Question

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

• A. $c^2{\left[G\frac{e^2}{4\pi {\epsilon }_0}\right]}^{1/2}$
• B. $\frac{1}{c^2}{\left[\frac{e^2}{G4\pi {\epsilon }_0}\right]}^{1/2}$
• C. $\frac{1}{c^2}G\frac{e^2}{4\pi {\epsilon }_0}$
• D. $\frac{1}{c^2}{\left[G\frac{e^2}{4\pi {\epsilon }_0}\right]}^{1/2}$

Answer 1

Answer: (D)

Let $\frac{e^2}{4\pi {\epsilon }_0}=A=M{L}^3{T}^{-2}$
$I=C^x{G}^y(A{)}^z$
$L={\left[L{T}^{-1}\right]}^x{\left[M^{-1}{L}^3{T}^{-2}\right]}^y{\left[M{L}^3{T}^{-2}\right]}^z$
$-y+z=0\Rightarrow y=z$...(i)
$x+3y+3z=1$...(ii)
$-x-4z=0$...(iii)
From (i), (ii) & (iii)
$z=y=\frac{1}{2},x=-2$