Question

Verify that the correct value for the speed of light \(c\)is obtained when numerical values for the permeability and permittivity of free space (\({\mu _0}\)and \({\varepsilon _0}\)) are entered into the equation \(c = \frac{1}{{\sqrt {{\mu _0}{\varepsilon _0}} }}\).

Short answer

The value obtained after equating, for the speed and light is 3\times 10^{8}m/s.

The given statement \(c = \frac{1}{{\sqrt {{\varepsilon _0}{\mu _0}} }}\)is verified.

Step-by-step solution

Definition of electromagnetic wave

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.

Find the value for the speed and light

We know that in a vacuum, the speed of light is given by: \(c=3\times 10^{8}m/s\)

The relationship between the speed of light \((c)\), the permittivity of free space \(\left( {{\varepsilon _0}} \right)\), and the permeability of free space is:

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

Here \({\varepsilon _0} = 4\pi \times {10^{ - 7}}T.m/A,{\rm{ }}{\mu _0} = 8.85 \times {10^{ - 12}}{C^2}/N.{m^2}\)

Fill in the values for \({\varepsilon _0}\) and \({\mu _0}\) in the equation above and solve it for \(c\):

\(\begin{aligned} c &= \frac{1}{{\sqrt {\left( {4\pi \times {{10}^{ - 7}}\;T \times m/A} \right)\left( {8.85 \times {{10}^{ - 12}}{C^2}/N \times {m^2}} \right)} }}\\c &= 3 \times {10^8}\;m{s^{ - 1}}\end{aligned}\)

So, that we can see \(c = \frac{1}{{\sqrt {{\varepsilon _0}{\mu _0}} }}\) verified.

Therefore, the value for the speed and light \(c=3 \times {10^8}\;m{s^{ - 1}}\)

The given statement is verified\(c = \frac{1}{{\sqrt {{\varepsilon _0}{\mu _0}} }}\)is verified.