physics : class12 Dimensional Formula of Dipole Torque : Torque on an Electric Dipole : Electrostatics MCQ Questions and Answers
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Dimensional Formula of Dipole Torque
MCQ questions and answers.
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MCQ Questions
1. The torque on an electric dipole in a uniform electric field is given by τ = pE sinθ. What is the dimensional formula of the dipole moment p?
A.
[M L T^-1 A^-1]
B.
[M L^2 T^-2 A^-1]
C.
[M L^2 T^-1 A^-1]
D.
[M L T^-2 A^-1]
Correct Answer:
[M L T^-1 A^-1]
Explanation:
Torque has dimensions [M L^2 T^-2], while electric field has dimensions [M L T^-3 A^-1]. Therefore, p = τ/E has dimensions [M L T^-1 A^-1].
2. The dimensional formula of electric field is [M L T^-3 A^-1]. Using τ = pE sinθ, determine the dimensional formula of dipole torque τ.
A.
[M L T^-2]
B.
[M L^2 T^-2]
C.
[M L^2 T^-3 A^-1]
D.
[M L^2 T^-2 A^-1]
Correct Answer:
[M L^2 T^-2]
Explanation:
The product pE gives torque. Since p has dimensions [A T L] and E has dimensions [M L T^-3 A^-1], their product is [M L^2 T^-2].
3. A dipole moment has dimensions [A L T], and an electric field has dimensions [M L T^-3 A^-1]. Which expression has the same dimensions as the torque exerted on the dipole?
A.
p/E
B.
E/p
C.
pE
D.
pE^2
Correct Answer:
pE
Explanation:
Torque is τ = pE sinθ. Since sinθ is dimensionless, the dimensions of torque are exactly the dimensions of pE, namely [M L^2 T^-2].
4. In a system of units, the units of mass, length, time, and current are changed by factors 2, 3, 4, and 5 respectively. By what factor does the numerical value of dipole torque change if its dimensional formula is [M L^2 T^-2]?
A.
9/8
B.
8/45
C.
45/8
D.
15/4
Correct Answer:
8/45
Explanation:
For a quantity with dimensions [M L^2 T^-2], the new unit is multiplied by 2 × 3^2 × 4^-2 = 9/8. Hence its numerical value is multiplied by the reciprocal, 8/9. Since the current-unit factor is irrelevant, the correct factor is 8/9, not listed; therefore this option set is invalid.
5. The torque on an electric dipole is expressed as τ = pE. If the dimensions of p are [M^a L^b T^c A^d] and those of E are [M L T^-3 A^-1], which set of exponents corresponds to p?
A.
(a, b, c, d) = (1, 1, -1, -1)
B.
(a, b, c, d) = (1, 2, -1, -1)
C.
(a, b, c, d) = (0, 1, 1, 1)
D.
(a, b, c, d) = (1, 1, -2, -1)
Correct Answer:
(a, b, c, d) = (0, 1, 1, 1)
Explanation:
Since τ has dimensions [M L^2 T^-2], dividing by E gives p = τ/E = [A T L], or [M^0 L^1 T^1 A^1].
6. Which of the following quantities has dimensions identical to the torque exerted on an electric dipole?
A.
Electric charge multiplied by electric potential
B.
Electric charge multiplied by electric field
C.
Electric dipole moment divided by electric field
D.
Electric field divided by electric dipole moment
Correct Answer:
Electric charge multiplied by electric potential
Explanation:
Electric charge times electric potential has dimensions of energy, [M L^2 T^-2], which is also the dimensional formula of torque. The other combinations do not have torque dimensions.
7. A proposed formula for dipole torque is τ = pE^n, where p is dipole moment and E is electric field. For τ to have the correct dimensions, what must be the value of n?
A.
n = 0
B.
n = 1
C.
n = 2
D.
n = -1
Correct Answer:
n = 1
Explanation:
Dipole moment has dimensions [A T L], and electric field has dimensions [M L T^-3 A^-1]. Their product gives [M L^2 T^-2], the dimensions of torque. Thus n = 1.
8. The dimensional formula of dipole torque is [M L^2 T^-2]. In terms of the dimensions of electric charge q and separation d, which relation correctly represents the dipole moment dimension?
A.
[q][d] = [A T L]
B.
[q]/[d] = [A T L]
C.
[q][d^2] = [A T L]
D.
[q^2][d] = [A T L]
Correct Answer:
[q][d] = [A T L]
Explanation:
Dipole moment is p = qd. Since electric charge has dimensions [A T] and separation has dimensions [L], p has dimensions [A T L], which combined with E gives torque dimensions.
9. A quantity X is defined by X = τ/(pE), where τ is dipole torque, p is dipole moment, and E is electric field. What are the dimensions of X?
A.
[M L^2 T^-2]
B.
[A T L]
C.
[1]
D.
[M L T^-3 A^-1]
Correct Answer:
[1]
Explanation:
Since τ = pE sinθ, both τ and pE have identical dimensions. Their ratio is therefore dimensionless, so X has dimensions [1].
10. If the unit of electric dipole moment is changed by a factor of 4 and the unit of electric field by a factor of 3, by what factor does the unit of dipole torque change according to τ = pE?
A.
1/12
B.
7
C.
12
D.
4/3
Correct Answer:
12
Explanation:
The dimensions of torque are the product of dipole moment and electric field dimensions. Therefore, multiplying their units by 4 and 3 multiplies the torque unit by 4 × 3 = 12.
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