\(\text {Dimension of permittivity } \varepsilon=A^{2} C^{2} N^{-1} m^{-2}\)
\(\text {Dimension of Boltzmann constant } k_{B}=\left[\mathrm{ML}^{2} \mathrm{~T}^{-2} \mathrm{~K}^{-1}\right]\)
\(\text {Dimension of Absolute Temperature }=T\)
No. of charged particles per unit
Volume = No. of dimensions = a
Dimension of charge = AT
Dimension of length = L
\(\text { Dimension of }=\frac{\sqrt{\varepsilon k_{B} T}}{n q^{2}}\)
\(=\frac{\sqrt{\left[\mathrm{A}^{2} \mathrm{C}^{2} \mathrm{~N}^{-1} \mathrm{~m}^{-2}\right]}\left[\mathrm{ML}^{2} \mathrm{~T}^{-2} \mathrm{~K}^{-1}\right][\mathrm{T}]}{[\mathrm{AT}]^{2}}\)
\(=\sqrt{L^{2}}=L\)
\(\therefore \text { Dimension of length }=\mathrm{L}\)

\(\text { Velocity of light } c=\frac{1}{\sqrt{\mu_{0} \varepsilon_{0}}}\)
\(c=\left(\mu_{0} \varepsilon_{0}\right)^{-\frac{1}{2}}\)
\(\text { Hence dimension }\left(\mu_{0} \varepsilon_{0}\right)^{-\frac{1}{2}} \text { is that of velocity. }\)

\(v=a t+b t^{2}\)
\(\text { Dimensional equation is } \mathrm{LT}^{-1}\)
\(=a T=b T^{2}\)
\(\therefore \text { The dimension of } b=\frac{\mathrm{LT}^{-1}}{\mathrm{~T}^{2}}=\mathrm{LT}^{-3}\)

\(\text { Torque }=\text { Force } \times \mathrm{r}\)
\(\text { Dimension of torque }\)
\(=\left[\mathrm{MLT}^{-2}\right][\mathrm{L}]=\left[\mathrm{ML}^{2} \mathrm{~T}^{-2}\right]\)
\(\text { Energy }=\text { Force } \times \text { displacement }\)
\(\text { Dimension of Energy }\)
\(=\left[\mathrm{MLT}^{-2}\right][\mathrm{L}]=\left[\mathrm{ML}^{2} \mathrm{~T}^{-2}\right]\)