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A length-scale $(l)$ depends on the permittivity $(\varepsilon)$ of a dielectric material. Boltzmann constant $\left(k_B\right)$, the absolute temperature $(T)$, the number per unit volune $(n)$ of certain charged particles, and the charge $(q)$ carried by each of the particless. Which of the following expression($s$) for $l$ is(are) dimensionally correct?
($A$) $l=\sqrt{\left(\frac{n q^2}{\varepsilon k_B T}\right)}$
($B$) $l=\sqrt{\left(\frac{\varepsilon k_B T}{n q^2}\right)}$
($C$)$l=\sqrt{\left(\frac{q^2}{\varepsilon n^{2 / 3} k_B T}\right)}$
($D$) $l=\sqrt{\left(\frac{q^2}{\varepsilon n^{1 / 3} k_B T}\right)}$
$B,A$
$B,C$
$C,A$
$B,D$
Solution
The correct options are
$B I=\sqrt{\left(\frac{\epsilon k_{ B } T}{n q^2}\right)}$
$D I=\sqrt{\left(\frac{q^2}{ en ^{1 / 3} k_B T}\right)}$
As given condition $\rightarrow$
$1 \propto \in k_B T$ and $l \propto \frac{1}{q^2}$
So, finding the dimension of the function
${\left[\frac{\varepsilon_0 k_B T}{q^2}\right]=\left[\frac{q^2 F r}{F r^2 q}\right]=\left[\frac{1}{r}\right]=\left[L^{-1}\right]}$
${\left[\frac{q^2}{\epsilon_0 k_B T}\right]=[L]}$
For option $A :\left[\sqrt{\frac{n q^2}{\varepsilon_0 k _{ B } T}}\right]=\left[\sqrt{ L ^{-3} L}\right]=\left[L^{-1}\right]$
For option $B :\left[\sqrt{\frac{\varepsilon_0 k _{ B } T}{ nq ^2}}\right]=\left[\sqrt{\frac{ L ^{-1}}{L^{-3}}}\right]=[ L ]$
For option C: $\left[\sqrt{\frac{ q ^2}{\varepsilon_0 K_{ B } T n^2}}\right]=\left[\sqrt{\frac{ L }{ L ^{-2}}}\right]=\left[ L ^{3 / 2}\right]$
For option D: $\left[\sqrt{\frac{q^2}{\varepsilon_0 K_{ B } n ^2}}\right]=\left[\sqrt{\frac{L}{L^{-1}}}\right]=[L]$
Hence, answer $B$ and $D$ matches correctly.
