Physics 106N - How Things Work - Spring, 1997 

Problem Set 2 - Answers



Problem 1:  Chapter 12, Case 10a-d (Pg. 461) 

A ground fault interrupter is a device that senses when some of the current flowing out one side of an electric outlet isn't returning through the other side of that outlet. The only way such an imbalance can occur is if some current is returning to the electric company through the ground. Because that accidental current path might include your body, the interrupter shuts off current as soon as it senses trouble. 

a. Inside the interrupter, the two wires from the electric company pass together around the iron core of a single transformer, forming two identical primary coils. If the AC current passing through each coil is equal in magnitude but opposite in direction, how does the magnetization of the transformer's core change with time? 
Answer: There is no magnetization of the transformer's core. 
Why: Since the two currents passing around the core are equal in amount but opposite in direction, the magnetic fields produced by two currents cancel one another perfectly. 

b. If the AC current passing through each coil isn't equal (perhaps because some current is escaping from a hair dryer plugged into the outlet), how does the magnetization of the transformer's core change with time? 
Answer: The magnetization reverses its direction rapidly with time (120 reversals per second). 
Why: Since the two currents passing around the core aren't equal in amount, their magnetic fields don't cancel and the core aquires an overall magnetization. However, because the two currents reverse directions every 120th of a second, the magnetization of the core reverses its direction 120 times a second. 

c. The transformer's core has a secondary coil wrapped around it. If all of the current flowing to the hair dryer through one wire returns from it through the other wire, current passing through this secondary coil will experience no change in voltage. But if some of the current flowing to the hair dryer doesn't return, current in the secondary coil will experience a change in voltage. Explain. 
Answer: If the currents through the two primary coils are equal, there is no overall magnetic field in the core and no electric fields around it to push or pull on current in the secondary coil. That current experiences no change in voltage. But if the currents through the two primary coils are unequal, there is an overall magnetization of the core, a magnetization that changes with time and is thus accompanied by an electric field. This electric field push or pulls on the current in the secondary coil, changing its voltage. 
Why: When the currents in the two primaries aren't equal in amount but opposite in direction, energy is transferred between those currents and the current in the secondary coil. The voltage of the current in the secondary coil changes as the result of this energy transfer. 

d. The current from the secondary coil is used to trigger a switch that disconnects the outlet from the electric company. This switch has manual "test" and "reset" buttons. To test the ground fault interrupter, you press the "test" button. What can this button do to simulate a real current accident? 
Answer: The test button allows current to flow through one of the wires in the ground fault interrupter without returning through the other wire. 
Why: By connecting on side of the electric outlet to ground, typically through an electric resistor that limits the current flow to a small value, the interrupter demonstrates that it is very sensitive to currents that don't come back from an appliance. 

Problem 2: Chapter 13, Case 15a-e (Pg. 488) 

Your new electric guitar and amplifier sound great, although all the neighbors seem to have gone on vacation since you bought it last week. After 6 straight hours of jamming you decide to take a break and figure out how it works. 

a. You begin by examining the strings and pickups--the devices that sense the strings' motions and represent them as electric currents. The pickup near each string is a coil of wire wrapped around a small permanent magnet. The permanent magnet magnetizes the steel string so that it induces an alternating current in the pickup coil as it vibrates back and forth. What provides the power for this alternating current? 
Answer: The vibrating string (or you, as the person who plucked the string to make it vibrate). 
Why: The permanent magnet is only there to magnetize the string. This vibrating magnetic string is what generates the electric currents in the pickup coil. 

b. The electric power provided by the pickup is much too small to drive a speaker, so it must be amplified. The first step is to send it through a preamplifier. Current from the pickup flows through the preamplifier's input circuit and an amplified version of that current flows through the preamplifier's output circuit. The voltage rise at the preamplifier's output is 10 times as large as the voltage drop at its input, and the current passing through its output circuit is 10 times as large as the current passing through its input circuit. The power the preamplifier is providing is how many times as large as the power it's receiving from the pickup? 
Answer: 100 times. 
Why: The output circuit carries 10 times as many charges per second (10 times the current) as the input circuit and the amplifier gives each charge passing through the output circuit 10 times as much energy (10 times the voltage) as it takes from each charge passing through the input circuit. Overall, the amplifier gives the current passing through its output circuit 100 times as much energy each second (100 times the power) as it takes from the current passing through its input circuit. 

c. The output of the preamplifier is connected to the main power amplifier. You open a side panel of the amplifier and notice several large MOSFETs bolted to a heat sink. Since the amplifier was running recently (you wisely turned it off and unplugged it before opening it up), the heat sink is quite warm. These MOSFETs have been controlling the flow of current from the amplifier's power supply to your speaker, so their electric resistances have been fluctuating up and down. What has the amplifier been doing to the MOSFETs to make their electric resistances change? 
Answer: Changing the amounts of charge on their gates. 
Why: As the charge on a MOSFET's gate changes, so does its electric resistance. It can vary from being a good conductor and a good insulate, along with anything in between. 

