Question

If c = the velocity of light, which of the following is correct?
\[\begin{align}
  & A.\,{{\mu }_{0}}{{\varepsilon }_{0}}=c \\
 & B.\,{{\mu }_{0}}{{\varepsilon }_{0}}={{c}^{2}} \\
 & C.\,{{\mu }_{0}}{{\varepsilon }_{0}}=\dfrac{1}{c} \\
 & D.\,{{\mu }_{0}}{{\varepsilon }_{0}}=\dfrac{1}{{{c}^{2}}} \\
\end{align}\]

Answer 1

Hint: The speed of light in vacuum can be represented in terms of the permittivity of free space or the vacuum permittivity and the permeability of the free space or the vacuum permeability as 1 by the root of the product of the vacuum permittivity and the vacuum permeability. Using this we will obtain the desired expression.
Formula used:
\[c=\dfrac{1}{\sqrt{{{\varepsilon }_{0}}{{\mu }_{0}}}}\]

Complete step-by-step solution:
From the given information, we have the data as follows.
c = the velocity of light
The speed of light in a vacuum can be represented in terms of the permittivity of free space or the vacuum permittivity and the permeability of the free space or the vacuum permeability as 1 by the root of the product of the permittivity of free space and the permeability of the free space.
The mathematical representation of the same is given as follows.
\[c=\dfrac{1}{\sqrt{{{\varepsilon }_{0}}{{\mu }_{0}}}}\]
Square on both the sides of the equation.
\[{{c}^{2}}=\dfrac{1}{{{\varepsilon }_{0}}{{\mu }_{0}}}\]
Represent the above equation in terms of the product of the permittivity of free space and the permeability of the free space.
\[{{\varepsilon }_{0}}{{\mu }_{0}}=\dfrac{1}{{{c}^{2}}}\]
Therefore, the product of the permittivity of free space and the permeability of the free space equals 1 by the square of the speed of light in a vacuum.
\[\therefore \] If c = the velocity of light, then, the correct expression is, \[{{\mu }_{0}}{{\varepsilon }_{0}}=\dfrac{1}{{{c}^{2}}}\], thus, option (D) is correct.

Note: As in this problem, we have represented the speed of light in vacuum as the product of the permittivity of free space and the permeability of the free space, we can also represent the speed of light in any material medium as the product of the relative permittivity and the relative permeability. The speed of light in medium equals 1 by the root of the product of relative permittivity and the relative permeability.