Question

A physical quantity of the dimensions of length that can be formed out of c, G and $\frac{e^2}{4 \pi \varepsilon_0}$ is [c is velocity of light, G is universal constant of gravitation and e is charge] :-

• A. $c^{2}\left[G \frac{e^{2}}{4\pi \varepsilon_{0}}\right]^{1/ 2 }$
• B. $\frac{1}{c^{2}}\left[\frac{e^{2}}{G 4\pi \varepsilon_{0}}\right]^{1/ 2 }$
• C. $\frac{1}{c^{2}} G \frac{e^{2}}{4\pi \varepsilon_{0}}$
• D. $\frac{1}{c^{2}}\left[G\frac{e^{2}}{ 4\pi \varepsilon_{0}}\right]^{1/ 2 }$

Answer 1

Answer: (D)

$\frac{1}{c^{2}}\left[G\frac{e^{2}}{ 4\pi \varepsilon_{0}}\right]^{1/ 2 }$

Answer 2

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