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)

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Correct answer: B. l = ( ε k _ B T n q ^ 2 )

We know, F = 1 4π E _ 0 q ^ 2 r ^ 2 ⇒ q ^ 2 E _ 0 = (F r ^ 2 )4π So, dimension q ^ 2 E _ 0 = dim (F r ^ 2 ) = ML T ^ - 2 × L ^ 2 = M L ^ 3 T ^ - 2 Similarly; E = (3)/(2) K _ B T ⇒ dim ( K _ B T) = dim (Energy) = M L ^ 2 T ^ - 2 (i) n q ^ 2 E k _ B T = L ^ - 3 × M L ^ 3 T ^ - 2 M L ^ 2 T ^ - 2 = (1)/(L) (ii) (E) × vol F r ^ 2 = ML T ^ - 2 × L ^ 3 ML T ^ - 2 × L ^ 2 = L (iii) F r ^ 2 (vol) ^ (2)/(3) ( Kℇ ) = ML T ^ - 2 × L ^ 2 × L ^ 2 M L ^ 2 T ^ - 2 = L ^ 3 = L ^ (3)/(2) (iv) F r ^ 2 (vol) ^ (1)/(3) Energy = ML T ^ - 2 L ^ 2 × L M L ^ 2 T ^ - 2 = L [ ∴ dimension n = dim ((1)/(vol)) = L ^ - 3 ]

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