The magnetic component of an electromagnetic wave in vacuum has an amplitude of $\mathbf{85}\mathbf{.8}\mathbf{\text{\hspace{0.17em}nT}}$and an angular wave number of $\mathbf{4}\mathbf{.}\mathbf{00}\mathbf{}{\mathit{m}}^{\mathbf{-}\mathbf{1}}\mathbf{.}$What are (a) the frequency of the wave, (b) the rms value of the electric component, and(c) the intensity of the light?

The magnetic component of the wave$\text{B}=85.8\text{nT}\left(\frac{{10}^{-9}\text{\hspace{0.17em}T}}{1\text{\hspace{0.17em}nT}}\right)=85.8\times {10}^{-9}\text{T}$

The angular wave number,$\text{k}=4.00{\text{\hspace{0.17em}m}}^{-1}$

An electromagnetic wave is composed of an electric field as well as a magnetic field. It is described in terms of angular frequency and angular wave number.

Formulae:

The frequency of a wave,$f=\frac{\omega }{2\pi }$ (1)

The wavelength of the wave, $\lambda =\frac{2\pi }{k}$ (2)

The speed of an electromagnetic wave, $c=f\cdot \lambda =\frac{\omega }{k}$ (3)

The RMS value of the electric field, $E_{rms}=\frac{E_m}{\sqrt{2}}$ (4)

The RMS value of the magnetic field, $B_m=\frac{E_m}{c}$ (5)

The intensity of a radiated wave due to reflection, $I=\frac{1}{c{\mu }_o}{E}_{rms}^2$ (6)

The speed of the electromagnetic wave is c. Thus, using equation (3) and the given data, we can get the angular frequency of the wave as follows:

$\begin{array}{c}\omega =k\times c\\ =4.00m^{-1}\times 3\times {10}^{8}m/s\\ =12.0\times {10}^8\text{\hspace{0.17em}Hz}\end{array}$

Now, using this value in equation (1), we can get the frequency of the wave as follows:

$\begin{array}{c}\text{f}=\frac{12.0\times {10}^{8}Hz}{2\pi }\\ =1.91\times {10}^8\text{\hspace{0.17em}Hz}\end{array}$

Hence, the value of the frequency is.$1.91\times {10}^8\text{\hspace{0.17em}Hz}$

The rms value of the electric field component is calculated by substituting the value of the maximum electric field from equation (5) in equation (4) as follows:

$\begin{array}{c}{E}_{rms}=\frac{c\times B_m}{\sqrt{2}}\\ =\frac{3\times {10}^{8}m/s\times 85.8\times {10}^{-9}T}{\sqrt{2}}\\ =18.20\text{\hspace{0.17em}V/m}\end{array}$

Hence, the value of the electric field is.$18.20\text{\hspace{0.17em}V/m}$

The intensity of the wave is determined using the above value in equation (6) as follows:

$\begin{array}{c}I=\frac{{(18.20V/m)}^2}{3\times {10}^{8}m/s\times 4\pi \times {10}^{-7}H/m}\\ =879{\text{\hspace{0.17em}mW/m}}^{\text{2}}\end{array}$

Hence, the value of the intensity is.$879{\text{\hspace{0.17em}mW/m}}^{\text{2}}$