What is the area of a receiving antennae with an angular spread of θ = λ/d?

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The discussion revolves around calculating the area of a receiving antenna with an angular spread defined by θ = λ/d, where λ is the wavelength and d is the distance. The provided values are λ = 5*10^-2 m and d = 3*10^3 m, leading to an angular spread calculation. The spatial spread is determined by multiplying the angular spread by the distance the beam travels, resulting in an approximate area of 1 km². However, a more precise calculation suggests the area is closer to 0.54 km², indicating potential errors in the initial assumptions or calculations. The conversation emphasizes the importance of accuracy in deriving the area from the angular spread.
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Homework Statement



I just have a question on finding the area of a receiving antennae. The answer says the area is about 1km. I got an answer close to 1 km.

λ = 5*10-2 m
d = 3*103 m

The angular spread is θ = λ/d

θ= \frac{5*10<sup>-2</sup>}{3*10<sup>3</sup>}

The spatial spread is the diameter of the beam on the target. So that means it is the angular spread times the distance the beam travels. Correct?

S = r*θ
S = (5*10^7 m ) * θ
S ≈ 1 km The area of the receiving antennae is them [1km / 2 ]2 * ∏ = 0.78km which is about 1 km.
 
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The S ≈ 1 km approximation is rather rough. 830m is closer, and that leads to a final answer of about .54 km2 (note: km2, not km). This suggests you have an error somewhere else, but I'm not qualified to say where.
 
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