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. `l= sqrt(((nq^2)/(varepsilonk_(B)T)))`
B. `l= sqrt(((varepsilonk_(B)T)/(nq^2)))`
C. `l= sqrt(((q^2)/(varepsilon n^(2/3) k_(B)T)))`
D. `l= sqrt(((q^2)/(varepsilon n^(1/3)k_(B)T)))`

Answer

B, D

Explanation

Permittivity : `[varepsilon][M^(-1)L^(-3)T^(4)I^(2)]`
Boltzmann constant : `[k_B]=[ML^(2)T^(-2) theta^(-1)]`
Absolute temperature : `[T]= [theta]`
Number per unit volume : `[n]= [L^(-3)]`
Charge : `[theta]=[TI]`
Option (b) : `sqrt((varepsilon k_(B)T)/(n q^(2)))= sqrt((L^(-1)T^(2)I^(2))/(L^(-3)T^(2)I^(2)))=[L]`
Option (d) : `sqrt((q^2)/(varepsilon n^(1/3)k_(B)T)) = sqrt((A^(2)T^(2))/([L^(-1)T^(2)A^(2)]xx[L^(-3)]^(1/3)))`
`=sqrt((1)/([L^(-1)]xx[L^(-1)]))=[L]`
We can easily understand that dimensions of remaining options are not equal to that of length.
Hence options b and d are correct.