A length - scale `(l)` depends on the permittivity `(epsilon)` of a dielctric material. Boltzmann constant `(k_(B))`, the absolute tempreture `(T)`, the number per unit volume `(n)` of certain charged particles, and the charge `(q)` carried by each of the partcles. which of the following expression `(s)` for `I` is `(are)` dimensionally correct?

Options

A. (a) `l = sqrt(((nq^(2))/(epsilonK_(B)T)))`
B. (b)`l = sqrt(((epsilonK_(B)T)/(nq^(2)))`
C. ( c ) `l = sqrt(((q^(2))/(epsilonn^(2//3)K_(B)T)))`
D. ( d ) `l = sqrt(((q^(2))/(epsilonn^(1//3)K_(B)T)))`

Answer

B, D

Explanation

(b, d)
We know that, dimensionally `in = (q^(2)/(l^(2)F)`'
`k_(B)T = (RT)/(N_(A)) = PV = F xx l`
Now `sqrt((epsilonk_(B)T)/(nq^(2)) = [(q^(2))/(l^(2)F xx (F xx l)/(l^(-3)q^(2))]^1//2`
Also `sqrt((q^(2))/(inn^(1//3)k_(B)T)) = [(l^(2)F xx q^(2))/(q^(2)L^(-1) xx F xx l)]^(1//2) = l`