A dimensionless quantity is constructed in terms of electronic charge $e$, permittivity of free space $\varepsilon_0$, Planck's constant $h$, and speed of light $c$. If the dimensionless quantity is written as $e^\alpha \varepsilon_0^\beta h^7 c^5$ and $n$ is a non-zero integer, then $(\alpha, \beta, \gamma, \delta)$ is given by

Time period $T\,\propto \,{P^a}\,{d^b}\,{E^c}$  then value of $c$ is  given $p$ is pressure, $d$ is density and $E$ is energy

Einstein’s mass-energy relation emerging out of his famous theory of relativity relates mass $(m)$ to energy $(E)$ as  $E = mc^2$, where $c$ is speed of light in vacuum. At the nuclear level, the magnitudes of energy are very small. The energy at nuclear level is usually measured in $MeV$, where $1\,MeV = 1.6\times 10^{-13}\,J$ ; the masses are measured i unified atomicm mass unit (u) where, $1\,u = 1.67 \times 10^{-27}\, kg$

$(a)$ Show that the energy equivalent of $1\,u$ is $ 931.5\, MeV$.

$(b)$ A student writes the relation as $1\,u = 931.5\, MeV$. The teacher points out that the relation  is dimensionally incorrect. Write the correct relation.

The physical quantity which has dimensional formula as that of $\frac{{{\rm{Energy}}}}{{{\rm{Mass}} \times {\rm{Length}}}}$ is

The dimension of $\frac{1}{{\sqrt {{\varepsilon _0}{\mu _0}} }}$ is that of

Dimensional formula for thermal conductivity is (here $K$ denotes the temperature)