physics : class12 Unstable Equilibrium of a Dipole : Torque on an Electric Dipole : Electrostatics MCQ Questions and Answers
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Unstable 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 with its dipole moment exactly opposite to the field. If it is displaced through a very small angle φ, where φ is measured from the opposite-field direction, its angular equation of motion is:
A.
I(d²φ/dt²) = -(pE)φ
B.
I(d²φ/dt²) = +(pE)φ
C.
I(d²φ/dt²) = 0
D.
I(d²φ/dt²) = +(pE)φ²
Correct Answer:
I(d²φ/dt²) = +(pE)φ
Explanation:
For an exactly opposite orientation, the torque is zero, but for a small displacement φ, τ = -pE sin(π + φ) ≈ +pEφ. Hence I(d²φ/dt²) = pEφ, so the displacement grows rather than being restored.
2. A dipole in a uniform electric field is initially maintained at its unstable equilibrium orientation and then given a small angular displacement. Neglecting damping, the subsequent angular displacement varies as:
A.
φ = φ₀ cos(√(pE/I)t)
B.
φ = φ₀ sin(√(pE/I)t)
C.
φ = φ₀ exp(√(pE/I)t), if released from rest
D.
φ = φ₀ exp[-(pE/I)t]
Correct Answer:
φ = φ₀ exp(√(pE/I)t), if released from rest
Explanation:
Near the antiparallel orientation, the equation is d²φ/dt² = (pE/I)φ. Its solutions are exponential, and for release from rest with initial displacement φ₀, the growing component is proportional to exp(√(pE/I)t), demonstrating instability.
3. For a dipole of moment p in a uniform field E, the potential energy near the unstable equilibrium orientation θ = π, with θ measured from the field direction, is best represented by:
A.
U ≈ -pE + (pE/2)(θ - π)²
B.
U ≈ pE - (pE/2)(θ - π)²
C.
U ≈ pE + (pE/2)(θ - π)²
D.
U ≈ -(pE/2)(θ - π)²
Correct Answer:
U ≈ pE - (pE/2)(θ - π)²
Explanation:
The dipole potential energy is U = -pE cosθ. Writing θ = π + φ gives U = pE cosφ ≈ pE - (pE/2)φ². The negative quadratic term shows that θ = π is a maximum of potential energy and therefore unstable.
4. A dipole is placed exactly antiparallel to a uniform electric field. Which statement correctly describes this state?
A.
The torque and potential energy are both zero.
B.
The torque is maximum and the potential energy is minimum.
C.
The torque is zero, but the potential energy is maximum.
D.
The torque is zero, and the potential energy has an inflection point.
Correct Answer:
The torque is zero, but the potential energy is maximum.
Explanation:
At θ = π, τ = pE sinπ = 0. However, U = -pE cosπ = +pE, the maximum possible dipole potential energy. A small displacement lowers the energy, so the dipole moves farther from this orientation.
5. Two identical dipoles, each having moment p and moment of inertia I, are placed antiparallel to identical uniform electric fields of magnitude E. Dipole 1 is given a small angular displacement and dipole 2 is given twice that displacement. The ratio of their initial angular accelerations is:
A.
1:1
B.
1:2
C.
2:1
D.
1:4
Correct Answer:
1:2
Explanation:
For small displacement from the antiparallel orientation, angular acceleration is α = (pE/I)φ. Since p, E, and I are identical, angular acceleration is proportional to displacement. Therefore, the ratio is φ₁:φ₂ = 1:2.
6. A dipole is displaced by a small angle φ from its unstable equilibrium orientation in a uniform field. Compared with the potential energy at the unstable equilibrium, its potential energy after displacement is:
A.
Greater by approximately (pE/2)φ²
B.
Smaller by approximately (pE/2)φ²
C.
Unchanged to the first and second orders in φ
D.
Smaller by approximately pEφ
Correct Answer:
Smaller by approximately (pE/2)φ²
Explanation:
Near θ = π, U ≈ pE - (pE/2)φ². Thus the potential energy decreases by (pE/2)φ² when the dipole is displaced from its unstable equilibrium.
7. For small angular displacement from the antiparallel orientation, the effective rotational potential of a dipole behaves like an inverted harmonic oscillator. The characteristic time associated with the instability is:
A.
√(I/pE)
B.
√(pE/I)
C.
I/(pE)
D.
pE/I
Correct Answer:
√(I/pE)
Explanation:
The linearized equation is d²φ/dt² = (pE/I)φ. Therefore, the exponential growth rate is √(pE/I), and its reciprocal, √(I/pE), is the characteristic instability time.
8. A dipole is held exactly opposite to a uniform electric field and then released without any initial angular velocity. In the ideal absence of perturbations, it will:
A.
Begin rotating immediately toward the field direction.
B.
Rotate through 90° with constant angular acceleration.
C.
Remain at rest at the antiparallel orientation.
D.
Oscillate with angular frequency √(pE/I).
Correct Answer:
Remain at rest at the antiparallel orientation.
Explanation:
Although the antiparallel position is unstable, the torque is exactly zero there. With no initial displacement and no perturbation, the dipole remains there by ideal deterministic dynamics; instability means that arbitrarily small disturbances grow.
9. A dipole near its unstable equilibrium has a small angular displacement φ and is released from rest. Which expression gives the magnitude of its torque to first order in φ, and in which direction does it act?
A.
pEφ, opposite to the displacement
B.
pEφ, in the direction of the displacement
C.
pE, opposite to the displacement
D.
pEφ², in the direction of the displacement
Correct Answer:
pEφ, in the direction of the displacement
Explanation:
If θ = π + φ, then τ = -pE sinθ = -pE sin(π + φ) ≈ +pEφ. Thus the torque is proportional to the displacement and acts in the same direction, which is the defining behavior of unstable equilibrium.
10. A dipole of moment p is displaced through a small angle φ from its antiparallel equilibrium position in a uniform field E. The maximum kinetic energy it can acquire while moving toward the stable orientation, assuming no energy loss, is approximately:
A.
(pE/2)φ²
B.
pEφ
C.
pE(1 - φ²/2)
D.
2pEφ²
Correct Answer:
(pE/2)φ²
Explanation:
At the displaced position near θ = π, the energy is U ≈ pE - (pE/2)φ². The unstable equilibrium has energy pE, so the dipole has an available energy difference of (pE/2)φ², which becomes kinetic energy as it moves away.
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