Complete Question:
The Voyager 1 spacecraft is now beyond the outer reaches of our solar system, but earthbound scientists still receive data from the spacecraft s 20-W radio transmitter. Voyager is expected to continue transmitting until about 2025, when it will be some 25 billion km from Earth.
What s the diameter of a dish antenna that will receive 10−20W of power from Voyager at this time? Assume that the radio transmitter on Voyager transmits equally in all directions(isotropically). In fact, the antenna on Voyager focuses the signal in a beam aimed at the earth, so this problem over-estimates the size of the receiving dish needed.
Answer:
d = 2,236 m.
Explanation:
The received power on Earth, can be calculated as the product of the intensity (or power density) times the area that intercepts the power radiated.
As we assume that the transmitter antenna is ominidirectional, power is spreading out over a sphere with a radius equal to the distance to the source.
So, we can get the power density as follows:
I = P /A = P / 4*π*r², where P = 20 W, and r= 25 billion km = 25*10¹² m.
⇒ I = 20 W / 4*π* (25*10¹²)² m²
The received power, is just the product of this value times the area of the receiver antenna, which we assumed be a circle of diameter d:
Pr = I. Ar =( 20W / 4*π*(25*10¹²)² m²) * π * (d²/4) = 10⁻²⁰ W
Simplifying common terms, we can solve for d:
d= √(16*(25)²*10⁴/20) = 2,236 m.
