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?
($A$) $l=\sqrt{\left(\frac{n q^2}{\varepsilon k_B T}\right)}$
($B$) $l=\sqrt{\left(\frac{\varepsilon k_B T}{n q^2}\right)}$
($C$)$l=\sqrt{\left(\frac{q^2}{\varepsilon n^{2 / 3} k_B T}\right)}$
($D$) $l=\sqrt{\left(\frac{q^2}{\varepsilon n^{1 / 3} k_B T}\right)}$
$B,A$
$B,C$
$C,A$
$B,D$
Frequency is the function of density $(\rho )$, length $(a)$ and surface tension $(T)$. Then its value is
$A, B, C$ and $D$ are four different physical quantities having different dimensions. None of them is dimensionless. But we know that the equation $AD = C\, ln\, (BD)$ holds true. Then which of the combination is not a meaningful quantity ?
$\left(P+\frac{a}{V^2}\right)(V-b)=R T$ represents the equation of state of some gases. Where $P$ is the pressure, $V$ is the volume, $T$ is the temperature and $a, b, R$ are the constants. The physical quantity, which has dimensional formula as that of $\frac{b^2}{a}$, will be
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