Question

A length-scale $(\ell )$ depends on the permittivity $(\epsilon )$ of a dielectric material, Boltzmann constant $\left(k_B\right)$, 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 expressions (s) for $\ell$ is (are) dimensionally correct?

• A. $\ell =\sqrt{\left(\frac{n{q}^2}{\epsilon k_{B}T}\right)}$
• B. $\ell =\sqrt{\left(\frac{\epsilon k_{B}T}{n{q}^2}\right)}$
• C. $\ell =\sqrt{\left(\frac{q^2}{\epsilon n^{2/3}{k}_{B}T}\right)}$
• D. $\ell =\sqrt{\left(\frac{q^2}{{\mathrm{sn}}^{1/3}{k}_{B}T}\right)}$

Answer 1

Answer: (B), (D)

$\ell \alpha {\epsilon }^a{k}^b{T}^c{n}^d{q}^e$
(A) $\ell =\sqrt{\frac{L^{-3}\times A^2{T}^2}{M^{-1}{A}^2{T}^4{L}^{-3}{M}^1{L}^2{T}^{-2}{\theta }^{-1}\theta }}$
$\ell =\sqrt{\frac{1}{L^2}}=\frac{1}{L}$
(B) $\ell =\sqrt{\frac{\epsilon k_{B}T}{n{q}^2}}$
$=\sqrt{\frac{\left(M^{-1}{A}^2{T}^4{L}^{-3}\right)M^1{L}^2{T}^{-2}{\theta }^{-1}\theta }{L^{-3}{A}^1{T}^2}}$
$=\sqrt{L^2}=L$
(C) $\ell =\sqrt{\frac{A^2{T}^2}{M^{-1}{A}^2{T}^4{L}^{-3}{L}^{-2}{M}^1{L}^2{T}^{-2}{\theta }^{-1}\theta }}$
(D) $\ell =\sqrt{\frac{A^2{T}^2}{M^{-1}{A}^2{T}^4{L}^{-3}{L}^{-1}{M}^{+1}{L}^2{T}^{-2}{\theta }^{-1}\theta }}$
$=\sqrt{L^2}=L$