Dipole in a Uniform External Field

Angular acceleration is governed by rotational dynamics, τ = Iα, where τ is torque, I is the moment of inertia, and α is angular acceleration. In a uniform electric field, equal and opposite forces act on the two charges, so their resultant force is zero; however, the forces act along separated lines and form a torque. That torque produces angular acceleration.

An electric dipole consists of charges +q and −q separated by a distance. By convention, its dipole moment is p = qd, with the vector d directed from the negative charge toward the positive charge. This convention also makes the dipole’s stable orientation parallel to the external electric field. The reverse direction is not the standard electric dipole-moment convention.

For a dipole in a uniform field, τ = pE sin θ, while τmax = pE. Dividing gives τ/τmax = sin θ. At θ = 30°, sin 30° = 1/2, so the torque is half of its maximum value. The maximum is obtained only at 90°, whereas one-fourth and three-fourths are not the correct sine value for this angle. Thus option B is correct.

Electric dipole moment is defined as p = qd, where q is the magnitude of either charge and d is the separation vector from the negative to the positive charge. Charge has SI unit coulomb and distance has SI unit metre. Therefore, p has unit C m, or coulomb metre. Newton per coulomb is the unit of electric field, joule per coulomb is volt, and C/m is not the dipole-moment unit. Option A is correct.

The torque magnitude on a dipole is tau = pE sin theta, where p is the dipole moment, E is the field magnitude, and theta is the angle between them. The maximum value occurs when theta = 90 degrees because sin 90 degrees = 1. Therefore, tau_max = pE, the product of dipole moment and electric field. The ratio, sum, and difference in the other choices do not follow the torque equation, so option A is correct.

An ideal electric dipole consists of two equal and opposite charges, +q and -q, separated by a finite distance. Its net charge is therefore (+q) + (-q) = 0, independent of the external uniform electric field. The field can exert equal and opposite forces and can produce torque, but it cannot change the dipole's total charge. The separated charges still give a nonzero dipole moment p = qd, so option C is correct.

The potential energy of a dipole in a uniform electric field is U = −pE cos θ. At θ = 120°, cos 120° = −1/2. Hence U = −pE(−1/2) = +pE/2, which is positive when p and E are nonzero. Zero energy occurs at 90°, while negative energy occurs for an acute angle such as 0°; therefore those alternatives do not apply here.

For a dipole, p = qd, where q is the magnitude of either charge and d is the separation. The torque magnitude in a uniform field is τ = pE sin θ = qdE sin θ. If d is doubled while q, E, and θ remain unchanged, p doubles and therefore τ also doubles. It does not become four times because the dependence on separation is linear.

The torque magnitude is τ = pE sin θ. Its maximum value occurs at θ = 90°, where sin θ = 1, so τmax = pE. If the dipole moment remains fixed and the field changes from E to 2E, the new maximum torque is p(2E) = 2pE = 2τmax. Therefore it doubles, not quadruples, because the dependence on field is linear.

The potential energy of an electric dipole in a uniform field is U = −p·E. The dot product p·E equals pE cos θ, so it includes both magnitudes and their relative orientation. The negative sign means that parallel alignment has minimum energy and antiparallel alignment has maximum energy. The other pairs do not give the standard dipole potential-energy expression.

The torque on an electric dipole is expressed vectorially as τ = p × E. Its magnitude is pE sin θ, and its direction is perpendicular to the plane containing p and E, determined by the right-hand rule. This is why the cross product is required. A dot product would describe a scalar such as p·E, which appears in potential energy, not torque.

The torque on a dipole in a uniform electric field is τ = pE sin θ. Setting the external field E to zero makes τ = 0 for every orientation. However, the dipole’s charge magnitude, separation, and dipole moment p = qℓ are properties of the dipole itself and do not disappear merely because the external field is removed. Therefore only the field-produced torque becomes zero.

