Question

An electromagnetic wave with frequency$\mathbf{5}\mathbf{.}\mathbf{70}\mathbf{\times }{\mathbf{10}}^{\mathbf{14}}\mathbf{}\mathbf{Hz}$ propagates with a speed of$\mathbf{2}\mathbf{.}\mathbf{17}\mathbf{\times }{\mathbf{10}}^{\mathbf{8}}\mathbf{}\mathbf{m}\mathbf{/}\mathbf{s}$ in a certain piece of glass. Find (a) the wavelength of the wave in the glass; (b) the wavelength of a wave of the same frequency propagating in air; (c) the index of refraction of the glass for an electromagnetic wave with this frequency; (d) the dielectric constant for glass at this frequency, assuming that the relative permeability is unity.

Short answer

1. The wavelength of a wave of the wave in the glass is $3.81\times {10}^{-7}\mathrm{m}$.
2. The wavelength of a wave of the wave in the air is $5.26\times {10}^{-7}\mathrm{m}$.
3. The index of refraction of the glass for an electromagnetic wave with$5.70\times {10}^{14}\mathrm{Hz}$ frequency is 1.38 .
4. The dielectric constant for glass at$5.70\times {10}^{14}\mathrm{Hz}$ frequency is 1.90 .

Step-by-step solution

Step 1: Define frequency and write formulas.

he number of waves that moves a fixed point in unit time is known as frequency.

$\mathrm{f}=\frac{1}{\mathrm{T}}\mathrm{or}\frac{\mathrm{\omega }}{2\mathrm{\pi }}$

Where,T =Time period in seconds and is angular frequency in .

The frequency, wavelength, and speed of light of any wave are related by the wavelength-frequency relationship.

The wavelength-frequency relationship is:

$c=\lambda f$

Where, c is the speed of light which is equal to localid="1664349940163" $3\times {10}^8\mathrm{m}/\mathrm{s},\mathrm{\lambda }$is the wavelength and is the frequency.

The wavelength is inversely proportional to the frequency.

$\lambda =\frac{c}{f}or\frac{v}{f}$

Where, v is velocity of wave in m/s.

The relation between wave number and wavelength is:

$k=\frac{2\pi }{\lambda }$

Where, k is wavenumber in rad/m.

Determine the wavelength of glass and air.

Given that,

$$\begin{gathered} \mathrm{v}=2.17\times {10}^8\mathrm{m}/\mathrm{s} \\ \mathrm{f}=5.70\times {10}^{14}\mathrm{Hz} \end{gathered}$$

The wavelength of glass is

$$\begin{gathered} {\mathrm{\lambda }}_{\mathrm{glass}}=\frac{\mathrm{v}}{\mathrm{f}} \\ =\frac{2.17\times {10}^8}{5.70\times {10}^{14}} \\ =3.81\times {10}^{-7}\mathrm{m} \end{gathered}$$

The wavelength of air is

$$\begin{gathered} {\mathrm{\lambda }}_{\mathrm{air}}=\frac{\mathrm{c}}{\mathrm{f}} \\ =\frac{3\times {10}^8}{5.70\times {10}^{14}} \\ =5.26\times {10}^{-7}\mathrm{m} \end{gathered}$$

Hence, wavelength of a wave of the wave in the glass is $3.81\times {10}^{-7}$mand the wavelength of a wave of the wave in the air is $5.26\times {10}^{-7}\mathrm{m}$.

Determine the refractive index and dielectric constant.

The refractive index of glass is:

$$\begin{gathered} \mathrm{n}=\frac{\mathrm{c}}{\mathrm{v}} \\ =\frac{3\times {10}^8}{2.17\times {10}^8} \\ =1.38 \end{gathered}$$

And the dielectric constant for glass is:

$$\begin{gathered} K=n^2 \\ =1.{38}^2 \\ =1.90 \end{gathered}$$

Hence, index of refraction n of the glass for an electromagnetic wave with $5.70\times {10}^{14}\mathrm{Hz}$frequency is 1.38 and the dielectric constant for glass at $5.70\times {10}^{14}\mathrm{Hz}$frequency is 1.90 .