problem 22.42
a satellite beams microwave radiation with a power of 14 kw toward the earths surface, 550 km away. when the beam strikes earth, its circular diameter is about 1400 m.
part a
find the rms electric field strength of the beam at the surface of the earth. express your answer to two significant figures and include the appropriate units.
erms = value units

Question

Accepted Answer

# Explanation:
## Step1: Calculate the intensity $I$
The intensity formula is $I=\frac{P}{A}$, where $P = 14\times10^{3}\text{ W}$ (power) and $A=\pi r^{2}$, $r=\frac{d}{2}=\frac{1400}{2}=700\text{ m}$ (radius).
$$
\begin{align*}
A&=\pi\times(700)^{2}\
&= 490000\pi\text{ m}^{2}\
I&=\frac{14\times 10^{3}}{490000\pi}\
&=\frac{14\times10^{3}}{490000\times3.14}\
&\approx\frac{14000}{1538600}\
&\approx0.0091\text{ W/m}^{2}
\end{align*}
$$
## Step2: Relate intensity to $E_{rms}$
The formula relating intensity $I$ and $E_{rms}$ is $I = \epsilon_{0}cE_{rms}^{2}$, where $\epsilon_{0}=8.85\times 10^{-12}\text{ C}^{2}/(\text{N}\cdot\text{m}^{2})$ (permittivity of free space) and $c = 3\times10^{8}\text{ m/s}$ (speed of light).
$$
\begin{align*}
E_{rms}&=\sqrt{\frac{I}{\epsilon_{0}c}}\
&=\sqrt{\frac{0.0091}{8.85\times 10^{-12}\times3\times10^{8}}}\
&=\sqrt{\frac{0.0091}{2.655\times10^{-3}}}\
&=\sqrt{3.43}\
&\approx1.9\text{ V/m}
\end{align*}
$$

# Answer:
$E_{rms} = 1.9\text{ V/m}$

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QUESTION IMAGE

Question

Explanation:

Step1: Calculate the intensity \(I\)

Step2: Relate intensity to \(E_{rms}\)

Answer: