Electrostatic Potential and Potential Energy

The potential energy of a dipole in a uniform field is U = −pE cos θ, assuming p and E are nonzero. For 0° < θ < 90°, cos θ is positive, so the leading negative sign makes U negative. At 180° the energy is positive, while p = 0 or E = 0 gives zero energy. Hence option A is correct.

For a dipole, U = −pE cos θ. In the interval 90° < θ < 180°, cosine is negative. The negative sign in the energy formula therefore makes U positive, provided p and E are nonzero. At 0° energy is negative, at 90° it is zero, and if E is zero it is also zero. Thus option A is correct.

For a dipole in a uniform field, U = −pE cos θ. In the opposite, or antiparallel, position, θ = 180° and cos 180° = −1. Hence U = −(7)(3)(−1) = +21 J, so option B is correct. Option A has the sign appropriate to the parallel position, option C belongs to a perpendicular position, and D is not obtained from the formula.

The potential energy of a dipole in a uniform electric field is U = −pE cos θ, where θ is the angle between p and E. For 0° < θ < 90°, cos θ is positive; with the negative sign, U is negative, provided p and E are nonzero. At 90° the energy is zero, while between 90° and 180° it is positive. Hence option A is correct.

For a dipole in a uniform electric field, U = −pE cos θ. In the interval 90° < θ < 180°, cos θ is negative. The minus sign in the formula therefore makes U positive, assuming p and E are nonzero. At θ = 0° the energy is most negative, at θ = 90° it is zero, and a zero field also gives zero energy. Thus option A is the only correct choice.

The potential energy of a dipole in a uniform electric field is U = −pE cos θ. With p = 8 C m, E = 9 N/C, and cos 60° = 1/2, U = −8 × 9 × 1/2 = −36 J. Therefore option C is correct. The positive sign in option A reverses the energy convention, option B ignores the interaction, and option D omits the factor 1/2.

Use U = −pE cos θ for a dipole in a uniform field. Since cos 120° = −1/2, U = −(6)(11)(−1/2) = +33 J. Thus option D is correct. Option A has the wrong sign, option B would require cos θ = 0, and option C doubles the magnitude by failing to include the one-half factor.

For a dipole, U = −pE cos θ. The problem gives pE = 42 J, and cos 30° = √3/2. Therefore U = −42 × √3/2 = −21√3 J, so option A is correct. Option B has the wrong sign, option C incorrectly uses cos 60° or drops √3, and option D would apply only at 90°.

The potential energy of a dipole in a uniform electric field is U = −pE cosθ, where θ is the angle between the dipole moment and the field. For nonzero p and E, U becomes zero when cosθ = 0. In the range 0° to 180°, this occurs at θ = 90°. At 0° the energy is minimum, and at 180° it is maximum, so those alternatives are incorrect.

For a dipole in a uniform electric field, the potential energy is U = −pE cos θ, with the zero of energy chosen by convention. At θ = 90°, cos 90° = 0, so U = 0. This is a convenient reference level, not a unique physical requirement; another zero could be chosen by adding a constant. With the standard convention used in this question, the perpendicular position is correct.

The potential energy of a dipole in a uniform electric field is U = −pE cos θ. For an acute angle, 0° < θ < 90°, so cos θ is positive; multiplying by the minus sign makes U negative. At 90°, U is zero, while at an obtuse angle, cos θ is negative and U becomes positive. Therefore, the acute-angle condition is the correct one.

For a dipole in a uniform field, the torque magnitude is τ = pE sin θ, which is maximum when |sin θ| = 1; the standard principal angle is θ = 90°. The potential energy is U = −pE cos θ, and cos 90° = 0, so it equals the chosen reference zero there. At 0° or 180°, torque is zero, not maximum. Hence 90° satisfies both conditions.

The potential energy of a dipole in a uniform electric field is U = −pE cos θ. At 30°, cos 30° is positive, so U is negative. At 150°, cos 150° is negative, so the minus sign makes U positive. Thus the sign changes from negative to positive, making option A correct. The energy is not zero at either angle; it is zero only when θ is 90°.

For a slow, controlled rotation, the external work equals the change in the dipole’s potential energy: Wext = ΔU. Since U = −pE cos θ, U(0°) = −pE and U(60°) = −pE/2. Therefore ΔU = (−pE/2) − (−pE) = +pE/2, which is positive. The field opposes this displacement, so the external agent must supply energy. Hence option A is correct; the work is neither zero nor infinite.