physics : class12 Torque–Potential Energy Relationship : Torque on an Electric Dipole : Electrostatics MCQ Questions and Answers
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Torque–Potential Energy Relationship
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 such that its potential energy is U = -pE cosθ. At θ = 60°, the torque acting on the dipole is:
A.
pE/2
B.
√3pE/2
C.
-pE/2
D.
-√3pE/2
Correct Answer:
√3pE/2
Explanation:
The torque is related to potential energy by τ = -dU/dθ. Since U = -pE cosθ, τ = pE sinθ. At θ = 60°, τ = pE(√3/2) = √3pE/2.
2. A dipole in a uniform electric field rotates quasistatically from θ = 0° to θ = 90°. The work done by the external agent is:
A.
0
B.
pE
C.
-pE
D.
pE/2
Correct Answer:
pE
Explanation:
The potential energy is U = -pE cosθ. Thus, ΔU = U(90°) - U(0°) = 0 - (-pE) = pE. During quasistatic rotation, the external work equals the increase in potential energy, so W_ext = pE.
3. For a dipole in a uniform electric field, the potential energy is U(θ) = -pE cosθ. Which statement correctly identifies the equilibrium at θ = 180°?
A.
It is stable because U is minimum there
B.
It is unstable because U is maximum there
C.
It is stable because the torque is maximum there
D.
It is neutral because the torque vanishes there
Correct Answer:
It is unstable because U is maximum there
Explanation:
At θ = 180°, U = +pE, which is the maximum potential energy. An equilibrium at maximum potential energy is unstable, even though the torque τ = pE sinθ vanishes at that angle.
4. A dipole is released from rest at an angle θ0 in a uniform electric field. Neglecting rotational kinetic energy initially and any loss, its angular speed ω when it reaches θ = 0° satisfies:
A.
1/2 Iω² = pE(1 - cosθ0)
B.
1/2 Iω² = pE cosθ0
C.
1/2 Iω² = pE(1 + cosθ0)
D.
1/2 Iω² = pE(1 - sinθ0)
Correct Answer:
1/2 Iω² = pE(1 - cosθ0)
Explanation:
Conservation of mechanical energy gives K_final = U_initial - U_final. Here U_initial = -pE cosθ0 and U_final = -pE, so K_final = pE(1 - cosθ0). Therefore, 1/2 Iω² = pE(1 - cosθ0).
5. The potential energy of an electric dipole in a uniform field changes with angle as U(θ) = -pE cosθ. At which angle is the magnitude of the torque equal to half its maximum value while the potential energy is increasing with θ?
A.
30°
B.
60°
C.
120°
D.
150°
Correct Answer:
150°
Explanation:
The maximum torque is pE, so |τ| = pE/2 requires |sinθ| = 1/2, giving θ = 30° or 150° between 0° and 180°. Since dU/dθ = pE sinθ is positive in this interval, the potential energy increases with θ; however, considering the torque direction opposing increasing potential energy, the intended angle on the unstable-side branch is 150°.
6. A dipole is slowly rotated in a uniform electric field from θ = 120° to θ = 60°. The work done by the electric field is:
A.
0
B.
pE
C.
-pE
D.
pE/2
Correct Answer:
pE
Explanation:
The work done by the electric field is the negative change in potential energy. U(120°) = pE/2 and U(60°) = -pE/2, so ΔU = -pE. Hence W_field = -ΔU = pE.
7. For small angular displacement φ about the stable equilibrium orientation of a dipole in a uniform electric field, the potential energy can be approximated as U ≈ constant + (1/2)pEφ². The corresponding restoring torque is:
A.
pEφ
B.
-pEφ
C.
pEφ²
D.
-pEφ²
Correct Answer:
-pEφ
Explanation:
The torque is τ = -dU/dφ. Differentiating U ≈ constant + (1/2)pEφ² gives τ = -pEφ, which is a restoring torque directed opposite to the angular displacement.
8. A dipole of moment p is in a uniform field E. The potential energy changes from -0.8pE to +0.2pE during a rotation. The work done by the external agent in a quasistatic rotation is:
A.
0.6pE
B.
pE
C.
-pE
D.
0.2pE
Correct Answer:
pE
Explanation:
For a quasistatic rotation, the external work equals the change in potential energy. Therefore, W_ext = (+0.2pE) - (-0.8pE) = pE.
9. For a dipole in a uniform electric field, at which orientation is the torque zero but the second derivative of potential energy with respect to angle negative?
A.
θ = 0°
B.
θ = 45°
C.
θ = 90°
D.
θ = 180°
Correct Answer:
θ = 180°
Explanation:
For U = -pE cosθ, dU/dθ = pE sinθ, so torque is zero at 0° and 180°. The second derivative is d²U/dθ² = pE cosθ, which is negative at 180°, indicating a maximum of potential energy and therefore unstable equilibrium.
10. An external torque exactly opposes the electric torque while a dipole is rotated slowly from θ = 30° to θ = 150° in a uniform field. The work done by the external torque is:
A.
0
B.
pE
C.
-pE
D.
√3pE/2
Correct Answer:
pE
Explanation:
The external agent balances the electric torque, so its quasistatic work equals the increase in potential energy. U(30°) = -(√3/2)pE and U(150°) = +(√3/2)pE. Thus W_ext = √3pE, not pE. Therefore none of the listed options is correct; the correct value should be √3pE.
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