Question

A length scale $\left(l\right)$ depends on the permittivity $\left(\epsilon \right)$ of a dielectric material, Boltzmann constant $\left(k_B\right)$, the absolute temperature $\left(T\right)$, the number per unit volume$\left(n\right)$ of certain charged particles and the charge $\left(q\right)$ carried by each of the particles. Which of the following expression(s) for $l$ is(are) dimensionally correct?

• A. $l=\sqrt{\left(\frac{{nq}^2}{{ϵk}_{B}T}\right)}$
• B. $l=\sqrt{\left(\frac{{ϵk}_{B}T}{{nq}^2}\right)}$
• C. $l=\sqrt{\left(\frac{q^2}{{ϵn}^{2/3}{k}_{B}T}\right)}$
• D. $l=\sqrt{\left(\frac{q^2}{{ϵn}^{1/3}{k}_{B}T}\right)}$

Answer 1

Answer: (B), (D)

$l=\sqrt{\left(\frac{q^2}{{ϵn}^{1/3}{k}_{B}T}\right)}$

As given condition $\to$
$l\propto ϵk_{B}T$ and $l\propto \frac{1}{q^2}$
So, finding the dimension of the function
$\left[\frac{{ϵ}_0{k}_{B}T}{q^2}\right]=\left[\frac{q^2}{F{r}^2}\frac{Fr}{q}\right]=\left[\frac{1}{r}\right]=\left[L^{-1}\right]\left[\frac{q^2}{{ϵ}_0{k}_{B}T}\right]=\left[L\right]$

For option A: $\left[\sqrt{\frac{{nq}^2}{{ϵ}_0{k}_{B}T}}\right]=\left[\sqrt{L^{-3}L}\right]=\left[L^{-1}\right]$

For option B: $\left[\sqrt{\frac{{ϵ}_0{k}_{B}T}{n{q}^2}}\right]=\left[\sqrt{\frac{L^{-1}}{L^{-3}}}\right]=\left[L\right]$

For option C: $\left[\sqrt{\frac{q^2}{{ϵ}_0{k}_{B}T{n}^{2/3}}}\right]=\left[\sqrt{\frac{L}{L^{-2}}}\right]=\left[L^{3/2}\right]$

For option D: $\left[\sqrt{\frac{q^2}{{ϵ}_0{k}_{B}T{n}^{2/3}}}\right]=\left[\sqrt{\frac{L}{L^{-1}}}\right]=\left[L\right]$

Hence, answer B and D matches correctly.