The GPS network consists of 24 satellites, each of which...

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) simu- soidal electromagnetic signal at two frequencies, one of which is 1575.42 $\mathrm{MHz}$ . Assume that a satellite transmits half of its power at each frequency and that the waves travel uniformly 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 $\mathrm{cm}$ on a side that absorbs all of the beam, what average pressure does the signal exert on it? (d) What wavelength must the receiver be tuned to?

Key Concepts

Newtonian Orbital Mechanics

This concept involves using Newton's laws, particularly the balance between gravitational and centripetal forces, to determine the altitude and orbital parameters of a satellite. It is fundamental for understanding orbital motion and designing satellite trajectories in astrophysics and aerospace engineering.

Intensity of Electromagnetic Radiation

Intensity is defined as the power per unit area and describes how the energy emitted by a source spreads over space. It is crucial for calculating the received power at a given distance from an electromagnetic source and is applicable in areas ranging from antenna theory to optics.

Electromagnetic Field Amplitudes

This relates the strength of the electric and magnetic fields to the intensity of the wave. The amplitudes of these fields are interconnected via the impedance of free space, and determining them is essential in understanding the propagation and interaction of electromagnetic waves.

Signal Propagation

Understanding that electromagnetic signals travel at a constant speed (the speed of light) in a vacuum is key to determining the time delay between emission and reception. This concept is central to communications, radar, and various time-sensitive technologies.

Radiation Pressure

Radiation pressure is the pressure exerted by electromagnetic radiation on a surface due to momentum transfer when the radiation is absorbed or reflected. This concept is important in contexts such as solar sails, optical trapping, and the design of radiation detectors.

Frequency-Wavelength Relationship

This principle states that the product of a wave's frequency and its wavelength equals the speed of light in a medium. It is foundational in the study of electromagnetic waves, allowing conversions between frequency and wavelength, which is vital in communication and spectroscopy.

Transcript

00:01 Hello everyone.

00:01 In this problem, we're asked to find out a couple of details about the transmitted gps signal of a satellite.

00:08 So we're told that the satellite is broadcasting in a downward hemisphere, and its radiation is incident on a square panel of side length l.

00:20 And we're asked to find what the intensity is that this panel sees, what are the electric and magnetic field amplitudes that the position? of the panel.

00:32 How long does the signal take to reach the panel? what pressure does the radiation exert on the panel? and what wavelength the panel has to be tuned to in order to absorb the radiation coming from the satellite.

00:47 Okay, so here we're given a bunch of details.

00:50 We're told that the satellite is transmitting at a frequency or half of its power is transmitted at a frequency of 1575 .42 times the other 6 hz.

01:02 And we're told that the total power output is 50 watts and that it is split equally between the two frequencies at which it radiates, its electromagnetic radiation or signal.

01:12 And so the power of this wave with this given frequency is 25 watts.

01:18 We're told what the length of the side of the panel is, which eventually we're not going to use.

01:24 And then we're told that the satellites orbit the earth such that they make two revolutions a day around the surface of the earth, which means that the period for one revolution is half a day or 12 hours, which is 4 .32 times 10 to 4 seconds.

01:40 And then here, i've just collected a bunch of constants that we need to look up in order to complete our calculations, which are details about the earth.

01:48 So it has a mass of 5 .97 times 1024 kilograms, a radius of 6 .37 times 10 to 6 meters.

01:57 And newton's constant is 6 .67 times the 13 units meter squared per kilogram squared.

02:03 And then a bunch of other constants like speed of light and the primitivity of free space.

02:08 Okay, so what is the intensity? that's the first question that we have to ask.

02:13 And so the intensity is the radiated power divided by the area.

02:17 And because we are only, the satellite is only emitting its radiation in a downward hemisphere, which means that it's only half a sphere.

02:25 So the area of that hemisphere or half hemisphere is 2 pi times l squared, where l is the distance above the surface of the earth, whereas what we're going to be able to calculate using newton's second law and newton's gravitational law, the inverse square law is the distance between the center of the earth and the satellite.

02:47 But then using the radius of the earth, we can subtract that and we can find the altitude.

02:53 So how do we do that? well, we see that the forces on the satellite have to be equal.

02:58 So, or we have to say that the irritation, we have to set the gravitational force on the satellite equal to the force that is required for it to stay in orbit, which means that when it's going through centripetal motion, it has to have an acceleration that is v squared over r, where r is the distance between the two center of gravities or the two center of masses of the two objects.

03:25 And so we have for the force, g, mass of the satellite times mess of the earth over r squared is equal to mess of the satellite times v squared over r.

03:35 So this is just the consequence of circular motion.

03:40 And so v, what's the velocity? the velocity is 2 pi r divided by the period, which is, you know, it's just to say that in a circular motion, in one period, the object travels around one surface.

03:54 Which is equivalent to a distance of 2 pi times r...

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