Question

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?

Short answer

a) The frequency of the wave is.$1.91\times {10}^8\text{\hspace{0.17em}Hz}$

b) The rms value of the electric component of the wave is.$18.2\text{V/m}$

c) The intensity of the light is.$879{\text{\hspace{0.17em}mW/m}}^{\text{2}}$

Step-by-step solution

The given data

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

Understanding the concept of the radiated wave 

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)

a) Calculation of the frequency of the wave

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.$1.91\times {10}^8\text{\hspace{0.17em}Hz}$

b) Calculation of the rms value of the electric field component  

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.$18.20\text{\hspace{0.17em}V/m}$

c) Calculation of intensity of the wave

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.$879{\text{\hspace{0.17em}mW/m}}^{\text{2}}$