Lesson 13 Applications of Time-vaying Cicuits Lawence. Rees 7. You may make a single copy of this document fo pesonal use without witten pemission. 13. Intoduction In this lesson we ll look at a numbe of applications of time-vaying electic and magnetic fields to eveyday life. Of couse, we can only conside a small numbe of these applications and not teat any of them in geat depth. You will see, howeve, how the concepts we have leaned this semeste came togethe to help undestand seveal these applications. 13.1 Tansfomes We found in Lesson 6 that when you put a slab of dielectic mateial inside a capacito, it inceases its capacitance. We can do something simila with an inducto by putting a soft ion coe inside the inducto. Soft ion has a vey small coecive foce, so it becomes magnetized eadily when it is placed in the magnetic field of the inducto. When the domains of the ion ae aligned in the inducto s field, they can incease the stength of the magnetic field significantly. When the magnetic flux though the inducto is lage, the induced EMF is lage, and the inductance is lage. Futhemoe, the ion coe of an inducto tends to keep the magnetic field lines tapped inside the ion. What I mean by this can be seen in Fig. 13.1. Figue 13.1. The field of an inducto with and without an ion coe. If we want to keep the magnetic field of the inducto confined in space, we can even make the ion coe into a loop. If we do this, the magnetic field lines will follow aound the ion loop, as shown in Fig. 13.. Figue 13.. An inducto with a looped ion coe. 1
In an idealized inducto of this type, we can detemine the magnetic field by applying Ampèe s Law in much the same way that we did fo a tous. Let s assume that the magnetic field is unifom though the coe. The effect of the ion coe is to amplify the magnetic field in much the same way that the effect of a dielectic in a capacito is to educe the electic field. We then have: Λ kµ i l kµ N1i whee l is the (aveage) length of the path aound one field line. N 1 is the total numbe of tuns in the coil of wie. i is the cuent passing though the coil. k is the facto by which the magnetic field is multiplied due to the pesence of the feomagnetic coe. We can also use Faaday s Law to calculate the inductance of this inducto just as we did with an ai-coe solenoid in Lesson 1. The induced EMF is: dφ V1 N1 dt kµ N L l 1 enc kµ N1 N1A l A di dt L di dt Now, we ll take this same inducto and add a second coil aound the opposite side of the ion loop and then attach the bottom coil to a powe supply. The top coil now has a magnetic field passing though it and this magnetic field constantly changes in time. This, of couse, induces a voltage acoss the uppe coil. Figue 13.3. An ion-coe tansfome.
Faaday s Law gives us the EMF acoss the uppe coil. V N N V N 1 1 dφ dt since V 1 N 1 dφ dt (13.1 Voltage in a tansfome) V V 1 N N 1 This elation tells us that by adjusting the atio of tuns in the top coil to the bottom coil, we can get any voltage we want in the top coil. Such a device is called a tansfome. Tansfomes find uses in many diffeent applications. It is helpful to lean a few tems elating to tansfomes: pimay coil: the coil that is attached to the powe souce. seconday coil: the coil that is attached to a load. step-up tansfome: a tansfome in which the voltage of the seconday is geate than the voltage in the pimay. step-down tansfome: a tansfome in which the voltage of the seconday is less than the voltage in the pimay. The cuent in the pimay and seconday coils of tansfomes is a little hade to calculate; howeve, the powe povided by the pimay is the powe available fo a load in the seconday. If we assume that the powe factos ae nealy equal to one fo both pimay and seconday, this condition becomes: i. 1V1 iv The assumption that the powe factos ae appoximately equal to one is a athe poo assumption, but the elation does tell us something qualitative about the cuent available in the seconday coil. The cuent available fo the load in step-down tansfomes is geate in the cuent in the pimay. Convesely, the cuent available fo the load in step-up tansfomes is less than the cuent in the pimay. Step-down tansfomes ae used in applications whee a highe voltage is not vey desiable, such as in a doobell cicuit is a home, o in an adapte fo electonic devices. (Such adaptes also usually convet AC powe to DC powe.) Step-down tansfomes ae also used fo application like ac welding whee lage cuents ae equied. Step-up tansfomes ae used when high voltages ae needed in applications, such as in neon signs. 3
