Question

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?
A. `I = sqrt((nq^2)/(I"n" k_B T))`
B. `I =sqrt(("in" k_B T)/(nq^2))`
C. `I =sqrt((q^2)/("in" n^2//3 k_B T))`
D. `I = sqrt(q^2// "in" n^(1//3) k_B T)`

Answer 1

Correct Answer - (b,d)
Here, `I = [L], q = [AT], T = [K]`
`n =[L^(-3)], "in" = [M^(-1) l^(-3) A^2 T^4]`,
`k_B =[ML^2 T^(-2) K^(-1)]`
(b) `sqrt(("in" k_BT)/(nq^2))`
`=sqrt(((M^(-1) L^(-3)A^2 T^4) (ML^2T^(-2) K^(-1)).K)/(L^(-3) (AT)^2)`
`[m^0 LT^0 A^0] [L]`
(d) `sqrt((q^2)/("in" k^(1//3) k_BT))`
`= sqrt((A^2T^2)/((M^(-1) l^(-3)A^2 T^4)L^(-1)(ML^2 T^(-2) K^(-1))K))`
=[L]
Hence choices (b) and (d) are correct.