Question

A physical quantity of the dimension of length that can be formed out of c, G and $\frac{e^2}{4\pi {ϵ}_0}$ is [c is velocity of light, G is universal constant of gravitation and e is charge]

Answer 1

Given,

$C,G,\frac{e^2}{4\pi {\epsilon }_0}$

So,

$\left[L\right]=[C{]}^a,[G{]}^b[\frac{e^2}{4\pi {\epsilon }_0}{]}^c$

$\left[L\right]=[L{T}^{-1}{]}^a,[M{L}^3{T}^{-2}{]}^b[M{L}^3{T}^{-2}{]}^c$

$\left[L\right]=L^{a+3b+3c}{M}^{-b+c}{T}^{-a-2b-2c}$

on comparing we get,

$a=3b+3c=1,$

$-b+c=0$

$a+2b+2c=0$

On solving we get,

$a=-2,b=\frac{1}{2},c=\frac{1}{2}$

Thus,

$L=\frac{1}{C^2}\sqrt{\left[\frac{G{e}^2}{4\pi {\epsilon }_0}\right]}$