Dipole in a Uniform External Field

The conditions sinθ = 0 and cosθ = 1 give θ = 0°, meaning that the dipole moment and electric field are parallel. The torque is τ = pE sin0° = 0, and the potential energy is U = -pE cos0° = -pE, its minimum value. A small angular displacement increases the energy and creates a restoring torque toward θ = 0°. Therefore the position is stable equilibrium, so option C is correct.

By definition, the electric dipole moment points from the negative charge to the positive charge. In the stated situation, the electric field points from positive to negative, which is exactly opposite to the dipole-moment direction. The angle between two opposite vectors is 180°. Therefore option D is correct. Zero degrees would describe parallel directions, 90° would describe perpendicular directions, and 30° does not follow from the given opposite orientation.

The governing relations are τ = pE sin θ for torque and U = −pE cos θ for potential energy. Therefore, the first safe step is to establish the direction of the dipole moment, which is from the negative charge to the positive charge, and then measure θ from the electric-field direction. Adding values, setting energy to zero, or assuming maximum torque ignores the relevant angle and can produce an incorrect result.

For a dipole in a uniform electric field, the potential energy is U = −pE cos θ. Here pE = 36 J and cos 150° = −√3/2. Therefore, U = −36(−√3/2) = 18√3 J. Hence option B is correct. Option A has the wrong sign, option C ignores the trigonometric factor, and option D would apply only when θ = 90°.

The potential energy of a dipole is U = −pE cos θ. Initially, θ = 0°, so Ui = −54 J. Finally, at 60°, Uf = −54 cos 60° = −27 J. For a slow rotation without change in kinetic energy, the external work equals the increase in potential energy: Wext = Uf − Ui = −27 − (−54) = 27 J. Thus option B is correct.

Use U = −pE cos θ. At the initial parallel orientation, Ui = −30 cos 0° = −30 J. At 120°, Uf = −30 cos 120° = −30(−1/2) = +15 J. Since the rotation is slow, the required external work is the potential-energy increase: Wext = Uf − Ui = 15 − (−30) = 45 J. Therefore option C is correct.

For a dipole, U = −pE cos θ, and the work done by the conservative electric field is Wfield = Ui − Uf = −ΔU. At 90°, Ui = 0. At 0°, Uf = −48 J. Hence Wfield = 0 − (−48) = 48 J. The field does positive work because the dipole moves toward stable equilibrium. Thus option D is correct.

Unstable equilibrium corresponds to θ = 180°, while stable equilibrium corresponds to θ = 0°. Using U = −pE cos θ, the initial energy is Ui = +42 J and the final energy is Uf = −42 J. Therefore the work done by the field is Ui − Uf = 42 − (−42) = 84 J. It is positive because field energy decreases. Option A is correct.

At stable equilibrium θ = 0°, so Ui = −pE = −37 J. At unstable equilibrium θ = 180°, so Uf = +pE = +37 J. For a slow displacement, the external agent supplies the increase in potential energy: Wext = Uf − Ui = 37 − (−37) = 74 J. Option B is correct. The negative value would describe the field’s work, not the required external work.

The torque magnitude is τ = pE sin θ, while maximum torque is pE. Thus τ/τmax = sin θ = √2/2, giving θ = 45° or 135° in the range 0°–180°. Since U = −pE cos θ is negative, cos θ must be positive. That condition selects 45°, not 135°, because cosine is negative in the second quadrant. Hence option A is correct.

Because τ/τmax = sin θ = √2/2, the possible angles between 0° and 180° are 45° and 135°. The dipole energy is U = −pE cos θ. Positive energy requires cos θ < 0, which occurs in the second quadrant. Therefore θ = 135° is selected. The 45° option gives negative energy, while 30° and 60° do not produce the stated torque ratio.

The torque magnitude is τ = pE sin θ, and the magnitude of dipole potential energy is |U| = pE|cos θ|. Therefore τ/|U| = sin θ/|cos θ| = tan θ. At θ = 45°, tan 45° = 1. Thus the ratio is 1:1, represented by option B. The other options would result from using only one trigonometric factor or from an incorrect angle.

