Dipole in a Uniform External Field

Correct answer: B. When the dipole is antiparallel to the field

Explanation: For a dipole in a uniform field, U = −pE cos θ. The maximum value occurs when cos θ = −1, which is at θ = 180°. In that orientation the dipole moment is antiparallel to the electric field and U = +pE, so option B is correct. Parallel alignment gives minimum energy, perpendicular alignment gives zero energy, and free rotation does not define a fixed maximum position.

Correct answer: A. Right-hand rule

Explanation: The torque on an electric dipole is represented by the vector relation τ = p × E, where p is the dipole moment and E is the external electric field. Since torque is a cross product, its direction is perpendicular to both vectors and is determined by the right-hand rule. The left-hand rule concerns magnetic force, while Ohm’s and Kirchhoff’s laws concern electric circuits.

Correct answer: B. A couple of equal and opposite forces

Explanation: In a uniform electric field, the positive and negative charges of a dipole experience forces of equal magnitude, qE, in opposite directions. Their lines of action are separated, so the net translational force is zero but a turning effect remains. Thus they form a couple of equal and opposite forces. It is not a single force or a pair acting in the same direction.

Correct answer: A. It remains the same

Explanation: For a dipole in a uniform electric field, torque is τ = pE sinθ. With the angle fixed, the factor controlling the change is the product pE. After the changes, p′ = 2p and E′ = E/2, so τ′ = (2p)(E/2)sinθ = pE sinθ = τ. Hence the torque remains unchanged, although each individual quantity changes.

Correct answer: B. To align its dipole moment with the field

Explanation: The torque magnitude is τ = pE sinθ and the potential energy is U = −pE cosθ. A free dipole rotates under the torque so that θ decreases and the potential energy falls. The lowest energy occurs when θ = 0°, meaning the dipole moment points along the electric field. The antiparallel position is unstable, not the natural stable alignment.

Correct answer: B. It restores it toward stable position

Explanation: For a dipole at angle θ to the field, τ = pE sinθ. Near the stable position θ = 0, the torque acts to reduce the angular displacement and bring the dipole moment back toward the field direction. For a small angle, sinθ is approximately θ, so the torque is approximately proportional to and opposite the displacement. Therefore it is a restoring torque.

Correct answer: D. Total mass of the dipole

Explanation: The torque magnitude on an electric dipole is τ = pE sinθ. Therefore, it directly depends on the dipole moment p, the external electric field E, and the angle θ between them. The total mass does not appear in this expression and does not directly determine the torque. Mass can affect the resulting angular acceleration through τ = Iα, but that is a motion response, not a factor in the torque itself.

Correct answer: B. Torque is zero and potential energy is minimum

Explanation: 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. At θ = 0°, sin 0° = 0, so the torque is zero. Also, cos 0° = 1, giving U = −pE, the minimum possible value. This is stable equilibrium, so option B is correct; maximum torque occurs at 90°, not at 0°.

Correct answer: A. Torque is zero and potential energy is maximum

Explanation: The dipole relations are τ = pE sin θ and U = −pE cos θ. At θ = 180°, sin 180° = 0, so the torque vanishes. Since cos 180° = −1, the energy becomes U = +pE, its maximum value. The dipole is in unstable equilibrium: a small displacement can make it rotate away. Therefore option A is correct.

Correct answer: A. Torque is maximum and potential energy is zero

Explanation: For an electric dipole, τ = pE sin θ and U = −pE cos θ. At θ = 90°, sin 90° = 1, so τ = pE, the maximum torque. Meanwhile cos 90° = 0, giving U = 0 when the usual zero of energy is used. Thus both parts of option A follow directly from the formulas; the other choices assign an incorrect value to either torque or energy.

Correct answer: B. One over root two of the maximum

Explanation: The torque magnitude on a dipole is τ = pE sin θ. Its maximum value is τmax = pE, reached when θ = 90°. Therefore, at 45°, τ/τmax = sin 45° = 1/√2. The torque is consequently 1/√2 times its maximum value, not one-half; one-half would correspond to an angle whose sine is 0.5, such as 30°. Option B is correct.

Correct answer: C. It is negative and has smaller magnitude than the minimum value

Explanation: The potential energy of a dipole is U = −pE cos θ. At 60°, cos 60° = 1/2, so U = −pE/2. This is negative, but its magnitude is smaller than the minimum energy −pE at 0°. It is neither the maximum +pE, which occurs at 180°, nor zero, which occurs at 90°. Therefore option C is correct.

Correct answer: B. Because electrostatic forces are conservative

Explanation: Electrostatic forces are conservative, meaning the work done by the electric field between two configurations is independent of the path followed. Consequently, a scalar potential-energy function can be assigned, with ΔU = −W_field. For a dipole in a uniform field this leads to U = −pE cos θ, apart from an arbitrary reference constant. Thus option B is correct; path dependence would prevent a unique potential energy.

