A length –scale (l) depends on the permittivity $(ε)$ of a dielectric material ,Boltzmann constant $(k_{B})$ , 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?

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$l = \sqrt{(\frac{n q^{2}}{ε k_{B} T})}$

$l = \sqrt{(\frac{ε k_{B} T}{n q^{2}})}$

$l = \sqrt{(\frac{q^{2}}{ε n^{2 / 3} k_{B} T})}$

$l = \sqrt{(\frac{q^{2}}{ε n^{1 / 3} k_{B} T})}$

answer is X.

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Detailed Solution

Permittivity $ε = [M^{- 1} L^{- 3} T^{4} I^{2}]$

Boltzmen constant $K_{B} = [M^{1} L^{2} T^{- 2} K^{- 1}]$

Temperature T = K

No.of particles per unit volume $n = [L^{- 3}]$

Charge $q = [I^{1} T^{1}]$

For A: $\sqrt{\frac{n q^{2}}{ε K_{B} T}} = \sqrt{\frac{[L^{- 3}] I^{2} T^{2}}{M^{- 1} L^{- 3} T^{4} I^{2} M^{1} L^{2} T^{- 2} K^{- 1} k}} = \frac{1}{L} = L^{- 1}$

For B: $\sqrt{\frac{ε K_{B} T}{n q^{2}}} = L$ [ from option A]

For C: $\sqrt{\frac{q^{2}}{ε n^{2 / 3} K_{B} T}} = \sqrt{\frac{I^{2} T^{2}}{M^{- 1} L^{- 3} T^{4} I^{2} L^{- 2} M L^{2} T^{- 2} K^{- 1} K}} = L^{3 / 2}$

For D: $\sqrt{\frac{q^{2}}{ε n^{1 / 3} K_{B} T}} = \sqrt{\frac{I^{2} T^{2}}{M^{- 1} L^{- 3} T^{4} I^{2} L^{- 1} M L^{2} T^{- 2} K^{- 1} k}} = L$

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