Question

# The dimensions of ${{\left( {{\mu }_{0}}{{\varepsilon }_{0}} \right)}^{{-1}/{2}\;}}$are:(A) $\left[ {{L}^{-1}}T \right]$(B) $\left[ L{{T}^{-1}} \right]$(C) $\left[ {{L}^{{1}/{2}\;}}{{T}^{{1}/{2}\;}} \right]$(D) $\left[ L{{T}^{{1}/{2}\;}} \right]$

Hint: In the given equation, quantities ${{\mu }_{0}}$and ${{\varepsilon }_{0}}$are magnetic constant or the permeability of free space and electric constant or permittivity of free space. The direct dimensions of these quantities are not known but they were obtained by Maxwell from the electromagnetic laws. It is related to the velocity of the light in vacuum.

The oscillation of electric field and magnetic field in vacuum creates electromagnetic waves. The light is electromagnetic waves in free space. The velocity of the light is denoted by $c$.The velocity of light $c$ is constant therefore $c$ depends on electric and magnetic constant. The electric constant is known by the permeability of free space (${{\mu }_{0}}$) and magnetic constant is known by permittivity of free space (${{\varepsilon }_{0}}$).

$c=\dfrac{1}{\sqrt{\left( {{\mu }_{0}}{{\varepsilon }_{0}} \right)}}$
$\therefore {{({{\mu }_{0}}{{\varepsilon }_{0}})}^{{-1}/{2}\;}}=c$

The unit of the $c$ is ${m}/{s}\;$. So the unit of given quantity is ${m}/{s}\;$.
The dimensional equivalent of unit ${m}/{s}\;$ is $\left[ L{{T}^{-1}} \right]$

So, the correct answer is “Option B”.

Permeability of free space (${{\mu }_{0}}$) = $4\pi \times {{10}^{-7}}N{{A}^{-2}}$
Permittivity of free space (${{\varepsilon }_{0}}$) = $8.85419\times {{10}^{-12}}{{C}^{2}}{{N}^{-1}}{{m}^{-2}}$
Velocity of light ($c$) = $2.99792\times {{10}^{8}}m{{s}^{-1}}$
The dimensional relationship does not mean that the relation between both side quantities are linearly the same, it only compares the dimensions. Here the relation between the speed of light and $({{\mu }_{0}}{{\varepsilon }_{0}})$ is not only inversely proportional but the square root of $({{\mu }_{0}}{{\varepsilon }_{0}})$ inversely proportional to speed of light but the dimensional does not predict it.