Dipole in a Uniform External Field

The conditions sin θ = 0 and cos θ = −1 correspond to θ = 180°, meaning the dipole moment is antiparallel to the field. At this angle τ = pE sin 180° = 0, while U = −pE cos 180° = +pE, the maximum energy. A slight displacement lowers the energy and drives the dipole farther from this orientation, so it is unstable equilibrium. Therefore B is correct.

The conditions sin θ = 0 and cos θ = 1 identify θ = 0°, so the dipole moment is parallel to the electric field. The torque is τ = pE sin 0° = 0, and the potential energy is U = −pE cos 0° = −pE, its minimum value. A small displacement creates a restoring torque toward θ = 0°. Hence the position is stable equilibrium, so C is correct.

The dipole’s potential energy is U = −pE cos θ. In stable alignment, θ = 0° and U = −pE; in the unstable antiparallel position, θ = 180° and U = +pE. Thus the increase in potential energy is ΔU = 2pE. For a slow rotation without kinetic-energy change, the external agent does Wext = ΔU = 2pE, which is positive. Hence option C is correct.

During slow rotation, the external agent supplies work equal to the increase in the dipole’s potential energy, because the change in kinetic energy is negligible. Using U = −pE cos θ, U(0°) = −pE and U(90°) = 0. Therefore ΔU = 0 − (−pE) = +pE, so the external work is positive. Option A is correct; the field itself does negative work in this controlled rotation.

The electric field exerts a restoring torque that turns the dipole from 90° toward its stable alignment at 0°. The potential energy changes from U(90°) = 0 to U(0°) = −pE, so ΔU = −pE. For a conservative electric field, work done by the field is W_field = −ΔU = +pE. Hence the field does positive work, making option A correct.

A dipole contains equal and opposite charges. In a uniform field, the two charges experience equal forces in opposite directions, so the net force is zero, although a torque may remain. In a non-uniform field, the field strengths at the positive and negative charges can differ, so the two forces need not cancel. Consequently, the dipole can experience a net translational force as well as torque.

In a uniform electric field, the forces on the positive and negative charges have equal magnitudes and opposite directions, so their vector sum, the net force, is zero. However, the forces act along different parallel lines when the dipole is not aligned with the field. They can therefore form a couple with torque τ = pE sin θ. Thus zero net force does not require zero torque.

For a slow rotation, the change in kinetic energy is negligible, so the external work equals the change in potential energy: Wext = ΔU. With U = −pE cos θ, at 90° the energy is 0, while at 180° it is +pE. Thus ΔU = +pE, and the external agent does positive work. The field tends to oppose this rotation, so its work is negative, but that is not the quantity asked.

The dipole energy is U = −pE cos θ. At 180°, U = +pE, and at 0°, U = −pE, so the change in potential energy is ΔU = −2pE. For a conservative electric field, the work done by the field is Wfield = −ΔU = +2pE, which is positive. The field drives the dipole toward alignment, lowering its potential energy; therefore zero or negative work is incorrect.

Stable equilibrium occurs when the dipole is parallel to the field, θ = 0. For a small angular displacement θ, the torque is τ = −pE sin θ. Using the small-angle approximation sin θ ≈ θ, we obtain τ ≈ −pEθ. The negative sign means the torque is restoring and opposite to the displacement, exactly like simple harmonic rotational behavior. It is therefore linear, not quadratic or independent.

A uniform electric field exerts equal and opposite forces on the positive and negative charges of a dipole, so the net force is zero and translational acceleration does not begin. If the dipole is at a nonzero angle θ, these forces form a couple with torque τ = pE sin θ, which is nonzero except at 0° or 180°. Thus a nonzero angle allows rotation without translation. A zero field or zero dipole moment gives no torque.

Electrostatic forces are conservative, so the change in potential energy depends only on the initial and final orientations, not on the intermediate route. During slow rotation, angular acceleration is negligible and the external agent balances the electric torque. Hence external work equals the increase in dipole potential energy, ΔW_ext = ΔU, making it path independent. Option A is correct.

