physics : class12 Stable Equilibrium of a Dipole : Torque on an Electric Dipole : Electrostatics MCQ Questions and Answers
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Stable Equilibrium of a Dipole
MCQ questions and answers.
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MCQ Questions
1. An electric dipole of moment p is placed in a uniform electric field E. Its potential energy is U(θ) = -pE cos θ. Which condition correctly identifies its stable equilibrium orientation?
A.
θ = 0, because dU/dθ = 0 and d²U/dθ² > 0
B.
θ = π, because dU/dθ = 0 and d²U/dθ² > 0
C.
θ = π/2, because the torque is maximum there
D.
Every orientation is stable because the field is uniform
Correct Answer:
θ = 0, because dU/dθ = 0 and d²U/dθ² > 0
Explanation:
Equilibrium requires dU/dθ = 0, giving θ = 0 or π. Since d²U/dθ² = pE cos θ, it is positive at θ = 0 and negative at θ = π. Therefore, the dipole aligned parallel to the field is in stable equilibrium.
2. A dipole is initially in stable equilibrium in a uniform electric field. It is rotated through a small angle δ. To the lowest nonzero order in δ, the increase in its potential energy is:
A.
pEδ
B.
pEδ²/2
C.
-pEδ²/2
D.
pE sin δ
Correct Answer:
pEδ²/2
Explanation:
At stable equilibrium θ = 0, U = -pE cos θ. For a small displacement δ, cos δ ≈ 1 - δ²/2, so the energy increase is U(δ) - U(0) = pE(1 - cos δ) ≈ pEδ²/2, which is positive.
3. A dipole in a uniform electric field is at θ = π. A small angular displacement is given to it. The resulting torque tends to:
A.
Restore it to θ = π
B.
Rotate it farther away from θ = π toward θ = 0
C.
Become zero for every small displacement
D.
Reverse the electric field direction
Correct Answer:
Rotate it farther away from θ = π toward θ = 0
Explanation:
The torque is τ = -pE sin θ. For θ = π + δ, where δ is small, τ ≈ pEδ, which has the same sense as the displacement. Thus the torque increases the displacement, showing that the antiparallel orientation is unstable rather than stable.
4. For a dipole in stable equilibrium in a uniform electric field, the effective angular restoring torque for a small angular displacement δ is:
A.
τ ≈ +pEδ
B.
τ ≈ -pEδ
C.
τ ≈ -pE/δ
D.
τ ≈ 0 to first order in δ
Correct Answer:
τ ≈ -pEδ
Explanation:
Stable equilibrium occurs when the dipole is parallel to the field. With θ = δ and δ small, τ = -pE sin θ ≈ -pEδ. The negative sign indicates that the torque opposes the angular displacement and therefore acts as a restoring torque.
5. The potential energy of a dipole in a uniform field is minimum at stable equilibrium. If the dipole is turned from stable equilibrium through an angle θ, the exact energy required for this quasistatic rotation is:
A.
pE sin θ
B.
pE cos θ
C.
pE(1 - cos θ)
D.
pE(1 + cos θ)
Correct Answer:
pE(1 - cos θ)
Explanation:
Stable equilibrium is at θ = 0, where Umin = -pE. At angle θ, U = -pE cos θ. Hence the energy supplied quasistatically is U(θ) - Umin = -pE cos θ + pE = pE(1 - cos θ), which is nonnegative.
6. Two dipoles have moments p and 2p and are placed separately in the same uniform electric field E. For small angular displacements from their stable equilibria, the magnitudes of their restoring torque constants are in the ratio:
A.
1:1
B.
1:2
C.
2:1
D.
1:4
Correct Answer:
1:2
Explanation:
Near stable equilibrium, the restoring torque is τ ≈ -pEδ, so the torque constant is pE. For moments p and 2p in the same field, the constants are pE and 2pE, giving the ratio 1:2.
7. A dipole is in stable equilibrium in a uniform electric field. If the field magnitude is increased while the dipole remains parallel to the field, which statement is correct about its stability?
A.
The equilibrium becomes unstable because the torque increases
B.
The equilibrium remains stable, and the restoring torque constant increases
C.
The equilibrium orientation changes to perpendicular to the field
D.
The equilibrium remains stable, but the restoring torque constant becomes zero
Correct Answer:
The equilibrium remains stable, and the restoring torque constant increases
Explanation:
The stable orientation is determined by alignment of the dipole with the field, not by the field magnitude. For a small displacement, the restoring torque constant is pE, so increasing E makes the stable equilibrium stronger.
8. For a dipole of fixed moment p in a uniform field E, the curvature of its potential-energy graph with respect to θ at stable equilibrium is:
A.
-pE
B.
0
C.
pE
D.
p/E
Correct Answer:
pE
Explanation:
Since U(θ) = -pE cos θ, its second derivative is d²U/dθ² = pE cos θ. At stable equilibrium θ = 0, this becomes pE, a positive curvature that confirms a local minimum of potential energy.
9. A dipole is constrained to rotate about an axis perpendicular to its plane and passes through its center. Which small-displacement condition signifies stable equilibrium in a uniform field?
A.
The torque has the same sign as the angular displacement
B.
The torque is independent of the angular displacement
C.
The torque is opposite in sign to the angular displacement
D.
The potential energy decreases for every small displacement
Correct Answer:
The torque is opposite in sign to the angular displacement
Explanation:
For a stable equilibrium, a small displacement must produce a restoring torque. Mathematically, τ ≈ -κδ with κ > 0. Thus the torque must be opposite in sign to the angular displacement, as occurs when the dipole is parallel to the field.
10. A dipole has stable equilibrium at θ = 0 in a uniform electric field. At which angle, measured from the field direction, is the potential-energy difference from stable equilibrium equal to pE?
A.
θ = π/6
B.
θ = π/3
C.
θ = π/2
D.
θ = π
Correct Answer:
θ = π/2
Explanation:
The energy difference from stable equilibrium is ΔU = pE(1 - cos θ). Setting ΔU = pE gives 1 - cos θ = 1, so cos θ = 0. Therefore, θ = π/2.
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