Which of the following combinations has the dimension of electrical resistance ( ${ \varepsilon _0}$ is the permittivity of vacuum and ${\mu _0}$ is the permeability of vacuum) ?
$\sqrt {\frac{{{ \varepsilon _0}}}{{{\mu _0}}}} $
${\frac{{{\mu _0}}}{{{ \varepsilon_0}}}}$
$\frac{{{ \varepsilon_0}}}{{{\mu _0}}}$
$\sqrt {\frac{{{\mu _0}}}{{{\varepsilon_0}}}} $
The speed of light $(c)$, gravitational constant $(G)$ and planck's constant $(h)$ are taken as fundamental units in a system. The dimensions of time in this new system should be
An artificial satellite is revolving around a planet of mass $M$ and radius $R$ in a circular orbit of radius $r$. From Kepler’s third law about the period of a satellite around a common central body, square of the period of revolution $T$ is proportional to the cube of the radius of the orbit $r$. Show using dimensional analysis that $T\, = \,\frac{k}{R}\sqrt {\frac{{{r^3}}}{g}} $, where $k$ is dimensionless constant and $g$ is acceleration due to gravity.
Dimensional formula for thermal conductivity is (here $K$ denotes the temperature)
If energy $(E),$ velocity $(V)$ and time $(T)$ are chosen as the fundamental quantities, the dimensional formula of surface tension will be
The dimensions of the product $\mu_{0} \varepsilon_{0}$ are related to those of velocity as