Dipole in a Uniform External Field

The potential energy of a dipole in a uniform electric field is U = −pE cos θ. When the dipole is parallel to the field, θ = 0° and U = −pE, which is the minimum possible energy. Also, τ = pE sin 0° = 0, and a small displacement produces a restoring torque. Hence the parallel orientation is stable equilibrium, so option A is correct; the antiparallel orientation is the unstable one.

For a dipole, U = −pE cos θ. In the antiparallel orientation, θ = 180°, so U = +pE, the maximum potential energy. The torque is zero at this exact orientation because sin 180° = 0, but a small angular displacement produces torque that moves the dipole away from this position rather than restoring it. Therefore it is unstable equilibrium, making option A correct.

A restoring torque is the defining mechanical sign of stable equilibrium: after a small displacement, it acts opposite to the displacement and tends to return the system to its original position. For a dipole in a uniform electric field, the parallel orientation has θ = 0°, minimum potential energy, and zero torque at equilibrium. Therefore the described behavior identifies stable equilibrium, so option A is correct; an away-driving torque would indicate instability.

The potential energy of a dipole in a uniform electric field is U = −pE cos θ. At the opposite orientation, θ = 180° and the energy is maximum, U = +pE. A slight displacement creates a torque that drives the dipole toward orientations of lower energy, ultimately toward θ = 0°, where it is parallel to the field and the energy is minimum. Thus option A is correct; the charge, field, and mass do not change in the stated process.

The electric force on a charge is F = qE. For a positive charge, q is positive, so the force vector has the same direction as the electric-field vector. Therefore the positive charge of the dipole is pushed along the field direction, making option A correct. A negative charge would experience force opposite to the field, while the force is not always zero and has no universally fixed upward direction.

The force on a charge in an electric field is F = qE. Since a negative charge has q < 0, multiplication by q reverses the direction of the field vector. Consequently, the force on the negative charge of the dipole acts opposite to the electric field, so option A is correct. It is not necessarily rightward, upward, or zero; its direction depends on the actual field direction and its magnitude on |q|E.

The dipole moment p is directed from the negative charge to the positive charge, and its angle θ with the external field determines important results: torque is τ = pE sin θ and potential energy is U = −pE cos θ. Therefore identifying the direction of p and the angle with E should be the first step. Diagram colour, page number, and question length have no role in the physics, so option A is correct.

The magnitude of an electric dipole moment is p = qd, where q is the magnitude of either charge and d is the separation between the positive and negative charges. If q remains constant, increasing d increases p in direct proportion. Hence the separation should be increased, so option A is correct. Changing class time or colour has no physical effect, and changing field direction may alter torque but does not change the dipole moment itself.

The governing relation is τ = pE sin θ, where torque is the turning effect produced by the external electric field. It tends to rotate the dipole and thereby changes the angle θ, or orientation, between the dipole moment and the field. It does not alter the magnitude of either charge, create the field source, or change the identity of the dipole. Hence option A is correct.

For a dipole in a uniform electric field, the positive charge experiences a force qE and the negative charge experiences an equal force qE in the opposite direction. These two forces cancel vectorially, so the net translational force is zero, although a torque may still act if the dipole is not aligned with the field. Therefore option A is correct; the other choices confuse force with torque or field.

In a uniform field, the forces on the positive and negative charges are equal and opposite, so their resultant force is zero. However, these forces generally act along parallel but different lines when the dipole is inclined to the field. They form a couple with torque τ = pE sin θ, producing rotation without translation. Thus option B is correct; mass and charge remain unchanged.

The torque on an electric dipole in a uniform field is τ = pE sin θ, where θ is the angle between the dipole moment and the field. For fixed p and E, sine reaches its greatest value, 1, at θ = 90°. Hence τmax = pE when the dipole is perpendicular to the field. At 0° and 180° the torque is zero, so option B is correct.

For a dipole in a uniform electric field, torque is given by τ = pE sin θ. When the angle is zero, the dipole moment is parallel to the field and sin 0° = 0. Consequently, τ = pE × 0 = 0. The dipole may have zero torque in this aligned position even though forces act on its charges. Therefore option C is correct; it is not maximum or infinite.

