Question

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

Answer 1

There are many methods by which we can attempt to show the units of `1/sqrt(epsilon_0 cdotmu_0)`. Two are as shown below.

`epsilon_0` is permittivity of free space and its SI units are farad per meter `Fcdotm^-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 `Hcdotm^-1`
Reduced to base SI units of henry is `kg cdotm^2cdot s^-2cdotA^-2`.
As such SI units of permeability become `kg cdotmcdot s^-2cdotA^-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)`
`=>ms^-1`
Hence proved.

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

We know that by definition

`c-=1/sqrt(epsilon_0 cdotmu_0)`
where `c` is velocity of light in vacuum (free space).

Since units of velocity are `ms^-1`,
`:.` units of RHS are also `ms^-1`
Hence proved