Things to emembe: Tansfomes ae devices that change AC voltage by taking advantage of mutual induction between two coils wound aound an ion coe. V N The voltages in the pimay and seconday coils ae elated by the fomula. V1 N1 The powe in the pimay and seconday coils is appoximately elated by the fomula i V i. 1 1 V 13.. Getting Electic Powe to You Home One of the most impotant applications of electicity and magnetism in eveyday life is the electic powe we use in ou homes and buildings. We will lean most of the science you will need to know to wie a house. Much of what you will lean is based upon simple esistive cicuits. One diffeence between the cicuits we studied ealie and the cicuits in ou homes is that public utilities use altenating cuent (AC) athe than diect cuent (DC) powe. A battey poduces diect cuent. That is, the cuent in a simple esistive cicuit is constant: it has the same magnitude and the same diection at all times. Most electonic devices such as computes, digital clocks, adios, etc., equie DC powe. This means, of couse, that something must be done to convet the AC powe that comes fom the outlet into DC powe to be used in these devices. Some motos, such as those in kitchen most appliances, ae built to un on AC powe diectly. Incandescent lights, heates, etc., can un on eithe AC o DC powe. In the United States, the standad fo outlets is 11 1 V, 6 Hz, AC powe. ecause it is a convenient numbe, we will call line voltage 1 V. If we gaph the voltage in ou outlets as a function of time, we have something like: 4
15 1 5-5 -1-15 -.1..3.4.5.6.7.8.9 Figue 13.4. AC ω line π π voltage as a function of time. The fequency is f 6 Hz 6 cycles/second. The peiod is the time it takes fo the voltage to go though one complete cycle. This is just T 1/f.167 sec, as can be ead (oughly) off the gaph. The angula fequency is the numbe of adians pe second, o times the numbe of cycles pe second. In othe wods, it is f 377 adians/second. Finally, the maximum voltage is about 17 V. The functional fom fo an AC voltage is geneally of the fom: ad V ( t) 17V sin 377 t Vmax sin( ω t) s So why do we call this a 1 V outlet? You pobably ecall fom the last chapte that we geneally use the ms voltage o cuent when we deal with AC cicuits and 17 1. When electic powe companies wee fist begun in the United States, thee was a debate ove whethe DC powe o AC powe should be povided. Niccolo Tesla advocated AC while Thomas Edison advocated DC powe. The esolution of the issue came down to a question of enegy loss in powe lines. We can conside the geneato tansmission line load system to be a simple seies cicuit. The load, eveyone who is using the powe, is a esisto R l. The tansmission line is esisto R t. The geneato is (essentially) a battey with voltage V. 5
V + R t R l Figue 13.5. A simple cicuit we can use to model tansmission lines. Since the algeba can get a little messy, we ll assign some numbes to these quantities. Let V 1 V, and each esistance be 1. Fist, we wish to calculate the powe consumed by the load and the powe lost in the tansmission line. V I R + R V l IR P IV l t l l l 5A 5V, 5W, V t IR P IV t t 5V t 5W, P b IV 5W Now let us incease the voltage of the battey to 5 V while we keep the powe used by the load the same. Of couse, to keep the powe usage the same, we must change the esistance of the load. Leaving all the numbes in SI units (we won t wite the units explicitly), we now have: V 5 I R 1 + R tot P 5 I ( 1 + R l ) 5R l l 5 R l R 1 + l The solution of this quadatic equation leads to two possible esistances. Once we know the esistance, we can find the cuent and the powe loss in the tansmission line. One value of powe loss is vey lage, so we discad that solution. The second value of esistance gives the following esults: R l Voltage of the powe souce 1V 5V Resistance of the load 1 9.6 Powe loss in the load 5 W 5 W Powe loss in the line 5 W 1.9 W Powe povided by the battey 5 W 6.9 W 6
The conclusion is then that highe voltages ae best suited fo electical powe tansmission. On the othe hand, if you plug a vacuum cleane into a,v outlet, you would have to be athe caeful. y the time we use electical powe in ou homes, we eally need to have faily low voltages. As we fom the last section, we can easily change voltages when we use AC powe. We simply take a high voltage line into a tansfome and out comes whateve voltage we want. High voltage (o high tension tension is the Euopean tem fo voltage) powe lines can use voltages of seveal hunded kilovolts. Electical substations have tansfomes that typically educe voltages to 4 8 kv. If you look aound powe lines in esidential aeas you will see cylindes with powe lines going to two o thee homes. These cylindes ae tansfomes that take the voltage down to 1 V fo use in homes. Actually, thee wies go fom the tansfome to each house. One wie is a gound, V. The othe two wies have 1 V, but the phases of the voltage ae opposite, as shown in Fig. 13.6. 15 1 5-5 -1-15 -.5.1.15..5.3.35 Figue 13.6. Household powe with two voltages out of phase by 18º. If we attach a moto, fo example, between these two wies, the voltage acoss the moto will by 4 V athe than 1 V. This highe voltage is used fo majo appliances such as electic ovens and electic clothes dyes. (In pactice, the voltage is often called 8 V.) 7