The potential energy of a dipole is U = −pE cos θ. For fixed p and E, this energy is smallest when cos θ is largest, namely cos θ = 1 at θ = 0°. Thus the dipole moment aligns with the electric field. A free dipole experiences torque τ = pE sin θ that turns it toward this orientation; the opposite direction gives maximum energy and 90° gives intermediate energy.

The rotational effect is the electric torque, given for a dipole by τ = p × E, with magnitude τ = pE sin θ. Thus the dipole moment p, together with the external field and their angle, directly determines the torque. A dipole has zero net total charge, and its mass or color does not enter this electrostatic relation. Therefore dipole moment is the relevant property.

The torque magnitude on a dipole is τ = pE sin θ. Initially, τ = pE sin θ. After the changes, p′ = 3p and E′ = E/3, while θ is unchanged. Hence τ′ = (3p)(E/3) sin θ = pE sin θ = τ. The increase in dipole moment exactly compensates for the decrease in field strength, so the torque remains unchanged. Therefore option A is correct.

For a dipole, the dipole moment is p = qd, where q is the magnitude of either charge and d is their separation. The torque is τ = pE sin θ = qdE sin θ. If q becomes 3q and d becomes 2d, the new dipole moment is p′ = (3q)(2d) = 6p. Since the electric field and angle remain unchanged, τ′ = 6τ. Therefore option C is correct; the factors multiply rather than add.

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. Because p and E are positive magnitudes, the negative sign makes U negative. Thus option A is correct. The energy becomes zero at 90° and positive for an obtuse angle; it is neither always zero nor infinite.

For a dipole in a uniform electric field, U = −pE cos θ. At an obtuse angle, 90° < θ < 180°, so cos θ is negative. The minus sign therefore makes U positive. Option A is correct. For an acute angle the energy is negative, and at 90° it is zero. The energy also depends on p and E, so it is not independent of the field.

The torque on an electric dipole in a uniform field is given by the vector relation τ = p × E, with magnitude τ = pE sin θ. If the dipole moment p is zero, the entire expression is zero for every field strength and every orientation. Thus no rotational couple remains. Torque cannot be maximum, infinite, or dependent only on angle when p is zero. Hence option A is the only correct answer.

In a uniform electric field, the positive and negative charges of a dipole experience equal forces in opposite directions. These forces have zero resultant but act along different lines, forming a couple. The resulting torque is τ = pE sin θ and tends to align the dipole with the field. Thus option A is correct, although the torque becomes zero for parallel or antiparallel alignment.

For a dipole in a uniform electric field, the force on +q is qE along the field and the force on −q is qE opposite to it. Because the field has the same magnitude at both charges, these forces cancel vectorially: F_net = qE − qE = 0. Therefore option A is correct. A torque may still act because the equal forces are separated, but that does not produce net translation.

The torque on an electric dipole in a uniform field is τ = pE sin θ, where θ is the angle between dipole moment p and electric field E. It is zero when sin θ = 0, namely at θ = 0° or 180°. Thus the dipole moment is parallel or antiparallel to the field, so option A is correct. At 90° the torque is maximum, not zero.

For a dipole in a uniform electric field, torque is τ = pE sin θ. With fixed p and E, the largest possible value of sin θ is 1, which occurs at θ = 90°. Therefore the dipole moment must be perpendicular to the field and τ_max = pE; option A is correct. Parallel alignment gives zero torque, while zero field or zero charge also gives zero torque.

The potential energy of a dipole in a uniform electric field is U = −pE cos θ. It is minimum at θ = 0°, when the dipole moment is along the field. A small angular displacement then produces a restoring torque, so the dipole tends to return to alignment. Therefore option A is the stable equilibrium position. At θ = 180°, energy is maximum and the equilibrium is unstable.

For a dipole in a uniform field, U = −pE cos θ. At θ = 180°, the dipole moment is opposite to the field and the potential energy is maximum, U = +pE. A small displacement lowers the energy and produces a torque that carries the dipole farther from the opposite alignment. Hence option A is the unstable equilibrium position; alignment with the field is stable.