91Ó°ÊÓ

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 s.

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

The wavelength-frequency relationship is:

$$\begin{gathered} c=\mathrm{λ}f \\ ⇒f=\frac{c}{\mathrm{λ}} \end{gathered}$$

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

The formula used to determine the amplitude of electric and magnetic fields of the wave are:

$$\begin{gathered} E_{max}=\sqrt{\frac{2l}{{\mathrm{Î\mu }}_{0}c}} \\ {B}_{max}=\frac{E_{max}}{c} \end{gathered}$$

Where,${\mathrm{Î\mu }}_0=8.85×{10}^{-12}{\mathrm{C}}^2/\mathrm{N}â‹\dots {\mathrm{m}}^2$ and c is the speed of light that is equal to $3.0×{10}^8\mathrm{m}/\mathrm{s}$.

The equations of sinusoidal electromagnetic wave for electric and magnetic field are:

role="math" localid="1664345299622" $$\begin{gathered} \stackrel{→}{E}=E_{max}cos(kx-\mathrm{Ó\neg }t) \\ \stackrel{→}{B}=B_{max}cos(kx-\mathrm{Ó\neg }t) \end{gathered}$$

Given that,$\mathrm{λ}=432×{10}^{-9}\mathrm{m}$

The wavelength-frequency relationship is

$$\begin{gathered} c=\mathrm{λ}f \\ ⇒f=\frac{c}{\mathrm{λ}} \end{gathered}$$

Substitute the values of variables

$$\begin{gathered} f=\frac{3×{10}^8}{432×{10}^{-9}} \\ =6.94×{10}^{14}Hz \end{gathered}$$

Hence, the frequency of a sinusoidal electromagnetic wave is $6.94×{10}^{14}\mathrm{Hz}$.

Given that,$B_{max}=1.25×{10}^{-6}\mathrm{T}$

The formula used to determine the amplitude of electric and magnetic fields of the wave are:

$E_{max}=B_{max}.c$

Substitute the values of variables

$$\begin{gathered} E_{max}=\left(1.25×{10}^6\right).\left(3×{10}^8\right) \\ =375\mathrm{V}/\mathrm{m} \end{gathered}$$

Hence, the amplitude of the associated electric field is 375 V/m.

The angular frequency of wave is

$$\begin{gathered} \mathrm{Ó\neg }=2\mathrm{Ï€´Ú} \\ =2\mathrm{Ï€}\left(6.94×{10}^{14}\right) \\ =4.36×{10}^{15}\mathrm{rad}/\mathrm{s} \end{gathered}$$

And the wavenumber of wave is

$$\begin{gathered} \mathrm{k}=\frac{2\mathrm{π}}{\mathrm{λ}} \\ =\frac{2\mathrm{π}}{432×{10}^{-9}} \\ =1.45×{10}^7\mathrm{rad}/\mathrm{m} \end{gathered}$$

Substitute the variable in equations of sinusoidal electromagnetic wave for electric and magnetic field are:

role="math" localid="1664346679296" $$\begin{gathered} \stackrel{→}{E}=E_{max}\cos \left(kx-\mathrm{Ó\neg }t\right) \\ \stackrel{→}{E}=\left(375V/m\right)\cos \left(\left[1.45×{10}^{7}rad/m\right]x-\left[4.36×{10}^{15}rad/s\right]t\right) \\ \stackrel{→}{B}=B_{max}\cos \left(ks-\mathrm{Ó\neg }t\right) \\ \stackrel{→}{B}=\left(1.25×{10}^{-6}T\right)\cos \left(\left[1.45×{10}^{7}rad/m\right]x-\left[4.36×{10}^{15}rad/s\right]t\right) \end{gathered}$$

Hence, the equations for the electric and magnetic fields as function of x and t are and respectively.

$$\begin{gathered} \stackrel{→}{\mathrm{E}}=\left(375\mathrm{V}/\mathrm{m}\right)\cos \left(\left[1.45×{10}^7\mathrm{rad}/\mathrm{m}\right]\mathrm{x}-\left[4.36×{10}^{15}\mathrm{rad}/\mathrm{s}\right]\mathrm{t}\right)\mathrm{and} \\ \stackrel{→}{\mathrm{B}}=\left(1.25×{10}^{-6}\mathrm{T}\right)\cos \left(\left[1.45×{10}^7\mathrm{rad}/\mathrm{m}\right]\mathrm{x}-\left[4.36×{10}^{15}\mathrm{rad}/\mathrm{s}\right]\mathrm{t}\right)\mathrm{respectively} \end{gathered}$$