Question

A length-scale $(\ell)$ depends on the permittivity $(\varepsilon)$ 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. $\ell=\sqrt{\left(\frac{ nq ^{2}}{\varepsilon k _{ B } T }\right)}$
• B. $\ell=\sqrt{\left(\frac{\varepsilon k _{ B } T }{ nq ^{2}}\right)}$
• C. $\ell=\sqrt{\left(\frac{q^{2}}{\varepsilon n^{2 / 3} k_{B} T}\right)}$
• D. $\ell=\sqrt{\left(\frac{q^{2}}{\operatorname{sn}^{1 / 3} k_{B} T}\right)}$

Answer 1

Answer: (D)

$\ell=\sqrt{\left(\frac{q^{2}}{\operatorname{sn}^{1 / 3} k_{B} T}\right)}$

Answer 2

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

So, the correct option is (D): $\ell=\sqrt{\left(\frac{q^{2}}{\operatorname{sn}^{1 / 3} k_{B} T}\right)}$