91Ó°ÊÓ

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

$f=\frac{1}{T}$or$\frac{\mathrm{Ó\neg }}{2\mathrm{Ï€}}$

Where,T =Time period in seconds and$\mathrm{Ó\neg }$ is angular frequency in rad/s.

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

The wavelength-frequency relationship is:

$c=\mathrm{λ}f$

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 wavelength is inversely proportional to the frequency.

$\mathrm{λ}=\frac{c}{f}$or$\frac{v}{f}$

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

The relation between wave number and wavelength is:

$k=\frac{2\mathrm{π}}{\mathrm{λ}}$

Where, k is wavenumber in rad/m.

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

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

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

$$\begin{gathered} \stackrel{→}{E}=E_{max}\cos \left(kx-\mathrm{Ó\neg }t\right) \\ \stackrel{→}{B}=B_{max}\cos \left(kx-\mathrm{Ó\neg }t\right) \end{gathered}$$

Given that,$\stackrel{→}{E}(y,t)=(3.10×{10}^{5}V/m)\stackrel{Áåœ}{k}ncos3[ky-(12.65×{10}^{12}rad/s\left)t\right]$

The values of angular frequency$\left(\mathrm{Ó\neg }\right)$ is$12.65×{10}^{12}rad/s$ and the maximum electric field$E_{max}$ is $3.10×{10}^{5}V/m$.

Since the cross products of electric and magnetic fields give the $\stackrel{Áåœ}{k}$. Therefore, the wave travels in the +y- direction.

The frequency of wave is:

$$\begin{gathered} \mathrm{f}=\frac{\mathrm{Ó\neg }}{2\mathrm{Ï€}} \\ =\frac{12.65×{10}^{12}}{2\mathrm{Ï€}} \\ =2.014×{10}^{12}\mathrm{Hz} \end{gathered}$$

The wavelength of wave is:

$$\begin{gathered} \mathrm{λ}=\frac{\mathrm{c}}{\mathrm{f}} \\ =\frac{3×{10}^8}{2.014×{10}^{12}} \\ =1.49×{10}^{-4}\mathrm{m} \end{gathered}$$

Hence, wave is travelling in +y- direction and the wavelength of the wave is $1.49×{10}^{-4}\mathrm{m}$.

The wavenumber of wave is:

$$\begin{gathered} k=\frac{2\mathrm{Ï€´Ú}}{c} \\ =\frac{2\mathrm{Ï€}\left(2.014×{10}^{12}\right)}{3×{10}^8} \\ =4.21×{10}^{4}rad/m \end{gathered}$$

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

$B_{max}=\frac{E_{max}}{c}$

Substitute the values of variables

$$\begin{gathered} B_{max}=\frac{3.10×{10}^5}{3×{10}^8} \\ =1.03×{10}^{-3}\mathrm{T} \end{gathered}$$

The equation of sinusoidal electromagnetic wave for magnetic field is:

$$\begin{gathered} \stackrel{→}{B}=B_{max}\cos \left(kx-\mathrm{Ó\neg }t\right) \\ =\left(1.03×{10}^{-3}T\right)\cos \left[\left(4.21×{10}^{4}rad/m\right)y-\left(12.65×{10}^{12}rad/s\right)t\right] \end{gathered}$$

Hence, the vector equation for$\stackrel{→}{B}\left(y,t\right)$ is $\stackrel{→}{\mathrm{B}}=\left(1.03×{10}^{-3}\mathrm{T}\right)\cos \left[\left(4.21×{10}^4\mathrm{rad}/\mathrm{m}\right)\mathrm{y}-\left(12.65×{10}^{12}\mathrm{rad}/\mathrm{s}\right)\mathrm{t}\right]$.