1)

A length-scale (l) depends on the permittivity (ε) of a dielectric material, Boltzmann constant (kB) 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 / is (are) dimensionally correct?


A) l=(nq2ϵkBT)

B) l=(ϵkBTnq2)

C) l=(q2ϵn23kBT)

D) l=(q2ϵn13kBT)

Answer:

Option B,D

Explanation:

[n]=[L3];[q]=[AT]

[ϵ]=[M1L3A2T4]

[T]=[L]

[l]=[L]

[kB]=[M1L2T2K1]

(a)  RHS

[L3A2T2][M1L3T4A2][M1L2T2K1][K]

[L3A2T2][L1A2T2]

[L2]=[L1]   Wrong

(b) RHS

[M1L3T4A2][M1L2T2K1][K][L3A2T2]

[L1A2T2][L3A2T2]=[L]  Correct

(c) RHS

[A2T2][M1L3T4A2][L2][M1L2T2K1][K]

[L3] Wrong

(d) RHS

[A2T2][M1L3T4A2][L1][M1L2T2K1]

[A2T2][L2A2T2]= [L]   Correct