d. Why have the MOSFETs been producing thermal energy?
Answer: Because it has an electric resistance, it wastes some of the energy of the current passing through it. That energy becomes thermal energy.
Why: Like any normal conductor of electricity, the MOSFET wastes some of the energy of any current passing through it.

e. You reinstall the amplifier's side panel and take a look at the speaker cabinet. This cabinet contains several speakers, each of which has a coil of wire that becomes magnetic when current flows through it. A nearby permanent magnet pushes on the magnetic coil and this force moves a paper cone to create sound. The currents needed for high volume are large and they flow to and from the speaker through the speaker wires. You notice that the speaker wires are warm--they have been wasting some of the power from your amplifier! You knew it was a mistake to buy cheap thin speaker wire, but it's too late now. You have extra wire which you can use to reduce the wasted power. Should you add a second wire in parallel to each of the present wires or should you add a second wire in series with each of the present wires? Explain your answer.
Answer: You should connect the second wire in parallel to the first. The two wires will then share the current and, since each will carry less current than before, it will waste much less power from that current.
Why: By sharing the current between two wires, you'll reduce the current in each wire by a factor of 2. That will reduce the power wasted in each wire by a factor of 4. However, because there are two wires now, the total power wastage will be down by an overall factor of 2.

Problem 3: Chapter 14, Case 2a-e (Pg. 519) 

As a DJ at a very small local radio station, you often find yourself involved in technical issues for the station's two channels, one of which is AM at 1020 kHz and the other of which is FM at 89.5 MHz. The station is in the process of building a new transmitting system. 

a. To save money, the director wants to use a single antenna for both channels. You warn him that the AM channel needs a taller antenna than the FM channel. Why is that true?
Answer: The AM station involves a much lower wavelength radio wave and an efficient antenna for this longer wavelength wave must itself be longer.
Why: The ideal length for an antenna is 1/4 of the wavelength of the wave it transmits. Since the AM wave has a longer wavelength, it needs a longer antenna.

b. The director had planned to put the antennas next to the station, which is in a valley at the base of a small mountain. You suggest putting them at the top of the mountain, despite the extra cost of wires. Why is altitude important, particularly for the FM antenna?
Answer: Radio waves (particularly shorter wavelength ones) travel in straight lines. Increasing the altitude of the antenna allows the waves to reach more distant radios without hitting anything.
Why: Shorter wavelength radio waves follow line-of-sight paths to your radio. If you can't see the transmitting antenna, at least a little, you'll have trouble receiving its radio transmission.

c. A suggestion to orient the antennas horizontally is quickly dismissed as non-traditional. But it wouldn't work well, either. List two reasons why a horizontal antenna would produce a radio signal that most people would find hard to receive with their radios.
Answer: Its radio wave will be horizontally polarized (it will be hard to receive with a vertical antenna on your radio) and the radio wave will be extremely weak out along the ends of the transmitting antenna.
Why: To receive the transmission from a horizontal antenna, you do best with a horizontal receiving antenna. An antenna doesn't tranmit a wave out of its ends, so that the reception along the direction of the antenna itself will be poor.

d. The AM channel must be careful not to "overmodulate" the radio wave during very loud passages because it distorts the sound people hear in their radios. You explain this effect as due to moments when the transmitter actually turns itself completely off. Why would the transmitter stop transmitting any wave at all?
Answer: The AM station signals your radio to move its speaker toward and away from you by varying the amount of charge it moves up and down its antenna. If it tries to move too little charge up and down the antenna as part of this signalling, it may reach the point where it moves no charge up and down at all.
Why: In amplitude modulation, it is the amount of charge moving up and down the antenna that indicates how to move the radio's speaker. While there is no upper limit to the amount of charge that can be moved up and down the antenna, there is a lower limit: zero.

e. The FM channel must also avoid overmodulation during loud passages because it will get in trouble with the FCC. Other FM stations in your area will also be angry with your station for spoiling the reception of their transmissions. How can your FM station affect those other FM stations when they operate at different carrier wave frequencies?
Answer: The FM station signals your radio to move its speaker toward and away from you by varying the frequency with which it moves charge up and down its antenna. If it shifts that frequency too far above or below its normal operating frequency, it will enter the frequency range allocated to another station and the two stations will compete for the attention of your radio.
Why: In frequency modulation, it is the frequency with which charge moves up and down the antenna that indicates how to move the radio's speaker. If that frequency changes too much, it will reach frequency ranges that are supposed to be used only by other stations.