The torque on a dipole is τ = pE sin θ, so it vanishes at θ = 0° and 180°. Stability is decided by U = −pE cos θ: at θ = 0°, U is minimum (−pE), whereas at 180° it is maximum (+pE). A small displacement therefore produces a restoring torque only in the parallel position. Option A is correct.

For a dipole, τ = pE sin θ, so torque is zero at both 0° and 180°. However, U = −pE cos θ is maximum at θ = 180°, giving +pE. A small rotation lowers the energy and makes the dipole move farther from this orientation, so the antiparallel position is unstable. Thus option B is correct.

The external field, the dipole moment, and the charge magnitude are fixed properties in this comparison, so options A, B, and D remain unchanged. Torque magnitude is τ = pE sin θ. At the complementary angle it becomes pE sin(90° − θ) = pE cos θ, which is generally different from pE sin θ. Therefore option C is the unique correct answer.

For an electric dipole, the torque magnitude is |τ| = pE sin θ and the potential-energy magnitude is |U| = pE|cos θ|. Equality of their numerical factors requires sin θ = |cos θ|. For the usual acute angle between vectors, this becomes tan θ = 1, giving θ = 45°. Thus option B is correct. At 90° the energy magnitude is zero, and at 30° or 60° the sine and cosine magnitudes are unequal.

Slow rotation means the angular acceleration is kept negligible, so the net torque is approximately zero. The torque equation is τ_ext + τ_electric ≈ 0. Hence the external torque must oppose the electric torque and have the same instantaneous magnitude, although that magnitude can vary with θ as pE sin θ. Option A is correct.

For an infinitesimal angular displacement dθ, the work done by the field is dW_field = τ_field dθ = −pE sin θ dθ. At θ = 0° or 180°, sin θ = 0, so the instantaneous torque and infinitesimal work are zero. Maximum torque instead occurs at 90°, and zero potential energy occurs at 90°, not necessarily zero work. Thus option A is correct.

For slow rotation, the external work equals the increase in potential energy. Since U = −pE cos θ, the initial stable position θi = 0° has Uinitial = −pE. The largest final energy occurs at θf = 180°, where Ufinal = +pE. Hence the maximum work is ΔU = 2pE, obtained when the dipole is rotated to 180°, not 90°, 30°, or 0°.

For a dipole, torque magnitude is |τ| = pE|sin θ|, so it is zero at 0° and 180°. The potential energy is U = −pE cos θ. At 0° it is −pE, the minimum, whereas at 180° it is +pE, the maximum positive value. Therefore the angle satisfying both conditions is 180°; 90° gives maximum torque and zero energy, not the required combination.

For a dipole, τ = pE sin θ and the maximum torque is pE. Half-maximum torque requires sin θ = 1/2, giving θ = 30° or 150° in the usual 0°–180° range. Since U = −pE cos θ, the energy is negative when cos θ is positive, which occurs at the acute angle 30°. At 150° the energy is positive, while 90° gives maximum torque.

The torque relation is τ = pE sin θ, with maximum value pE. Therefore, half-maximum torque requires sin θ = 1/2, which gives 30° or 150°. The potential energy is U = −pE cos θ; it is positive when cos θ is negative, namely for the obtuse angle 150°. At 30° the energy is negative, at 90° torque is maximum, and 60° gives a different torque.

The dipole torque is τ = pE sin θ. For a small angular displacement θ, sin θ ≈ θ, so the torque magnitude is approximately pEθ. About the stable position, its direction is opposite to the displacement, giving a restoring torque τ ≈ −pEθ. Hence it depends on dipole moment p, field E, and angular displacement θ, not merely on charge, mass, or temperature.

For a dipole oscillating through a small angle about stable alignment, the restoring torque is approximately τ = −pEθ. Comparing this with I(d²θ/dt²) = −pEθ gives angular frequency ω = √(pE/I). Thus increasing the electric field E increases ω. Making E or p zero removes the restoring torque, while increasing I decreases the frequency rather than increasing it.

For small oscillations about the stable direction, the restoring torque is −pEθ. The equation Iθ¨ = −pEθ gives ω² = pE/I, or ω = √(pE/I). Therefore, when the moment of inertia I increases while p and E remain fixed, the angular frequency decreases. A larger rotational inertia resists angular acceleration; it cannot make the frequency unchanged or infinite.