Question

Consider electromagnetic waves propagating in air. (a) Determine the frequency of a wave with a wavelength of (i) 5.0 km, (ii) 5.0 mm, (iii) 5.0 nm , (b) What is the wavelength (in meters and nanometers) of (i) gamma rays of frequency$\mathbf{6}\mathbf{.}\mathbf{50}\mathbf{\times }{\mathbf{10}}^{\mathbf{21}}\mathbf{Hz}$ and (ii) an AM station radio wave of frequency ?

Short answer

a) The frequency of a wave with a wavelength of

i. For$\mathrm{\lambda }=5.0\mathrm{Km}$ is $6\times {10}^4\mathrm{Hz}$.

ii. For$\mathrm{\lambda }=5.0\mu m$ is $6\times {10}^{13}\mathrm{Hz}$.

iii. For$\mathrm{\lambda }=5.0nm$ is $6\times {10}^{16}\mathrm{Hz}$.

b) The wavelength of

i. For gamma rays of frequency $6.50\times {10}^{21}\mathrm{Hz}$is $4.62\times {10}^{-14}\mathrm{m}$.

ii. For an AM station radio wave of frequency 590 KHz is 508 m

Step-by-step solution

Step 1: Define frequency and write formulas.

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

$f=\frac{1}{T}$

Where,T =Time period in

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

The wavelength-frequency relationship is:

$$\begin{gathered} c=\lambda f \\ \Rightarrow f=\frac{c}{\lambda } \end{gathered}$$

Where,c is the speed of light which is equal to $3\times {10}^8\mathrm{m}/\mathrm{s}$,$\lambda$ is the wavelength andf 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.

Determine the frequency of wave.

The wavelength-frequency relationship is:

$$\begin{gathered} c=\lambda f \\ \Rightarrow f=\frac{c}{\lambda } \end{gathered}$$

Forrole="math" localid="1664348819222" $\mathrm{\lambda }=5.0\mathrm{Km}$

$$\begin{gathered} \mathrm{f}=\frac{3\times {10}^8}{5\times {10}^3} \\ =6\times {10}^4\mathrm{Hz} \end{gathered}$$

For$\mathrm{\lambda }=5.0\mu m$

$$\begin{gathered} \mathrm{f}=\frac{3\times {10}^8}{5\times {10}^{-6}} \\ =6\times {10}^{13}\mathrm{Hz} \end{gathered}$$

For$\mathrm{\lambda }=5.0nm$

$$\begin{gathered} \mathrm{f}=\frac{3\times {10}^8}{5\times {10}^{-9}} \\ =6\times {10}^{16}\mathrm{Hz} \end{gathered}$$

Hence, frequency of a wave with a wavelength of

i . For$\mathrm{\lambda }=5.0\mathrm{Km}$ is $6\times {10}^4\mathrm{Hz}$.

ii . For$\mathrm{\lambda }=5.0\mu m$ is $6\times {10}^{13}\mathrm{Hz}$.

iii . For$\mathrm{\lambda }=5.0nm$ is $6\times {10}^{16}\mathrm{Hz}$.

Determine the wavelength of wave.

The wavelength is inversely proportional to the frequency.

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

For gamma rays of frequency $6.50\times {10}^{21}\mathrm{Hz}$:

$$\begin{gathered} \lambda =\frac{3\times {10}^8}{6.50\times {10}^{21}} \\ =4.62\times {10}^{-14}m \end{gathered}$$

For AM station radio wave of frequency 590 KHz:

$$\begin{gathered} \mathrm{\lambda }=\frac{3\times {10}^8}{590\times {10}^3} \\ =508\mathrm{m} \end{gathered}$$

Hence, wavelength of

i. For gamma rays of frequency $6.50\times {10}^{21}\mathrm{Hz}$is $4.62\times {10}^{-14}\mathrm{m}$.

ii. For an AM station radio wave of frequency 590 KHz is 508 m