Electric and magnetic fields in many materials can be analyzed using the same relationships as for fields in vacuum, only substituting relative values of the permittivity and the permeability, \(\epsilon=\kappa \epsilon_{0}\) and \(\mu=\kappa_{\mathrm{m}} \mu_{0},\) for their vacuum values, where \(\kappa\) is the dielectric constant and \(\kappa_{\mathrm{m}}\) the relative permeability of the material. Calculate the ratio of the speed of electromagnetic waves in vacuum to their speed in such a material.

In vacuum, we have \(\epsilon = \epsilon_{0}\) and \(\mu = \mu_{0}\). Thus, the speed of electromagnetic waves \(v_{0}\) in vacuum can be written as: \(v_{0} = \frac{1}{\sqrt{\epsilon_{0}\mu_{0}}}\) In a material with permittivity \(\epsilon = \kappa\epsilon_{0}\) and permeability \(\mu = \kappa_{m}\mu_{0}\), the speed of electromagnetic waves \(v_{m}\) can be written as: \(v_{m} = \frac{1}{\sqrt{(\kappa \epsilon_{0})(\kappa_{m} \mu_{0})}}\)

We want to find the ratio \(\frac{v_{0}}{v_{m}}\). Using the expressions for \(v_{0}\) and \(v_{m}\), we can write the ratio as: \(\frac{v_{0}}{v_{m}} = \frac{\frac{1}{\sqrt{\epsilon_{0}\mu_{0}}}}{\frac{1}{\sqrt{(\kappa \epsilon_{0})(\kappa_{m} \mu_{0})}}}\)

Now, we will simplify the expression for the ratio \(\frac{v_{0}}{v_{m}}\): \(\frac{v_{0}}{v_{m}} = \frac{\sqrt{(\kappa \epsilon_{0})(\kappa_{m} \mu_{0})}}{\sqrt{\epsilon_{0}\mu_{0}}}\) We can further simplify this by taking the square root of the product of the dielectric constant and the relative permeability out of the square root, as follows: \(\frac{v_{0}}{v_{m}} = \frac{\sqrt{\kappa \kappa_{m}} \sqrt{\epsilon_{0}\mu_{0}}}{\sqrt{\epsilon_{0}\mu_{0}}}\) Since there is a common factor of \(\sqrt{\epsilon_{0}\mu_{0}}\) in both numerator and denominator, we can cancel it out: \(\frac{v_{0}}{v_{m}} = \sqrt{\kappa\kappa_{m}}\) Thus, the ratio of the speed of electromagnetic waves in vacuum to their speed in a material is given by the square root of the product of the dielectric constant and the relative permeability of the material.