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 $I$ is (are) dimensionally correct?

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

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

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

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

answer is B.

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Detailed Solution

Dimension of $[n] = L^{- 3} [q] = [A T]$

$[\frac{q^{2}}{\in}] = [F r^{2}] = [u \times r] = [M L^{3} T^{- 2}] [K_{B} T] = [U] = [M L^{2} T^{- 2}] D i m e n s i o n s l = \sqrt{[\frac{ε K_{B} T}{n q^{2}}]} = \sqrt{[\frac{K_{B} T}{n (\frac{q^{2}}{\in})}]} = \sqrt{\frac{[U]}{[n] [U \times r]}} = \sqrt{\frac{1}{[n] [r]}} = \sqrt{\frac{1}{[L^{- 3}] [L]}} = [L]$