Question

An airplane flying at a distance of$\mathbf{10}\mathbf{}\mathbf{km}$from a radio transmitter receives a signal of intensity$\mathbf{10}\mathbf{}\mathbf{mW}\mathbf{/}{\mathbf{m}}^{\mathbf{2}}$.(a) What is the amplitude of the electric component of the signal at the airplane and (b) What is the amplitude of the magnetic component of the signal at the airplane? (c) If the transmitter radiates uniformly over a hemisphere, what is the transmission power?

Short answer

(a) The amplitude of the electric field component at the airplane is $8.7\times {10}^{-2}\frac{\mathrm{V}}{\mathrm{m}}$.

(b) The amplitude of the magnetic field component at the airplane is $2.9\times {10}^{-10}\mathrm{T}$.

(c) The transmission power is $6.3\times {10}^3\text{W}$.

Step-by-step solution

Determine the given quantities

Intensity of sunlight $I=10\frac{\mathrm{\mu W}}{{\mathrm{m}}^2}$

Distance$r=10\mathrm{km}$

Determine the concepts of electric and magnetic component

Consider the intensity relation of wave, square both the sides, and solve for electric field component. Finally, substitutingtheelectric field component value inthemagnetic field component relation, find its amplitude. Using intensity and power relation, and find the transmission power.

Formulae:

$c=\frac{E_m}{B_m}$

$I=\frac{E_m^2}{2{\mu }_{0}c}$

$P=2\pi r^{2}I$

(a) Determine the amplitude of electric field component  

The time averaged rate per unit area at which energy is transported, $S_{avg},$is called the intensity $I$ of the wave.

$$\begin{gathered} I=\frac{E_m^2}{{24}_{n}c} \\ {E}_{rms}=\frac{E_m}{\sqrt{2}} \\ I=\frac{E_m^2}{2{\mu }_{0}c} \end{gathered}$$

Substitute the values in the equation and solve as:

role="math" localid="1662974648726" $\begin{array}{rcl}{E}_m& =& \sqrt{2{\mu }_{0}Ic}\\ & =& \sqrt{2\left(4\pi \times {10}^{-7}\frac{\mathrm{T}·\mathrm{m}}{\mathrm{A}}\right)\left(10\times {10}^{-6}\frac{\mathrm{W}}{{\mathrm{m}}^2}\right)\left(3\times {10}^8\frac{\mathrm{m}}{\mathrm{s}}\right)}\\ & =& 8.7\times {10}^{-2}\frac{\text{V}}{\mathrm{m}}\end{array}$

The amplitude of the electric field component at the airplane is$8.7\times {10}^{-2}\frac{\mathrm{V}}{\mathrm{m}}$

(b) Determine the amplitude of magnetic field component

The wave speed cand amplitudes of electric and magnetic fields are related as

$$\begin{gathered} c=\frac{E}{B} \\ c=\frac{E_m}{B_m} \end{gathered}$$

The amplitude of magnetic field component in the wave is given by

$\begin{array}{rcl}{B}_m& =& \frac{E_m}{c}\\ & =& \frac{8.7\times {10}^{-2}\frac{\mathrm{V}}{\mathrm{m}}}{3\times {10}^8\frac{\mathrm{m}}{\mathrm{s}}}\\ & =& 2.9\times {10}^{-10}\mathrm{T}\end{array}$

The amplitude of the magnetic field component at the airplane is$2.9\times {10}^{-10}\text{T}$

(c) Determine the transmission power P

The intensity I and power P are related as:

$$\begin{gathered} I=\frac{P}{4\pi r^2} \\ P=4\pi r^{2}I \end{gathered}$$

The intensity oftheemerging polarized light is half of the original intensity of light.

role="math" localid="1662975216508" $\begin{array}{rcl}P& =& \frac{4\pi r^{2}I}{2}\\ & =& 2\pi r^2\\ & =& 2\pi {\left(10\times {10}^3\mathrm{m}\right)}^2\left(10\times {10}^{-6}\mathrm{W}/{\mathrm{m}}^2\right)\\ & =& 6.3\times {10}^3\mathrm{W}\end{array}$

The transmission power is $6.3\times {10}^3\text{W}$.