Once the powe cables come to a house, they go into an electic mete. This mete measues the amount of enegy that is used. We ll lean late how these metes wok. Afte passing though the mete, the powe cables ente a sevice panel in the home. Sevice panels distibute the powe to a numbe of sepaate cicuits within the home. Figue 13.7. The oute of electic powe into a home: high voltage powe lines, substation, tansfome, sevice dop, electic mete, sevice panel (cicuit beakes). Things to emembe In the United States household powe is 1 V (ms), 6 Hz powe. Tansmission lines use high voltages in ode to minimize loss of enegy in the tansmission lines. Tansfomes ae used to educe high voltage powe to 1 V fo household use. Thee lines come into a house, V and two 1 V that ae out of phase by 18 º. 13.3 Cicuits and Cicuit eakes Each cicuit that comes fom the sevice panel sevices a paticula pat of the home s electical needs. Lage appliances, such as stoves and dyes equie thei own cicuits. Othe cicuits may include all the lights and outlets in a paticula pat of a house. When moe cuent flows in a wie, the wie has to be lage in diamete (as we shall soon see). Lage wie is moe expensive and hade to use because of its size and stiffness. Fo these easons, and also just fo convenience, houses typically have many diffeent cicuits. Cicuits fo majo appliance ae usually designed to cay about 4 5 A. Othe cicuits usually cay no moe than A. On each cicuit is a cicuit beake. This is a device that seves two puposes: 1) it is a switch to tun off powe to a paticula pat of a house, and ) it automatically shuts off powe to the cicuit if thee is too much cuent flowing though the cicuit. 8
Figue 13.8. A bimetallic stip. The metal on the bottom expands moe when it gets hot. In olde homes and in many applications such as in cas, fuses ae used instead of cicuit beakes. Fuses have stips of metal that melt if cuent eaches a given value, theeby beaking the cicuit. Cicuit beakes pefom the same function, but they do it in a moe sophisticated way that allows them to be simply eset by a switch. Cicuit beakes usually have two modes of opeation. One mode senses heat and one senses cuent. If thee is enough cuent to cause a esisto in the beake to get hot, it causes a bimetallic stip to bend and beak the cicuit. (A bimetallic stip is a thin stip made of one metal on top and anothe metal on the bottom. The two metals expand at diffeent ates as the stip heats, causing it to bend. imetallic stips ae often used in themostats.) The second mode of opeation kicks in when thee is a vey lage cuent. fom tansfome to cicuit to cicuit to cicuit gound 1 V Figue 13.9. Schematic epesentation of how wies ae connected in a sevice panel. The blue boxes ae the cicuit beakes. The top cicuits ae 1 V cicuits and the bottom is a 4 V cicuit. In this mode, a small solenoid poduces a magnetic field that epels a pemanent magnet connected to a switch. When the cuent gets lage, the magnetic field of the solenoid gets lage and the switch closes within a faction of a second. Cicuit beakes can be puchased in a vaiety of sizes, but 15 A and A beakes ae used in most home applications. 9 V 1 V
y appopiate choice of cicuit beakes, cicuits can be made to opeate at eithe 1 V o 4 V. As we descibed above, thee ae thee diffeent lines that ente the sevice panel: one is gounded and the othe two each cay 1 V in opposite phases. 1 V cicuit beakes connect one 1 V leg to the gound. 4 V cicuit beakes connect the two 1 V legs togethe. Things to emembe: Thee powe lines come into a house. One line is at V and two ae at 1 V. The 1 V lines ae out of phase by 18. y connecting to both 1 V lines, a 4 V cicuit is poduced. Cicuit beakes potect against excessive cuent in a cicuit. Home cicuit beakes have a bimetallic stip switch and a solenoid switch. 