$\text{Consider the Friis’ transmission equation}{\mathrm{ P}}_R=\frac{P_T{G}_T{G}_R{\lambda }^2}{{\left(4\pi D\right)}^2},\text{where}{\mathrm{ P}}_R\text{and}{\mathrm{ P}}_T\text{ are the received and the transmitted}$

$\text{powers, respectively.}{\mathrm{ G}}_T\text{and}{\mathrm{ G}}_R\text{ are the gain of transmitting and receiving antennas, respectively,}\mathrm{ D}\text{ is the distance}$

$\text{between the transmitting and receiving antennas, and}\lambda \text{is the wavelength in free space.}$

$\text{Given:}{\mathrm{ G}}_T=G_R=1.0,\lambda =0.30\text{m and}{P}_T=+10\text{dBm.}$

$\text{Choose the distance}\left(D\right)\text{, in km, from the following options at which the received power,}{P}_R=-90\text{dBm?}$

The correct option is 15/ 2π.

Step-by-step Explanation:

We are given Friis’ transmission equation:
$$P_R=\frac{P_T{G}_T{G}_R{\lambda }^2}{{\left(4\pi D\right)}^2}$$
where:
• $G_T=G_R=1.0$ (antenna gains)
• $\lambda =0.30\text{m}$ (wavelength in free space)
• $P_T=+10\text{dBm}$ (transmitted power)
• $P_R=-90\text{dBm}$ (received power)
• $D$ is the distance between the antennas in meters.

Step 1: Determine the ratio of received power to transmitted power in linear scale
First, let's find the power ratio in decibels (dB):
$$\frac{P_R}{P_T}\text{(in dB)}=P_R\text{(in dBm)}-P_T\text{(in dBm)}$$
Substituting the given values:
$$\frac{P_R}{P_T}\text{(in dB)}=-90\text{dBm}-10\text{dBm}=-100\text{dB}$$
Now, we convert this ratio from dB to the linear scale:
$$\frac{P_R}{P_T}={10}^{\frac{-100}{10}}={10}^{-10}$$

Step 2: Solve for the distance D in meters
Using Friis' equation, we can write:
$$\frac{P_R}{P_T}=\frac{G_T{G}_R{\lambda }^2}{{\left(4\pi D\right)}^2}$$
Substitute the known values into the equation:
$${10}^{-10}=\frac{1.0\times 1.0\times {\left(0.30\right)}^2}{{\left(4\pi D\right)}^2}$$
Taking the square root on both sides:
$${10}^{-5}=\frac{0.30}{4\pi D}$$
Rearranging to solve for D:
$$D=\frac{0.30}{4\pi \times {10}^{-5}}$$
$$D=\frac{0.30\times {10}^5}{4\pi }=\frac{30000}{4\pi }=\frac{7500}{\pi }\text{meters}$$

Step 3: Convert the distance D to kilometers
Since $1\text{km}=1000\text{m}$:
$$D=\frac{7500}{1000\pi }\text{km}=\frac{7.5}{\pi }\text{km}=\frac{15}{2\pi }\text{km}$$
Thus, the distance $D$ is indeed 15/ 2π km.