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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?

l=√(q²εn^(1/3)kBT)

l=√(εkBTnq²)

l=√(q²εn^(2/3)kBT)

l=√(nq²εkBT)

Solution:

[ε₀kBT/q²] = [q²F/r²F/q] = [1/r] = [L⁻¹][q²/ε₀kBT] = [L]Option A: [√(nq²ε₀kBT)] = [√L⁻¹L] = [L⁻¹]Option B: [√(ε₀kBTnq²)] = √[L⁻¹L] = [L]Option C: [√(q²ε₀kBTn^(2/3))] = [√L⁻¹L] = [L^(3/2)]Option D: [√(q²ε₀kBTn^(2/3))] = [√L⁻¹L] = [L]Answer is BD.