Problem 4: Chapter 14, Case 5a-e (Pg. 520) 

The Global Positioning System or GPS is a system of earth-orbiting satellites that provide position information to anyone with a GPS radio receiver. Each satellite transmits two microwaves, one at 1.57542 GHz and the other at 1.2276 GHz. These waves are modulated by carefully timed pulses so that by measuring exactly when it receives those pulses from four or more satellites, a GPS receiver can determine its position on earth to within 100 m. Recent changes in U.S. Government policy will allow GPS receivers to locate their positions even more accurately over the next few years. 

a. Because each satellite carries a time-standard cesium atomic clock, its pulses are emitted at precisely known times. The GPS receiver knows exactly when it receives the pulses, so it can tell exactly how far it is from the satellite. How?
Answer: Because it knows when a pulse left the satellite and when it arrived at the antenna, the GPS receiver can easily determine how long it took the pulses to travel from the satellite to the antenna. From this time and a knowledge that the pulses travel at the speed of light, the GPS receiver can determine the distance the pulses travelled, and how far the receiver is from the satellite.
Why: The GPS receiver is simply timing the flights of the pulses and using the fact that these pulses travel at the speed of light to determine the distances to satellites.

b. Because a GPS receiver only receives microwaves from satellites that are above the horizon, there must be at least 17 satellites in orbit at once. Why can't the receiver detect satellites that are below the horizon?
Answer: The microwaves travel in straight lines and can't go through the earth.
Why: If a satellite is below the horizon, where you can't see it with your eyes, you also can't detect its microwave transmissions.

c. What are the wavelengths of the two microwaves?
Answer: 19.03 cm and 24.42 cm.
Why: If you divide the speed of light by the frequency of each wave, you will find out how far the wave travels during each cycle and thus how far apart the wave crests are--the wavelength of the wave.

d. One of the reasons for using microwaves in the GPS is that microwaves can be received by small antennas with well-defined locations. How long should an antenna be to receive the lower frequency GPS microwave?
Answer: About 6.1 cm.
Why: The ideal length for an antenna is 1/4 of the wavelength of the wave it emits or receives. In this case, 1/4 of 24.42 cm (the lower frequency microwave) is about 6.1 cm.

e. Geologists are using the GPS to study motions of the earth's crust that are as small as a few millimeters. To do this, they actually study the electric and magnetic fields of the microwaves. Why would two GPS receivers located a few centimeters apart detect somewhat different electric and magnetic fields if they studied the microwave from one particular satellite at precisely the same moment?
Answer: At one instant in time, the electric field of the microwave points in different directions, depending on where you look along the path of the microwave.
Why: The microwave's electric field points back and forth along its travel path, with its direction of pointing reversing every 10 or 12 cm.

Problem 5: Chapter 15, Case 1a-e (Pg. 553) 

The color of the sky depends on the weather and the air. 

a. On a very humid day, when the air contains many water droplets that are about 1 micron (0.000001 m) in diameter, the air appears hazy and white. Why?
Answer: The water droplets are large enough that they Rayleigh scatter all colors of light about equally and appear white.
Why: The light passing through the water droplets is often absorbed and reemitted in new directions. Since there is no color preference for this large droplets, the light is scattered evenly and remains white.

b. On a very clear, dry day when there is no dust or water in the air, the sky is very dark blue. Why?
Answer: When only the tiniest particles are present in the atmosphere, Rayleigh scatter is weak and only blue or violet light scatters significantly. The sky appears very deep blue as a result.
Why: Since red light is rarely Rayleigh scattered by the tiniest particles, it travels through the atmosphere unaffected. Only bluish light is scattered significantly and you see only this bluish light in the sky overhead.

c. Why is the sky black on the airless moon?
Answer: With no atmosphere to Rayleigh scatter sunlight, there is no blue sky.
Why: The earth's atmosphere is what gives the sky a blue appearance. Without any particles overhead, the moon's sky doesn't affect sunlight and has no blue coloring.

d. When you look at the surface of a calm lake, you see a reflection of the sky. The water is not a metal, yet you see a reflection. What change occurs as light passes from air to water that causes some of the light to be reflected?
Answer: The light slows down.
Why: As light enters water, it slows down and some of it reflects.

e. Mirages appear on sunny days in the desert when very hot air from the earth's surface bends light from the horizon upward so that you see it as though it were coming from the ground in the distance. Hot air has fewer air molecules in it than colder air at the same pressure. Why does light bend slightly as it moves from colder air to hotter air?
Answer: The difference in densities of the two airs (hot and cold) causes light to change speeds as it goes from one to the other and the light bends.
Why: Colder air is more dense than hotter air (at a given pressure), so light speeds up as it moves from colder air to hotter air. It often bends when this happens.