A radar for tracking aircraft broadcasts a 12 GHz microwave beam from a 2.0-m-diameter circular radar antenna. From a wave perspective, the antenna is a circular aperture through which the microwaves diffract. If the antenna emits 100 kW of power, what is the average microwave intensity at 30 km?

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Step 1: Understand the problem. The radar antenna emits microwaves at a frequency of 12 GHz and has a diameter of 2.0 m. The power emitted is 100 kW, and we need to calculate the average microwave intensity at a distance of 30 km. Intensity is defined as power per unit area, and the area at 30 km corresponds to the surface area of a sphere with radius 30 km.

Step 2: Calculate the wavelength of the microwave beam. The relationship between frequency \( f \) and wavelength \( \lambda \) is given by \( \lambda = \frac{c}{f} \), where \( c \) is the speed of light (\( 3 \times 10^8 \; \text{m/s} \)). Substitute \( f = 12 \; \text{GHz} \; (12 \times 10^9 \; \text{Hz}) \) into the formula.

Step 3: Determine the surface area of the sphere at 30 km. The formula for the surface area of a sphere is \( A = 4 \pi r^2 \), where \( r \) is the radius. Here, \( r = 30 \; \text{km} \; (30 \times 10^3 \; \text{m}) \). Substitute this value into the formula to find \( A \).

Step 4: Calculate the average intensity. Intensity \( I \) is defined as \( I = \frac{P}{A} \), where \( P \) is the power emitted (100 kW or \( 100 \times 10^3 \; \text{W} \)) and \( A \) is the surface area calculated in Step 3. Substitute the values of \( P \) and \( A \) into the formula.

Step 5: Interpret the result. The average intensity represents the power per unit area of the microwave beam at a distance of 30 km. This value can be used to understand the distribution of energy from the radar antenna over a large distance.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Diffraction

Diffraction is the bending of waves around obstacles and the spreading of waves when they pass through an aperture. In the context of the radar antenna, the circular aperture causes the emitted microwave beam to spread out as it travels, affecting how the intensity of the signal diminishes with distance.

Intensity of Waves

Intensity is defined as the power per unit area carried by a wave. For electromagnetic waves like microwaves, intensity can be calculated by dividing the total power emitted by the area over which the power is distributed. As the distance from the source increases, the intensity decreases due to the spreading of the wavefront.

Antenna Gain and Effective Aperture

Antenna gain refers to the ability of an antenna to direct radio frequency energy in a particular direction compared to an isotropic radiator. The effective aperture is a measure of how well an antenna can receive power from incoming waves. These concepts are crucial for understanding how the radar antenna's size and design influence the intensity of the microwaves at a given distance.

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