A physical quantity of the dimensions of length that can be formed out of \(c, G,~\text{and}~\dfrac{e^2}{4\pi\varepsilon_0}\)is [\(c\) is the velocity of light, \(G\) is the universal constant of gravitation and \(e\) is charge]:
1. \(c^2\left[G \dfrac{e^2}{4 \pi \varepsilon_0}\right]^{\dfrac{1}{2}}\)
2. \(\dfrac{1}{c^2}\left[\dfrac{e^2}{4 G \pi \varepsilon_0}\right]^{\dfrac{1}{2}}\)
3. \(\dfrac{1}{c} G \dfrac{e^2}{4 \pi \varepsilon_0}\)
4. \(\dfrac{1}{c^2}\left[G \dfrac{e^2}{4 \pi \varepsilon_0}\right]^{\dfrac{1}{2}}\)

Planck's constant (\(h\)), speed of light in the vacuum (\(c\)), and Newton's gravitational constant (\(G\)) are the three fundamental constants. Which of the following combinations of these has the dimension of length?

If dimensions of critical velocity \({v_c}\) of a liquid flowing through a tube are expressed as \(\eta^{x}\rho^yr^{z}\), where \(\eta, \rho~\text{and}~r\) are the coefficient of viscosity of the liquid, the density of the liquid, and the radius of the tube respectively, then the values of \({x},\) \({y},\) and \({z},\) respectively, will be:

If force (\(F\)), velocity (\(\mathrm{v}\)), and time (\(T\)) are taken as fundamental units, the dimensions of mass will be:

The dimensions of ${\left({\mu }_0{\epsilon }_0\right)}^{-1/2}$ are:

1.  $[$ $L^{-1}T$ $]$

2.  $[$ $L{T}^{-1}$ $]$

3.  $[$ $L^{1/2}{T}^{1/2}$ $]$

4. $[$ $L^{1/2}{T}^{-1/2}$ $]$