13.4. Wies As noted in Fig. 13.9, thee ae thee wies that leave the sevice panel in each 1 V cicuit. One wie is connected to the 1 V line. This is called the hot wie. A second wie is connected to the V line fom the tansfome, so it gounded at the tansfome. This is called the neutal wie. The thid wie is connected to a gound in the house and is called the gound wie. These wies ae typically bundled togethe in a single cable with electical insulation aound the entie bundle. Nea the sevice panel and each outlet, light switch, lamp, etc., the outside insulation is stipped back so the individual wies in the cable can be accessed. Inside the bundle thee is one wie with black insulation, one with white insulation, and one that is bae. The wie with black insulation is the hot wie, the one with white insulation is the neutal wie, and the bae wie is the gound. gound hot neutal Figue 13.1. Wies bundled in a cable. Just as a note of caution: when cuent is flowing though a cicuit though a vacuum cleane plugged into an outlet, fo example cuent passes though the hot (black) wie into the vacuum cleane, and out the neutal (white) wie, and back to the tansfome whee it passes on to gound. This means that although we call the black wie hot, thee can be cuent passing though the white wie and it can be just as deadly as the black wie! Of couse, if you inset a metal object into the neutal side of an outlet, you will be OK, because thee is no voltage on the neutal side of the outlet unless thee is cuent in the cicuit. ut to be on the safe side, you should always teat neutal o white as if it wee hot. In the cable of a 4V cicuit, thee is an additional wie connected to the second hot wie of the sevice panel. This fouth wie is coloed ed to distinguish it fom the fist wie. 1
Table 13.1. Colo Code fo Wies black ed white no insulation o geen white with black tape hot hot, opposite phase neutal gound hot when both black and white need to be hot. When we select wie fo ou cicuits, we need to conside two pincipal things: mateial and size. The only types of wie that ae nomally used in wiing ae aluminum and coppe. Aluminum has the advantage that it is less expensive, but it has seveal disadvantages: it is moe bittle than coppe, making it easie to beak when you have to bend wies to fit into electical boxes; aluminum coodes moe easily than coppe; aluminum has geate esistivity than coppe, so aluminum wies must be of lage diamete than coppe; and aluminum wie cannot be safely connected diectly to coppe wie, as the connection can have high esistance and can oveheat. The bottom line is that aluminum wie is often used fo the lage wie needed in high ampeage cicuits, but coppe is geneally used eveywhee else. If you do use aluminum wie, howeve, be sue that all switches, outlets, etc. ae designed fo use with aluminum. The next thing to conside is the size of the wie. The concen hee is that if we put too much cuent though a wie, the wie can oveheat and cause electical fies. Since we use cicuit beakes in a house, we do know the maximum cuent that can pass though a wie. The basic physics is faily simple, but the math is a bit messy. You don t need to woy too much about the details, but ty to follow them the best you can. Fo a fixed amount of cuent, the powe lost as heat in a section of wie is P IV I R whee V is the voltage acoss the length of wie and R is its esistance. This then epesents the amount of heat enegy pe unit time added to the wie. Heat leaves the length of wie though conduction. The electical insulation also acts as a themal insulation that helps hold the heat in the wie. As you may have leaned in Physics 13 (Don t woy if you haven t had 13.), the amount of heat pe unit time that passes out of the wie is whee: P T ka T d k is the themal conductivity of the insulation A is the suface ae of the length of wie T is the tempeatue of the wie, which we take to be unifom T is the tempeatue outside the insulation of the wie d is the thickness of the insulation 11
Finally, the change in tempeatue in the wie is elated to the net heat flow into the wie by the expession also fom Physics 13): whee: we have: Q mc T Q is the flowing into the wie m is the mass of the wie c is the specific heat of the coppe o aluminum T is the change in tempeatue of the wie Let us take the wie to be cylindical of adius and length L with a esistivity ρ. Then Q t I T mc t T T R ka d ka T + I d T mc t ka R + T d This is a diffeential equation that tells us how the tempeatue of the wie changes in time. It can be solved to yield: I Rd T( t) T + ka 1 As time inceases, the final tempeatue becomes kat / mcd ( e ) I Rd T ( t ) T + ka T f To see how this depends on the adius of the conducto,, we may ewite this as: T T f T I Rd ka L I ρ d π k π L I ρ d 3 kπ Fom this we conclude that the tempeatue incease due to cuent in the wie is: I (13.) T 3 1
