# Show that units of 1/sqrt(epsilon_o*mu_o)= ms^(-1) ?

Apr 12, 2017

There are many methods by which we can attempt to show the units of $\frac{1}{\sqrt{{\epsilon}_{0} \cdot {\mu}_{0}}}$. Two are as shown below.

${\epsilon}_{0}$ is permittivity of free space and its SI units are farad per meter $F \cdot {m}^{-} 1$.
We know that derived unit of farad in SI base units‎ are ‎s^4⋅A^2⋅m^-2⋅kg^-1
As such SI units of permittivity become s^4⋅A^2⋅m^-3⋅kg^-1 .....(1)

Now ${\mu}_{0}$ is permeability of free space and its SI units are henry per meter $H \cdot {m}^{-} 1$
Reduced to base SI units of henry is $k g \cdot {m}^{2} \cdot {s}^{-} 2 \cdot {A}^{-} 2$.
As such SI units of permeability become $k g \cdot m \cdot {s}^{-} 2 \cdot {A}^{-} 2$ .....(1)

Now we that LHS of equation has units as of
1/sqrt((s^4⋅A^2⋅m^-3⋅kg^-1)(kg cdotmcdot s^-2cdotA^-2))
=>1/sqrt(s^2⋅m^-2)
$\implies m {s}^{-} 1$
Hence proved.

.-.-.-.-.-.-

We know that by definition

$c \equiv \frac{1}{\sqrt{{\epsilon}_{0} \cdot {\mu}_{0}}}$
where $c$ is velocity of light in vacuum (free space).

Since units of velocity are $m {s}^{-} 1$,
$\therefore$ units of RHS are also $m {s}^{-} 1$
Hence proved