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?
($A$) $l=\sqrt{\left(\frac{n q^2}{\varepsilon k_B T}\right)}$
($B$) $l=\sqrt{\left(\frac{\varepsilon k_B T}{n q^2}\right)}$
($C$)$l=\sqrt{\left(\frac{q^2}{\varepsilon n^{2 / 3} k_B T}\right)}$
($D$) $l=\sqrt{\left(\frac{q^2}{\varepsilon n^{1 / 3} k_B T}\right)}$
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?
($A$) $l=\sqrt{\left(\frac{n q^2}{\varepsilon k_B T}\right)}$
($B$) $l=\sqrt{\left(\frac{\varepsilon k_B T}{n q^2}\right)}$
($C$)$l=\sqrt{\left(\frac{q^2}{\varepsilon n^{2 / 3} k_B T}\right)}$
($D$) $l=\sqrt{\left(\frac{q^2}{\varepsilon n^{1 / 3} k_B T}\right)}$
- [IIT 2016]
- A
$B,A$
- B
$B,C$
- C
$C,A$
- D
$B,D$
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