Discussion Forum



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) $l=\sqrt{\left(\frac{nq^{2}}{\epsilon k_{B}T}\right)}$

B) $l=\sqrt{\left(\frac{\epsilon k_{B}T}{nq^{2}}\right)}$

C) $l=\sqrt{\left(\frac{q^{2}}{\epsilon n^{\frac{2}{3}} k_{B}T}\right)}$

D) $l=\sqrt{\left(\frac{q^{2}}{\epsilon n^{\frac{1}{3}} k_{B}T}\right)}$

Answer:

Option B,D

Explanation:

$[n]=[L^{-3}]; [q]=[AT]$

$[\epsilon]=[M^{-1}L^{-3}A^{2}T^{4}]$

$[T]=[L]$

$[l]=[L]$

$[k_{B}]=[M^{1}L^{2}T^{-2}K^{-1}]$

(a)  RHS

$\sqrt{\frac{[L^{-3}A^{2}T^{2}]}{[M^{-1}L^{-3}T^{4}A^{2}][M^{1}L^{2}T^{-2}K^{-1}][K]}}$

$\sqrt{\frac{[L^{-3}A^{2}T^{2}]}{[L^{-1}A^{2}T^{2}]}}$

$\sqrt{[L^{-2}]}$=$[L^{-1}]$   Wrong

(b) RHS

$\sqrt{\frac{[M^{-1}L^{-3}T^{4}A^{2}][M^{1}L^{2}T^{-2}K^{-1}][K]}{[L^{-3}A^{2}T^{2}]}}$

$\sqrt{\frac{[L^{-1}A^{2}T^{2}]}{[L^{-3}A^{2}T^{2}]}}$=[L]  Correct

(c) RHS

$\sqrt{\frac{[A^{2}T^{2}]}{[M^{-1}L^{-3}T^{4}A^{2}][L^{-2}][M^{1}L^{2}T^{-2}K^{-1}][K]}}$

$\sqrt{[L^{3}]}$ Wrong

(d) RHS

$\sqrt{\frac{[A^{2}T^{2}]}{[M^{-1}L^{-3}T^{4}A^{2}][L^{-1}][M^{1}L^{2}T^{-2}K^{-1}]}}$

$\sqrt{\frac{[A^{2}T^{2}]}{[L^{-2}A^{2}T^{2}]}}$= [L]   Correct