Question

Question1. A U.S. or Canadian broadcasting licence for a (large) FM radio station stipulates a maximum broadcast power of 100 kW. At any distance from the broadcast antenna the intensity doesn’t quite follow the inverse square law that is described for spherical waves in the videos lectures, because of reflection of the signal off of the ground. But the inverse square law is not a bad approximation, so let’s use it

a) Find the intensity of the radio signal from a 100 kW FM radio transmitter at the following distances from the broadcast tower:•

1.0 km

•

5.0 km

•

90 km (this is about the maximum range at which a typical radio would be able to pick up the signal reliably)

(b) We don’t usually use a radio dish to listen to FM radio (an antenna is good enough), but there is no reason we couldn’t, other than that it would make your radio big and bulky. For technical reasons the dish would need to be about 1.0 m in diameter to work well. At a range of 90 km from the transmitter, what total power would be received by a 1.0 m radio dish?

(c) Let’s contrast this with a rather extreme case of radio communications. The Voyager 2 probe is currently 19 billion km away from Earth. Its radio transmitter has a broadcast power of 22 W (that’s right, not kW! Just 22 W! That’s about the power of the light bulb inside a refrigerator...). When NASA communicates with Voyager 2 it is typically using a 70 m diameter radio dish. What is the power received by this dish from Voyager 2? NB EACH QUESTION SOLVED SHOULD BE PROPERLY NUMBER CORRESPONDING TO THE CORRECT ANSWER

Answer 1

(a) Intensity follows the inverse square law :

I1 = Io / r^2

so, at 1 km I1 = 100 kW/ 1000*1000 = 0.1 Watt/m^2

at 5 km I2 = 100kW / 5000*5000 = 0.02 watt/m^2

at 90 km I3 = 100 kW / 90000*90000 = 1.23*10^(-5) Watt/m^2

(b) Intensity at 90 km is calculated in part (a) which is I3

now area of the 1m diameter dish is   = pi*r^2

thus Power received is P = I3*pi*r^2

P = 1.23*10^(-5)*3.14*1/4 = 0.96*10^(-5) Watts.

(c) Intensity of the waves at Earth I = Io/r^2

where Io = 22 W

and r = 19 billion km

thus Intensity I = Io/r^2 = 0.0609*10^(-30) Watt/m^2

Now for the 70m dish power received :

P = I*pi*r^2 = 0.0609*10^(-30)*3.14*70*70/4 = 2.34*10^(-28) watts.