Cp global positioning system (gps). the gps network consists of 24 satellites, each of which makes two orbits around the earth per day. each satellite transmits a 50.0-w (or even less) sinusoidal electromagnetic signal at two frequencies, one of which is 1575.42 mhz. assume that a satellite transmits half of its power at each frequency and that the waves travel uni- formly in a downward hemisphere. (a) what average intensity does a gps receiver on the ground, directly below the satellite, receive? (hint: first use newton’s laws to find the altitude of the satellite.) (b) what are the amplitudes of the electric and magnetic fields at the gps receiver in part (a), and how long does it take the signal to reach the receiver? (c) if the receiver is a square panel 1.50 cm on a side that absorbs all of the beam, what average pres- sure does the signal exert on it? (d) what wavelength must the receiver be tuned to?

(a)

First of all, we need to find the altitude of the satellite. The gravitational attraction between the Earth and the satellite is equal to the centripetal force that keeps the satellite in circular motion, so

(1)

where

G is the gravitational constant

m is the satellite's mass

M is the earth's mass

r is the distance of the satellite from the Earth's centre

is the angular frequency of the satellite

The satellite here makes two orbits around the Earth per day, so its frequency is

And by solving eq.(1) for r, we find

The radius of the Earth is

So the altitude of the satellite is

The average intensity received by a GPS receiver on the Earth will be given by

where

P = 50.0 W is the power

A is the area of a hemisphere, which is:

So the intensity is

(b) , 0.068 s

The relationship between average intensity of an electromagnetic wave and amplitude of the electric field is

where

c is the speed of light

is the vacuum permittivity

E is the amplitude of the electric field

Solving for E, we find

Instead, the amplitude of the magnetic field is given by:

The signal travels at the speed of light, so the time it takes to reach the Earth is the distance covered divided by the speed of light:

(c)

In case of a perfect absorber (as in this case), the radiation pressure exerted by an electromagnetic wave on a surface is given by

where

I is the average intensity

c is the speed of light

In this case, we have

So the average pressure is

(d) 0.190 m

The wavelength of the receiver must be tuned to the same wavelength as the transmitter (the satellite), which is given by

where

c is the speed of light

f is the frequency of the signal

For the satellite in the problem, the frequency is

So the wavelength of the signal is: