Show that, when SI units for \({\mu _0}\) and \({\varepsilon _0}\) are entered, the units given by the right-hand side of the equation in the problem above are \(m/s\).

Examples of waves transmitted by concurrent periodic changes in the strength of the electric and magnetic fields include radio waves, infrared, visible light, ultraviolet, X-rays, and gamma rays.

The speed of light permittivity and the permeability of open space are known to be connected by a formula.

\(c = \frac{1}{{\sqrt {{\varepsilon _0}{\mu _0}} }}\)

Where\({\varepsilon _0} = \)permeability of free space,\({\mu _0} = \)the permittivity of free space, and\(c = \)speed of light

Here SI unit of\({\varepsilon _0} = \frac{{{C^2}}}{{\;N \times {m^2}}}\)and SI unit\({\mu _0} = \frac{{T \times m}}{A}\).

In the above equation, replace\({\varepsilon _0}\) and\({\mu _0}\).

\(\begin{aligned} \frac{1}{{\sqrt {{\mu _0}{\varepsilon _0}} }}{\rm{ }} &= \frac{1}{{\sqrt {\left( {\frac{{{C^2}}}{{\;N \times {m^2}}}} \right)\left( {\frac{{T \times m}}{A}} \right)} }}\\ &= \frac{1}{{\sqrt {\left( {\frac{{{C^2}}}{{\;N \times m}}} \right)\left( {\frac{T}{A}} \right)} }}\\ &= \sqrt {\frac{A}{{T \times F}}} \end{aligned}\)

Now substitute,\(F = \frac{{{C^2}}}{J},T = \frac{{N \times s}}{{C \times m}}\)and\(A = \frac{C}{s}\)in the above equation –

\(\begin{aligned} \frac{1}{{\sqrt {{\mu _0}{\varepsilon _0}} }} &= \sqrt {\frac{{\frac{C}{{\;s}}}}{{\left( {\frac{{N \times s}}{{C \times m}}} \right)\left( {\frac{{{C^2}}}{{\;J}}} \right)}}} \\ &= \sqrt {\frac{{J \times m}}{{N \times {s^2}}}} \\ &= \sqrt {\frac{{(N \times m)m}}{{N \times {s^2}}}} \\ &= \sqrt {\frac{{{m^2}}}{{{s^2}}}} \\ &= \frac{m}{s}\end{aligned}\)

Therefore, the unit of\(\frac{1}{{\sqrt {{\mu _0}{\varepsilon _0}} }}\)is same as the unit of speed.