The torque on an electric dipole is τ = pE sin θ. At θ = 180°, the dipole moment is antiparallel to the electric field, and sin 180° = 0. Thus τ = pE × 0 = 0. This position is an equilibrium orientation, although it is unstable under a small angular displacement. Hence option D is correct; the torque is neither maximum nor double.

For a dipole in a uniform electric field, the potential energy is U = −pE cos θ. Equilibrium occurs when the torque pE sin θ is zero, at θ = 0° or 180°. At θ = 180°, U = +pE, its maximum value, so a small displacement lowers the energy and produces a restoring-away effect. Hence the antiparallel, or opposite, position is unstable. The parallel position has minimum energy and is stable.

The potential energy of an electric dipole in a uniform field is U = −pE cos θ, where θ is the angle between p and E. For a fixed p and E, the smallest value occurs when cos θ = 1, that is, θ = 0°. Then U = −pE. Therefore the dipole has minimum potential energy when its moment is parallel to the field. The antiparallel position gives maximum energy, while the perpendicular position gives zero energy.

For a dipole in a uniform electric field, U = −pE cos θ. The maximum value is obtained when cos θ = −1, which occurs at θ = 180°. Thus Umax = +pE and the dipole moment is antiparallel to the electric field. This is also the unstable equilibrium position. Parallel alignment gives the minimum value −pE, whereas perpendicular alignment gives U = 0, so neither is the maximum.

The potential energy of a dipole in a uniform electric field is U = −pE cos θ. If the dipole is perpendicular to the field, the angle is θ = 90°. Since cos 90° = 0, substitution gives U = −pE(0) = 0. Therefore option A is correct. This value is neither the minimum nor the maximum: the minimum is −pE at 0°, and the maximum is +pE at 180°, for nonzero p and E.

A dipole in a uniform electric field experiences a torque τ = pE sin θ. This torque rotates the dipole toward θ = 0°, where its dipole moment is parallel to the field. At that orientation, the potential energy U = −pE is minimum, so it is the stable equilibrium direction. The opposite orientation is unstable, and a perpendicular orientation generally experiences maximum torque rather than a preferred final alignment.

The torque on an electric dipole placed in a uniform electric field is τ = pE sin θ, where p is the dipole-moment magnitude, E is the field magnitude, and θ is the angle between them. Thus torque changes with all three quantities. It is maximum, pE, at 90° and zero at 0° or 180°. Mass, temperature, and time alone do not determine the electrostatic torque in this ideal situation.

For an electric dipole in a uniform field, the potential energy is U = −p · E = −pE cos θ, where θ is the angle between the dipole moment and the field. Therefore the angular factor is cosine. Sine belongs to the torque magnitude, τ = pE sin θ, which is a related but different quantity. Tangent does not occur in this formula, and the angle is essential except in a special fixed orientation.

The magnitude of torque on a dipole is τ = pE sin θ. If p and θ remain constant, then sin θ is constant and τ is directly proportional to E. Consequently, increasing the electric-field magnitude increases the torque in the same proportion. It does not necessarily become zero; that happens only when θ is 0° or 180°. A decrease or no change would contradict the proportional relation under the stated conditions.

The torque magnitude on an electric dipole is τ = pE sin θ. With E and θ fixed, the factors E and sin θ are constant, so τ is directly proportional to the dipole moment p. Therefore increasing p increases the torque by the same factor. It is zero only for a parallel or antiparallel orientation, not for every angle, and no finite increase in p makes the torque infinite in this ideal relation.

For an electric dipole in a uniform field, the torque magnitude is τ = pE sin θ. If E = 0, the product pE sin θ is zero for every orientation, regardless of the value of p or θ. Equivalently, no electric force acts on either charge because the applied field is absent, so there is no turning tendency. Therefore the torque is zero, not maximum, negative, or equal to p.

The torque on an electric dipole in a uniform field is given by τ = pE sin θ, where p is the dipole moment, E is the field strength, and θ is the angle between them. If p = 0, then τ = 0 × E × sin θ = 0 for every orientation. Thus the torque is zero; it is neither undefined nor equal to the field.