This means that if two wies have the same T and one wie has twice the cuent of the othe wie, the diamete of the lage wie must be a facto of /3 1.59 times that of the smalle wie and the coss-sectional be a facto of 4/3.5 lage. Table 13. is a useful summay of wie gauges commonly used in esidential wiing. (AWG stands fo Ameican Wie Gauge.) Thee is a geneal ule of thumb that you should not exceed 4. A /mm in coppe wie o.3 A/mm in aluminum wie. Maximum cuents obtained using this ule ae indicated in Column 4. Since 1 AWG wie is typically used fo A cicuits in homes, the last column, based on Eq. (13.3) pobably pesents moe ealistic values of maximum cuents. AWG Table 13.. Data fo common coppe wie sizes diamete (mm) Typical use I max (A) (4 A/mm ) 6 4.1 electic stoves 53.3 57. 1.59 wate heates, electic dyes 1.5 kitchen, dining oom, bathoom, utility aeas (best fo most household cicuits) 14 1.63 low cuent household cicuits (best fo thee-way lights) 1.1 8.4 13.. 8.3 14. I max (A) Eq. (C.1) nomalized to A fo 1 AWG 16 1.9 Low voltage wiing: 5.. 1. 18 1. doobells and themostats 3.3 7. Things to emembe: In home wiing: black and ed ae hot, white is neutal, and bae o geen is gound. Coppe wie conducts bette, is moe flexible, and coodes less than aluminum wie. Aluminum wie is less expensive. Lage wie must be used in cicuits that daw moe cuent. 6 AWG coppe wie is used fo stoves, 1 AWG wie is used fo most household cicuits. 13
13.5. Switches and Outlets The pupose of switches, of couse, is to stat and stop the flow of cuent though cicuits. Switches ae chaacteized as single-pole o double pole, depending on whethe one o two wies ae connected. They ae also temed single-thow o double-thow, depending on whethe the switch is just open-closed o if the switch can tansfe input cuent to two diffeent output cicuits. Schematic diagams of vaious types of switches ae illustated in Fig. 13.11. in out single-pole single-thow out out in single-pole double-thow in out in out double-pole single-thow out out out in out in double-pole double-thow Figue 13.11. Common types of switches. The most common switch used in house wiing is the single-pole single-thow (SPST) switch. It is often called just a single-pole o SP switch. SP switches ae placed in the hot line to switch lights, outlets, o had-wied appliances (such as bathoom fans) on and off. elow is a diagam of two ways switches can be wied. Note that the switch is enclosed in a eceptacle fo safety. Wie nuts ae used to make a secue connection between wies. In these diagams, hot will be black, neutal will be white, and gound will be gay. 14
in wie nut gound hot neutal hot Figue 13.1. Wiing a simple light switch. If two switches ae in the same box, the inlet hot and neutal wies can be connected with wie nuts to both switches. Although this is the simplest way to wie a switch, sometimes it is moe convenient to connect the incoming powe to the light. If this is the case, the switch can be wied as shown in Fig. 13.13. in white hot (white with black tape) Figue 13.13. An altenative method of wiing a switch. 15
It is often desiable to have two diffeent switches contol a light, so that changing the position of eithe switch will change the on-off state of the light. In this case, simple SP switches will not suffice. The switches that ae used ae called 3-way switches. These ae just singlepole double-thow switches. A hot wie comes in to the common, o COM, connecto of the switch. The switch then selects one of two output wies, called taveles. Two 3-way switches ae connected to a light as shown in Fig. 13.14. Look though the diagam and be sue you undestand how the cicuit woks. ed hot in COM ed hot COM white hot ed hot Figue 13.14. Wiing a 3-way switch. Standad outlets have thee diffeent holes in them. The ounded hole is the gound, the long slot is neutal, and the shot slot is hot. Just in case a plug is not in tight and something conductive is dopped on the plug, it is safest not to have the hot plug on top. Outlets should be installed in one of the configuations shown in Fig. 13.15. Figue 13.15. The pope diections to install outlets. 16
The connections on the side of the outlet neaest the hot slot ae fo the hot wies. These two connections ae joined by a coppe stip, so they ae electically equivalent to each othe. Thee ae two common ways that a seies of outlets can be joined in a cicuit. These ae called seies and paallel, although the teminology is a bit misleading. Seies is most common, as it is easie to install; howeve, paallel has the advantages that if a wie comes loose in an outlet box (which seldom happens), othe outlets ae not affected. These ae depicted in Fig. 13.16 and Fig. 13.17. in out Figue 13.16. Outlets connected in seies. 17
