Dipole in a Uniform External Field

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.

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.

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.

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.

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.

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.

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.

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°.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.