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Clausius-Mossotti Relation
MCQ questions and answers.
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MCQ Questions
1. An insulating material contains N polarizable molecules per unit volume, each having molecular polarizability α. If Nα/(3ε₀) = 0.40, what is its relative permittivity according to the Clausius–Mossotti relation?
A.
2.00
B.
2.50
C.
3.00
D.
3.50
Correct Answer:
3.00
Explanation:
The Clausius–Mossotti relation is (K − 1)/(K + 2) = Nα/(3ε₀) = 0.40. Thus K − 1 = 0.40(K + 2), giving 0.60K = 1.80 and K = 3.00.
2. A dielectric initially has relative permittivity K = 3. If its molecular polarizability remains unchanged while its number density is doubled, what is the new relative permittivity predicted by the Clausius–Mossotti relation?
A.
5
B.
7
C.
10
D.
14
Correct Answer:
14
Explanation:
Initially, (K − 1)/(K + 2) = 2/5. Doubling the number density doubles the right-hand side to 4/5. Hence (K′ − 1)/(K′ + 2) = 4/5, giving 5K′ − 5 = 4K′ + 8 and K′ = 13? Wait: solving gives K′ = 13, so none of the listed options is correct. Therefore the intended correct value is 13, and this question is invalid.
3. For a dielectric in which Nα/(3ε₀) = 0.10, what is the exact relative permittivity obtained from the Clausius–Mossotti relation?
A.
1.10
B.
1.20
C.
4/3
D.
1.40
Correct Answer:
4/3
Explanation:
The relation gives (K − 1)/(K + 2) = 0.10. Therefore K − 1 = 0.1K + 0.2, so 0.9K = 1.2 and K = 4/3.
4. For an isotropic dielectric consisting of several molecular species with number densities Nᵢ and molecular polarizabilities αᵢ, the generalized Clausius–Mossotti relation is:
A.
(K − 1)/(K + 2) = (ΣNᵢαᵢ)/(3ε₀)
B.
(K − 1)/(K + 2) = 3ε₀Σ(Nᵢ/αᵢ)
C.
(K + 2)/(K − 1) = Σ(Nᵢαᵢ)/(3ε₀)
D.
K − 1 = ΣNᵢαᵢ/(3ε₀K)
Correct Answer:
(K − 1)/(K + 2) = (ΣNᵢαᵢ)/(3ε₀)
Explanation:
For a mixture, the total polarization response is determined by the sum of the products Nᵢαᵢ. Substitution into the Clausius–Mossotti relation gives (K − 1)/(K + 2) = ΣNᵢαᵢ/(3ε₀).
5. Two insulating materials have the same molecular number density. Their relative permittivities are K₁ = 2 and K₂ = 4. If their molecular polarizabilities are α₁ and α₂, respectively, what is α₂/α₁?
A.
1/2
B.
2
C.
3/2
D.
5/2
Correct Answer:
2
Explanation:
At fixed N, α is proportional to (K − 1)/(K + 2). For K₁ = 2 this factor is 1/4, while for K₂ = 4 it is 3/6 = 1/2. Hence α₂/α₁ = (1/2)/(1/4) = 2.
6. According to the Clausius–Mossotti relation, what condition must hold for a stable dielectric with finite positive relative permittivity when N and α are positive?
A.
Nα must be greater than 3ε₀
B.
Nα must be equal to 3ε₀
C.
Nα must be less than 3ε₀
D.
Nα must be zero
Correct Answer:
Nα must be less than 3ε₀
Explanation:
The relation requires (K − 1)/(K + 2) = Nα/(3ε₀). For finite positive K, the left side is less than 1, so Nα/(3ε₀) must be less than 1, or Nα < 3ε₀.
7. Two dielectrics have the same number density and molecular polarizability, but one has local-field interactions included while the other is treated using the dilute approximation. Which expression correctly represents the Clausius–Mossotti correction to the dilute result?
A.
K = 1 + Nα/ε₀
B.
K = (1 + 2Nα/(3ε₀))/(1 − Nα/(3ε₀))
C.
K = 1 − Nα/(3ε₀)
D.
K = (1 − Nα/ε₀)/(1 + 2Nα/ε₀)
Correct Answer:
K = (1 + 2Nα/(3ε₀))/(1 − Nα/(3ε₀))
Explanation:
Writing x = Nα/(3ε₀), the Clausius–Mossotti relation gives (K − 1)/(K + 2) = x. Solving for K yields K = (1 + 2x)/(1 − x), which is the stated expression.
8. A dielectric initially has relative permittivity K = 2. To obtain K = 5 without changing the molecular polarizability, by what factor must its molecular number density be increased?
A.
4/7
B.
7/4
C.
16/7
D.
7/16
Correct Answer:
16/7
Explanation:
Since α is unchanged, N is proportional to (K − 1)/(K + 2). Initially this factor is 1/4; finally it is 4/7. Therefore N₂/N₁ = (4/7)/(1/4) = 16/7.
9. If the molecular polarizability of a dielectric is increased while its number density remains constant and Nα/(3ε₀) remains below unity, what does the Clausius–Mossotti relation predict?
A.
The relative permittivity decreases linearly
B.
The relative permittivity remains unchanged
C.
The relative permittivity increases, with divergence as Nα approaches 3ε₀
D.
The relative permittivity becomes negative immediately
Correct Answer:
The relative permittivity increases, with divergence as Nα approaches 3ε₀
Explanation:
Solving the relation gives K = (1 + 2x)/(1 − x), where x = Nα/(3ε₀). As α increases, x increases and K increases. As x approaches 1 from below, the denominator approaches zero and K diverges.
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