physics : class12 Torque in Nonuniform Electric Fields : Torque on an Electric Dipole : Electrostatics MCQ Questions and Answers
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Torque in Nonuniform Electric Fields
MCQ questions and answers.
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MCQ Questions
1. A point electric dipole with dipole moment p = p x̂ is placed with its centre at the origin in the nonuniform electrostatic field E = (E0 + αx)x̂ − αy ŷ. What are the net force and torque about the dipole centre?
A.
F = αp x̂ and τ = 0
B.
F = 0 and τ = αp x̂
C.
F = αp ŷ and τ = pE0 ẑ
D.
F = αp x̂ and τ = pE0 ẑ
Correct Answer:
F = αp x̂ and τ = 0
Explanation:
For a small dipole, the force is F = ∇(p·E), giving F = αp x̂. The torque about its centre is p × E at the centre. Since both p and E(0) = E0 x̂ are parallel, the torque is zero, although the nonuniform field produces a net force.
2. An electric dipole of moment p = p ŷ is centred at the origin in the field E = (E0 + αx)x̂ − αy ŷ. What is the torque about its centre?
A.
−pE0 ẑ
B.
pE0 ẑ
C.
−αp ẑ
D.
Zero because the field is nonuniform
Correct Answer:
−pE0 ẑ
Explanation:
The torque about the dipole centre is determined by the field at the centre: τ = p × E(0). Thus τ = p ŷ × E0 x̂ = −pE0 ẑ. Nonuniformity affects the net force but does not add a first-order torque for an ideal point dipole.
3. A finite dipole consists of charges +q and −q at positions ±a û about its centre. It is placed in E = (E0 + κx²)x̂, where û makes an angle θ with the x-axis. What is the exact torque about the dipole centre?
A.
−2qaE0 sinθ ẑ
B.
−2qa(E0 + κa²)sinθ ẑ
C.
−2qa(E0 + κa²cos²θ)sinθ ẑ
D.
−2qa(E0 + κa²sin²θ)cosθ ẑ
Correct Answer:
−2qa(E0 + κa²cos²θ)sinθ ẑ
Explanation:
The charges are at x = ±a cosθ. Both experience x-directed fields of magnitude E0 + κa²cos²θ. Summing their torques about the centre gives τz = −2qa(E0 + κa²cos²θ)sinθ.
4. A dipole with charges ±q separated by 2a is aligned along the x-axis and centred at x = 0 in the one-dimensional field E(x) = E0 + βx + γx² along x̂. What is its torque about its centre?
A.
2qaE0 ẑ
B.
−2qaβ ẑ
C.
Zero
D.
−2qaγ ẑ
Correct Answer:
Zero
Explanation:
Both the dipole axis and the electric field are along x̂ at the charge positions. Therefore each individual torque r × F vanishes, regardless of the field's nonuniformity. The nonuniform field can produce a net force but not a torque for this collinear arrangement.
5. A dipole of moment p = p(cosθ x̂ + sinθ ŷ) is centred at the origin in E = (E0 + κx²)x̂. Which expression gives its exact torque about the centre, treating the dipole as a finite separation 2a?
A.
p(E0 + κa²)sinθ ẑ
B.
−p(E0 + κa²cos²θ)sinθ ẑ
C.
−p(E0 + κa²sin²θ)cosθ ẑ
D.
−p(E0 + κa²cosθ)sinθ ẑ
Correct Answer:
−p(E0 + κa²cos²θ)sinθ ẑ
Explanation:
Using p = 2qa and the charge coordinates x = ±a cosθ, the field at both charges has x-component E0 + κa²cos²θ. The torque is therefore −2qa(E0 + κa²cos²θ)sinθ ẑ, which equals the stated expression.
6. A dipole in a nonuniform field experiences a net force F and torque τ about its centre. If the torque is instead calculated about a point displaced from the centre by vector R, what is the correct relation?
A.
τnew = τ − R × F
B.
τnew = τ + R × F
C.
τnew = τ + F × R with no sign convention needed
D.
τnew = τ because torque is independent of the reference point
Correct Answer:
τnew = τ + R × F
Explanation:
The torque about a new point equals the torque about the original point plus the moment of the resultant force about the new reference displacement, τnew = τ + R × F when R is the vector from the old point to the new point. In a nonuniform field, F is generally nonzero, so the reference point matters.
7. An ideal point dipole is placed at a point where the electric field is E0 and the field has a nonzero gradient. Which statement about its torque about its centre is correct to first order in the dipole size?
A.
It is always zero because the field is nonuniform
B.
It is p × E0, while the field gradient primarily determines the net force
C.
It is always parallel to the field gradient
D.
It is ∇(p·E0) × p
Correct Answer:
It is p × E0, while the field gradient primarily determines the net force
Explanation:
For an ideal point dipole, the leading torque about its centre is τ = p × E evaluated at the centre. The spatial gradient gives the force F = ∇(p·E), so nonuniformity does not by itself imply an additional first-order torque.
8. A finite dipole is centred at the origin in the field E = κx² x̂, with no uniform component. Its axis makes an angle θ with the x-axis. For which nontrivial orientations is the torque about its centre zero?
A.
Only θ = 0
B.
Only θ = π/2
C.
θ = 0 and θ = π/2
D.
All orientations except θ = π/4
Correct Answer:
θ = 0 and θ = π/2
Explanation:
The exact torque magnitude is 2qaκa²cos²θ sinθ. It vanishes when sinθ = 0, corresponding to θ = 0 or π, and when cosθ = 0, corresponding to θ = π/2. Thus the dipole is torque-free when parallel or perpendicular to the x-axis.
9. A dipole of moment p = p ŷ is placed with its centre at x = x0 in the field E = (E0 + κx²)x̂. What is its torque about its centre?
A.
−p(E0 + κx0²) ẑ
B.
p(E0 + 2κx0) ẑ
C.
−2pκx0 ẑ
D.
Zero because the field gradient is along x̂
Correct Answer:
−p(E0 + κx0²) ẑ
Explanation:
At the dipole centre, the field is (E0 + κx0²)x̂. The torque is τ = p ŷ × (E0 + κx0²)x̂ = −p(E0 + κx0²)ẑ. The gradient changes the force distribution but the point-dipole torque uses the local field.
10. A finite dipole with charges ±q at ±a û is placed in E = (E0 + κx²)x̂. For θ = 45° and E0 = κa², where θ is the angle between û and x̂, the magnitude of its torque is:
A.
qaE0
B.
√2 qaE0
C.
3qaE0/(2√2)
D.
3qaE0/2
Correct Answer:
3qaE0/(2√2)
Explanation:
The torque magnitude is 2qa(E0 + κa²cos²θ)sinθ. With θ = 45°, cos²θ = 1/2, sinθ = 1/√2, and κa² = E0, this becomes 2qa(3E0/2)(1/√2) = 3qaE0/(2√2).
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