Question

Consider the most recent generation of residential satellite dishes that are a little less than half a meter in diameter. Construct a problem in which you calculate the power received by the dish and the maximum electric field strength of the microwave signals for a single channel received by the dish. Among the things to be considered are the power broadcast by the satellite and the area over which the power is spread, as well as the area of the receiving dish.

Short answer

The power received by the dish is$1.91\times {10}^{-5}\mathrm{W}$ and the maximum electric field strength of the microwave signals is$0.15\mathrm{V}/\mathrm{m}$.

Step-by-step solution

Definition of Concept

Microwave signals: A microwave is an electromagnetic wave with a wavelength of less than one meter, according to the definition. The fundamental distinction between microwave signals and radio broadcasting signals is that radio waves are longer than a meter. They have a higher frequency than radio signals, indicating that they are higher in frequency.

Find the power received by the dish and the maximum electric field strength of the microwave signals

Considering the given information,

Power broadcast by the satellite is${\mathrm{P}}_5=3\mathrm{kW}\left(\frac{1000\mathrm{W}}{1\mathrm{kW}}\right)=3\times {10}^3\mathrm{W}$.

The area over which the power spread is ${\mathrm{A}}_{\mathrm{s}}=100{\mathrm{km}}^2$.

The diameter of the dish is $\mathrm{d}=0.45{\mathrm{m}}^2$.

Apply the formula,

The intensity formula is as follows:

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

The signal intensity when it reaches the dish can be calculated as follows:

$$\begin{gathered} \mathrm{I}=\frac{{\mathrm{P}}_{\mathrm{s}}}{{\mathrm{A}}_{\mathrm{s}}} \\ =\frac{3\times {10}^3\mathrm{W}}{100\times {10}^6{\mathrm{m}}^2} \\ =3\times {10}^{-5}\mathrm{W}/{\mathrm{m}}^2 \end{gathered}$$

$$\begin{gathered} \mathrm{I}=\frac{{\mathrm{P}}_{\mathrm{s}}}{{\mathrm{A}}_{\mathrm{s}}} \\ =\frac{3\times {10}^3\mathrm{W}}{100\times {10}^6{\mathrm{m}}^2} \\ =3\times {10}^{-5}\mathrm{W}/{\mathrm{m}}^2 \end{gathered}$$

The dish's power can be computed using the formula:

$$\begin{gathered} {\mathrm{P}}_{\mathrm{dis}\mathrm{h}}=\mathrm{I}\times {\mathrm{A}}_{\mathrm{dish}} \\ =\mathrm{I}\times 4\mathrm{\pi }{\left(\frac{\mathrm{d}}{2}\right)}^{\frac{1}{2}} \\ =\left(3\times {10}^{-5}\mathrm{W}/{\mathrm{m}}^2\right)\times 4\times 3.14\times {\left(\frac{0.45}{2}\right)}^2 \\ =1.91\times {10}^{-5}\mathrm{W} \end{gathered}$$

The maximal electric field strength can be computed using the following formula:

$$\begin{gathered} {\mathrm{E}}_0={\left(\frac{2\mathrm{I}}{{\mathrm{C\epsilon }}_0}\right)}^{\frac{1}{2}} \\ ={\left(\frac{2\times \left(3\times {10}^{-5}\mathrm{W}/{\mathrm{m}}^2\right)}{\left(3\times {10}^8\mathrm{m}/\mathrm{s}\right)\times \left(8.85\times {10}^{-12}{\mathrm{C}}^2/\mathrm{N}.{\mathrm{m}}^2\right)}\right)}^{\frac{1}{2}} \\ =0.15\mathrm{V}/\mathrm{m} \end{gathered}$$

Therefore, the required power received by the dish is$1.91\times {10}^{-5}\mathrm{W}$ and the maximum electric field strength of the microwave signals is$0.15\mathrm{V}/\mathrm{m}$.