Correct answer: B. Because the forces act along different lines and form a couple

Explanation: In a uniform electric field, the +q and −q charges of a dipole experience forces of equal magnitude and opposite direction, so their vector sum is zero. However, the forces act at different points and generally along parallel, distinct lines. Their separation produces a couple with torque τ = pE sin θ, which can rotate the dipole without translating its centre of mass. Therefore option B is correct.

Correct answer: B. Opposite to the field

Explanation: For a dipole in a uniform electric field, the torque is τ = pE sin θ. Equilibrium requires τ = 0, so θ must be 0° or 180°. At 0°, the potential energy U = −pE is minimum and the equilibrium is stable. At 180°, U = +pE is maximum; a small displacement lowers the energy and produces a restoring-away effect. Therefore, the opposite-to-field position is unstable.

Correct answer: B. In the direction of decreasing displacement

Explanation: Stable equilibrium occurs when the dipole moment is aligned with the electric field, at θ = 0°. If it is turned through a small angle, the torque is τ = pE sin θ and acts so that the angle decreases. Equivalently, it tends to reduce the displacement and restore alignment. It does not increase the displacement; that behavior belongs to unstable equilibrium.

Correct answer: C. It starts rotating toward stable equilibrium

Explanation: Unstable equilibrium for an electric dipole occurs when it is opposite to the field, at θ = 180°, where U = +pE is maximum. A slight displacement makes the torque τ = pE sin θ act so that the dipole moves away from the inverted orientation. It therefore rotates toward θ = 0°, the stable, minimum-energy alignment. It does not return to the original position or retain zero torque after displacement.

Correct answer: B. Dipole-moment vector cross electric-field vector

Explanation: The torque on an electric dipole in a uniform field is given by the vector equation τ⃗ = p⃗ × E⃗. Its magnitude is pE sin θ, where θ is the angle from the dipole moment to the electric field, and its direction follows the right-hand rule. The order matters: E⃗ × p⃗ would give the opposite direction because the cross product is anti-commutative.

Correct answer: B. Change in potential energy

Explanation: The governing principle is conservation of energy. If the dipole is rotated slowly, its kinetic energy does not appreciably change, so the external work changes only its electric potential energy. For a rotation from angle theta1 to theta2, W_external = U2 - U1, where U = -pE cos theta. Therefore, option B is correct. Torque is related to the rate of change of energy, not generally equal to work.

Correct answer: B. It decreases

Explanation: A dipole in a uniform electric field experiences torque tau = pE sin theta. When released, it generally rotates toward the stable equilibrium position, with its dipole moment parallel to the field. Since U = -pE cos theta, alignment makes cos theta larger and lowers U. Thus the potential energy decreases as the dipole moves toward equilibrium, so option B is correct. The decrease is converted mainly into rotational kinetic energy.

Correct answer: A. Equal

Explanation: For a dipole, the charges are +q and -q, so both have the same charge magnitude q. In a uniform electric field, the force magnitude on either charge is |F| = |q|E. The forces act in opposite directions, but their magnitudes are equal; their separation affects the torque, not the individual force magnitude. Therefore, option A is correct, while B, C, and D confuse direction or torque with force magnitude.

Correct answer: A. When the dipole is parallel or antiparallel to the field

Explanation: In a uniform field, the forces on the positive and negative charges are equal in magnitude, opposite in direction, and parallel to the field. Their lines of action coincide only when the dipole axis is parallel or antiparallel to the field, corresponding to theta = 0 or 180 degrees. Then the perpendicular separation of the force lines is zero and tau = pE sin theta = 0. Hence option A is correct.

Correct answer: B. Between dipole moment and electric field

Explanation: The torque on an electric dipole in a uniform field is expressed vectorially as tau = p cross E, with magnitude tau = pE sin theta. Here theta is the angle between the dipole-moment vector p, directed from negative to positive charge, and the electric-field vector E. It is not an angle between scalar charge, distance, or field-line length. Thus option B is correct.

Correct answer: A. It increases energy from minimum to maximum

Explanation: For a dipole in a uniform field, the potential energy is U = -pE cos theta. At theta = 0 degrees, U = -pE, its minimum value; at theta = 180 degrees, U = +pE, its maximum value. Therefore, slow external rotation requires positive work equal to Delta U = 2pE, raising the energy from minimum to maximum. Option A is correct; the change does depend on p and E.

Correct answer: B. When the dipole moment is parallel or antiparallel to the field

Explanation: The rotational tendency is measured by torque, whose magnitude is tau = pE sin theta. Torque is zero when sin theta = 0, which occurs at theta = 0 degrees or 180 degrees. Thus the dipole moment must be parallel or antiparallel to the electric field. At 90 degrees the torque is maximum, and at 60 degrees it is nonzero. Therefore, option B is the complete correct condition.