Question

A sinusoidal electromagnetic wave emitted by a cellular phone has a wavelength of 35.4 cm and an electric-field amplitude of$\mathbf{5}\mathbf{.}\mathbf{40}\mathbf{\times }{\mathbf{10}}^{\mathbf{-}\mathbf{2}}\mathbf{}\mathbf{V}\mathbf{/}\mathbf{m}$ at adistance of 250 m from the phone. Calculate (a) the frequency of the wave; (b) the magnetic-field amplitude; (c) the intensity of the wave.

Short answer

a. The frequency of wave is $8.74\times {10}^8\mathrm{Hz}$.

b. The amplitude of magnetic-field is $1.8\times {10}^{-10}\mathrm{T}$.

c. The intensity of wave is $3.87\times {10}^{-6}\mathrm{W}/{\mathrm{m}}^2$.

Step-by-step solution

Define the intensity ( I ) and the formulas.

The power transported per unit area is known as the intensity ( I ).

The formula used to calculate the intensity( I ) is:

$I=\frac{P}{A}$

Where,$A$ is area measured in the direction perpendicular to the energy and$P$ is the power in watts.

The relation between the frequency of wave $\left(f\right)$, wavelength$\left(\lambda \right)$ and speed of light$\left(c\right)$ is:

$c=f\lambda$

And also, $I=\frac{1}{2}{\epsilon }_{0}c{E}_{max}^2$ . Where, I is the intensity in $\mathrm{W}/{\mathrm{m}}^2$and is the speed of light that is equal to $3.0\times {10}^8\mathrm{m}/\mathrm{s}$and ${\epsilon }_0=8.85\times {10}^{-12}{\mathrm{C}}^2/\mathrm{N}·{\mathrm{m}}^2$.

The relation between the maximum electric field and maximum magnetic field is:

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

Determine frequency of wave.

Given that,$\lambda =35.4\times {10}^{-2}\mathrm{m}$

The frequency of wave is:

$$\begin{gathered} f=\frac{c}{\lambda } \\ =\frac{3\times {10}^8}{35.4\times {10}^{-2}} \\ =8.47\times {10}^8\mathrm{Hz} \end{gathered}$$

Hence, the frequency of wave is $8.47\times {10}^8\mathrm{Hz}$.

Determine the amplitude of magnetic field.

Given that,$E_{max}=5.4\times {10}^{-2}V/m$

The amplitude of magnetic field is:

$$\begin{gathered} B_{max}=\frac{E_{max}}{c} \\ =\frac{5.4\times {10}^{-2}}{3\times {10}^8} \\ =1.8\times {10}^{-10}\mathrm{T} \end{gathered}$$

Hence, the amplitude of magnetic-field is $1.8\times {10}^{-10}\mathrm{T}$.

Determine the intensity of wave.

The intensity of wave:

$$\begin{gathered} I=\frac{1}{2}{\epsilon }_{0}c{E}_{max}^2 \\ =\frac{1}{2}\times \left(8.854\times {10}^{-12}\right)\left(3\times {10}^8\right)\left(5.4\times {10}^{-2}\right) \\ =3.87\times {10}^{-6}\mathrm{W}/{\mathrm{m}}^2 \end{gathered}$$

Hence, intensity of wave is $3.87\times {10}^{-6}\mathrm{W}/{\mathrm{m}}^2$.