correct.

The torque magnitude is τ = pE sin θ, while the magnitude of potential energy is |U| = pE|cos θ|. Because the angle remains unchanged, the trigonometric factors stay constant. Doubling p multiplies each quantity by 2, and tripling E multiplies each by 3. The combined factor is 2 × 3 = 6. Therefore both torque and the magnitude of potential energy become six times their original values. Option A is correct.

For a fixed angle, the magnitude of dipole potential energy is |U| = pE|cos θ|. The angular factor does not change. The new product of dipole moment and field is p′E′ = (p/2)(4E) = 2pE. Hence the energy magnitude is multiplied by 2. It is not merely halved or quadrupled, and it does not remain unchanged. Therefore option B, double, is the correct answer.

The vector expression for torque on an electric dipole is τ⃗ = p⃗ × E⃗. A cross product produces a vector perpendicular to both vectors being multiplied, with direction given by the right-hand rule. Thus τ⃗ is perpendicular to both the dipole moment p⃗ and electric field E⃗, while its magnitude is pE sin θ. Therefore option A is correct; torque is a directed vector, not necessarily parallel to either vector.

When p⃗ and E⃗ are in the same direction, the angle between them is θ = 0°. The torque magnitude is τ = pE sin θ, so τ = pE sin 0° = 0. The potential energy is U = −pE cos θ, giving U = −pE cos 0° = −pE, its minimum value. This is stable equilibrium. Hence option B is correct; maximum torque occurs at 90°, not at 0°.

Let the angle between the dipole moment and electric field be θ. The condition sin θ = cos θ gives tan θ = 1, provided cos θ is nonzero. For the usual angle measured between two vectors, 0° ≤ θ ≤ 180°, the relevant solution is θ = 45°. At this angle, the torque magnitude pE sin θ and the magnitude of the energy factor pE cos θ are equal. Thus option B is correct.

The potential energy of a dipole is U = −pE cos θ. Zero energy occurs at θ = 90°, because cos 90° = 0. A small rotation toward the parallel orientation makes θ slightly less than 90°, so cos θ becomes positive. The negative sign in the formula then makes U negative. The dipole moves toward the stable minimum-energy orientation, so its energy decreases rather than becoming positive or remaining zero.

For a dipole, U = −pE cos θ, and U is zero at θ = 90°. Rotating it toward the opposite direction increases the angle beyond 90°, making cos θ negative. Consequently, the minus sign in U makes the potential energy positive. At the exactly opposite orientation, θ = 180° and U reaches its maximum value +pE, not its minimum. Therefore the energy becomes positive.

For a dipole in a uniform electric field, the torque magnitude is τ = pE sin θ. Substituting the given values gives τ = 15 × 4 × sin 30° = 15 × 4 × 1/2 = 30 N m. Thus option A is correct. Option B omits the sine factor, option C uses an incorrect arithmetic result, and option D would apply only when the dipole is parallel or antiparallel to the field.

The torque on a dipole is τ = pE sin θ. Here, pE = 9 × 8 = 72 N m and sin 60° = √3/2. Therefore, τ = 72 × √3/2 = 36√3 N m. Option B is correct. Option A incorrectly replaces sin 60° by 1/2, option C omits the angular factor, and option D uses only half of the dipole moment-field product.

For a dipole, torque magnitude is τ = pE sin θ. Since sin 150° = sin 30° = 1/2, τ = 12 × 5 × 1/2 = 30 N m. Hence option C is correct. Option A is pE without the sine factor, option B does not follow the formula, and option D would be correct only for an angle of 0° or 180°, not 150°.

The governing equation is τ = pE sin θ. For θ = 120°, sin 120° = sin 60° = √3/2. Thus τ = 10 × 6 × √3/2 = 30√3 N m. Option D is therefore correct. Option B is the unmodified product pE, while options A and C use incorrect numerical or angular factors.