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Question

A length scale (l) depends on the permittivity (ε) of a dielectric material, Boltzmann constant (kB), the absolute temperature (T), the number per unit volume(n) of certain charged particles and the charge (q) carried by each of the particles. Which of the following expression(s) for l is(are) dimensionally correct?

A
l=(nq2ϵkBT)
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B
l=(ϵkBTnq2)
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C
l=(q2ϵn2/3kBT)
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D
l=(q2ϵn1/3kBT)
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Solution

The correct options are
C l=(q2ϵn1/3kBT)
D l=(ϵkBTnq2)
[ϵ0kBTq2]=[q2Fr2Frq]=[1r]=[L1][q2ϵ0kBT]=[L]

Option A: [nq2ϵ0kBT]=[L3L]=[L1]

Option B: [ϵ0kBTnq2]=L1L3=[L]

Option C: [q2ϵ0kBTn2/3]=[LL2]=[L3/2]

Option D: [q2ϵ0kBTn2/3]=[LL1]=[L]

Answer is BD.

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