Since τ = pE sin θ and τmax = pE, half maximum torque requires sin θ = 1/2. In the range 0°–180°, this gives θ = 30° or 150°. The potential energy U = −pE cos θ is positive only when cos θ is negative, which occurs at 150°. Therefore option C is correct. At 30° the energy is negative, and 90° gives maximum rather than half torque.

For an electric dipole in a uniform field, torque is τ = pE sin θ and potential energy is U = −pE cos θ. Half maximum torque gives sin θ = 1/2, so θ can be 30° or 150°. Negative energy requires cos θ to be positive; this is true at 30°, not 150°. Therefore option D is correct. The 90° option gives maximum torque but zero energy.

The dipole torque relation is τ/τmax = sin θ, while its potential energy is U = −pE cos θ. Since τ/τmax = √3/2, the possible angles in the given range are 60° and 120°. Positive energy requires cos θ < 0 because of the minus sign, which occurs at 120°. Hence option B is correct; 60° gives negative energy.

For a dipole, τ = pE sin θ, so torque equal to √3/2 of its maximum value requires sin θ = √3/2. The relevant possibilities are 60° and 120°. The energy is U = −pE cos θ; it is negative when cos θ is positive. At 60° the cosine is positive, whereas at 120° it is negative. Thus option A, 60°, is the unique answer.

Torque on a dipole is τ = pE sin θ. Thus τnew/τold = (3p)(4E)sin 60° divided by pE sin 30° = 12 × (√3/2)/(1/2) = 12√3. The factor 12 comes from tripling p and quadrupling E, while the angular factor is √3. Therefore option A is correct; omitting the angle change would incorrectly give 12.

Using τ = pE sin θ, the ratio is τnew/τold = [(p/2)(10E)sin 30°]/[pE sin 90°]. The change in p and E gives a factor of 5, while sin 30°/sin 90° = 1/2. Consequently the total ratio is 5 × 1/2 = 2.5. Hence the new torque is two and a half times the old torque, so option B is correct.

For a dipole, τ = pE sin θ. The product pE changes by 4 × 1/2 = 2. Also, sin 60° = sin 120° = √3/2, so changing the angle does not alter the sine factor. Therefore τnew/τold = 2 × 1 = 2. The new torque is twice the old torque, making option C correct; the angle change does not cancel the pE change.

For a dipole in a uniform electric field, the torque magnitude is τ = pE sin θ and the potential energy is U = −pE cos θ. Since sin 30° = sin 150° = 1/2, the torque magnitude is unchanged. However, cos 30° is positive whereas cos 150° is negative, so U changes sign. Thus option A is correct; the other choices confuse sine and cosine dependence.

The relevant relations are τ = pE sin θ and U = −pE cos θ. At 45° and 135°, sin 45° = sin 135° = √2/2, so the torque magnitude is equal at both angles. Initially cos 45° is positive, giving negative energy; at 135°, cosine is negative, giving positive energy. Therefore B is correct, while the other options misapply the trigonometric relations.

By convention, the electric dipole moment points from the negative charge to the positive charge. Its energy in a uniform field is U = −p·E = −pE cos θ. Reversing the assigned direction changes p to −p and effectively changes θ to its supplementary angle; cosine changes sign. Consequently, the predicted energy sign can reverse, making A correct.

The potential energy of a dipole in a uniform field is U = −pE cos θ. For opposite directions, θ = 180° and U = +pE, the maximum value. The torque τ = pE sin 180° is nevertheless zero at the exact orientation, so it is an equilibrium position. A small angular displacement lowers the energy and produces a torque away from the original orientation; therefore the equilibrium is unstable. A is correct.

For a dipole in a uniform electric field, U = −pE cos θ. When the dipole is parallel to the field, θ = 0° and U = −pE, the minimum possible energy. After a small displacement, θ becomes slightly positive and the torque τ = pE sin θ acts in the restoring direction, reducing θ toward zero. Thus A is correct. The torque is zero exactly at θ = 0°, not maximum, and the net force remains zero in a uniform field.

A positive sine with zero cosine identifies θ = 90° for the angle between p and E. The torque magnitude is τ = pE sin θ, so τ = pE, its maximum value. The potential energy is U = −pE cos θ, giving U = 0. Therefore A is correct. At 0° or 180° the torque is zero, while the net force is not determined as maximum by this angular information and is zero for an ideal dipole in a uniform field.