An Electric Dipole Antenna Is Located On The Roof Of A 213 M
An Electric Dipole Antenna Is Located On The Roof Of A 213 M High Buil
An electric-dipole antenna is located on the roof of a 213-m-high building and is oriented vertically. An observer on the ground measures the intensity of the radiation from the antenna and finds 8.49 mW/m². This observer is 1.03 km away from the antenna. A second observer hovers in a blimp at an altitude of 561 m and measures the radiation intensity from the antenna when she is 1.37 km from the antenna. What intensity, in milliwatts per square meter, does she measure?
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The problem involves determining the radiation intensity received by a second observer positioned at a different altitude and distance from a vertically oriented electric dipole antenna located atop a building. The key goal is to analyze how the intensity of electromagnetic radiation from a dipole antenna varies with observer position, considering factors such as distance, antenna height, and orientation.
To approach this problem, it’s essential to understand the inverse square law of electromagnetic radiation and the specific characteristics of a dipole antenna’s radiation pattern. The inverse square law states that the intensity of radiation decreases proportionally to the square of the distance from the source in free space. However, since the antenna is located on top of a building and the second observer is at an altitude in a blimp, these factors influence the calculations.
First, the initial data reveal that the ground observer measures a radiation intensity of 8.49 mW/m² at a distance of 1.03 km from the antenna. The antenna’s height is 213 meters, which influences the propagation of the electromagnetic wave and the effective distance the wave travels from the antenna to the observer.
Second, the second observer in the blimp is at an altitude of 561 meters and a horizontal distance of 1.37 km. To compute the total distance from the antenna in this case, we use the three-dimensional distance measurement considering both horizontal distance and altitude difference.
The distance \( r_2 \) from the antenna to the second observer can be computed using the Pythagorean theorem:
\[
r_2 = \sqrt{(d_x)^2 + (d_y)^2}
\]
where \( d_x \) is the horizontal distance and \( d_y \) is the difference in altitude.
The horizontal distance \( d_x \) is 1.37 km, which is 1370 meters, and the altitude difference \( d_y \) is \( 561 - 213 \) meters, equaling 348 meters. Therefore, the total distance from the antenna to the second observer is:
\[
r_2 = \sqrt{(1370)^2 + (348)^2} \approx \sqrt{1,876,900 + 121,104} \approx \sqrt{2,098,004} \approx 1449.83\, \text{meters}
\]
Next, knowing that intensity \( I \) for radiating sources follows an inverse square relation:
\[
I \propto \frac{1}{r^2}
\]
we can set up a ratio between the known intensity \( I_1 = 8.49\, \text{mW/m}^2 \) at distance \( r_1 = 1030\, \text{meters} \) and the unknown intensity \( I_2 \) at distance \( r_2 = 1449.83\, \text{meters} \):
\[
\frac{I_2}{I_1} = \left( \frac{r_1}{r_2} \right)^2
\]
Solving for \( I_2 \):
\[
I_2 = I_1 \times \left( \frac{r_1}{r_2} \right)^2
\]
Plugging in the values:
\[
I_2 = 8.49\, \text{mW/m}^2 \times \left( \frac{1030}{1449.83} \right)^2
\]
\[
I_2 = 8.49\, \text{mW/m}^2 \times \left( 0.710 \right)^2
\]
\[
I_2 = 8.49 \times 0.504 \approx 4.28\, \text{mW/m}^2
\]
Thus, the second observer, in the blimp, measures approximately 4.28 milliwatts per square meter.
This calculation assumes free-space propagation and no additional environmental effects like reflection, absorption, or interference, which could otherwise influence the actual measurement. Understanding these principles is critical for applications such as radar, wireless communication, and antenna design, where signal strength and coverage are fundamental concerns. This analysis illustrates the application of inverse square law physics to real-world electromagnetic signals emanating from a dipole antenna positioned atop a tall building.
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