in out Figue 13.17. Outlets connected in paallel. Things to emembe: Switches ae named by the numbe of wies being switched (poles) and the numbe of output options (thows). Switches ae placed in the hot wie of a cicuit. A white wie with black tape wapped aound it is hot (white hot). The hot connection of an outlet is the naow one. Outlets should be installed with the hot connection down. Outlets can be connected in seies and paallel. e able to ecognize a 3-way switch cicuit, and seies and paallel outlet cicuits. 13.6. Safety Devices To potect people fom electical hazads, a numbe of pecautions ae taking in wiing homes. The most impotant of these is simple gounding. If, fo example, the insulation on the hot wie in an appliance beaks down and the case of the appliance is a conducto, the entie appliance becomes hot; that is, the case of the appliance is at 1 V potential. If the appliance is not gounded and someone touches it, cuent can flow to gound though the peson. (Since wate pipes and funace ducts ae usually good gounds, you must be especially caeful with electicity aound them.) This cuent can cause buns o disupt the heat. If the appliance is gounded, howeve, the gound wie and the peson both povide paths to gound. That is, the wie and the peson ae two esistos in paallel. Since the wie has a much smalle esistance than a human body, almost all the cuent flows though the gound wie. 18
If a hai dye, fo example, is opeating nomally, the same cuent flows into the hot wie of the cicuit as flows out the neutal wie of the cicuit. Howeve, if cuent flows though a gound wie o a peson to gound, the neutal wie has less cuent in it than the hot wie. A simple device called a GFCI o gound fault cicuit inteupte compaes the cuent in the hot and neutal legs of a cicuit. If they eve become unequal, a switch, much like a cicuit beake, immediately opens. Do note, howeve, that if cuent flows fom the hot wie though you body, and back to neutal, the GFCI will do nothing. The heat of a GFCI is a diffeential tansfome, a small tansfome that poduces cuents in opposite diections fom the hot wie and fom the neutal wie. As long as the cuents in the hot and neutal wies ae the same, the net voltage in the thid banch of the diffeential tansfome is zeo. hot wie neutal wie solenoid switch Figue 13.18. A Gound Fault Cicuit Inteupte GFCI s can be incopoated into cicuit beakes; howeve, moe often they eplace a nomal outlet in a cicuit. A GFCI looks like a nomal outlet, except it has a TEST and a RESET button on its face. GFCI s should outinely be installed in kitchens, bathooms, and othe locations whee shock hazads ae high. Since a GFCI may need to be manually eset if thee ae gound faults o powe outages, it is impotant not to opeate life-suppot equipment on GFCI potected cicuits. GFCI s ae wied in seies. Note that all outlets downsteam fom a GFCI ae also potected (and need not be gounded), so you eally need only one GFCI pe cicuit. A simila device is an AFCI o ac fault cicuit inteupte. If insulation weas in a cicuit, an ac can esult fom a hot wie to a neutal o gound wie. Acs don t daw enough cuent to shut off a nomal cicuit beake, but they can poduce dangeous levels of heat. AFCI s ae 19
typically incopoated into cicuit beakes. Most building codes now equie AFCI cicuit beakes fo any cicuits that have bedoom outlets. Things to emembe: Know why gounding is impotant. Know what a GFCI is haw it opeates. Know whee GFCI s need to be installed and that they need to be installed in seies. Know why AFCI s ae used and whee AFCI cicuit beakes should be installed. 13.7. Waves: A Review In the next sections we ae going to lean moe about electomagnetic adiation. To undestand some of the teminology, howeve, we should fist eview a few things about waves. The electical field in an electomagnetic wave that vaies in both space and time has the faily geneal fom: (13.3) E ( x, t) E sin( kx ± ω t + φ) This is a sine wave moving in the m x diection whee: E is the value of the electic field at x and t. It is measued in volts/mete (V/m). E is the amplitude (maximum electic field stength) of the wave in V/m. π k is called the wavenumbe. k whee λ is the wavelength. Its units ae m 1. λ ω is the angula fequency of the wave. It is elated to fequency, f, and peiod, T, by the π expessions ω π f. It is measued in ad/s. T φ is the phase angle. It tells us how the wave stats. That is, at xt E (,) E sinφ. Fo ou puposes, we can let φ. mean. Let s look at the wave in a little moe detail to undestand what these diffeent quantities Equation (13.3) gives the electic field we measue at a position x and time t. This equation only tells us the magnitude of the electic field, not its diection (except fo + and signs). We know that sine vaies between 1 and +1, so the lagest value the wave can attain is the amplitude, E. Let s daw a pictue of a wave on a sting at a paticula time. (Waves on stings ae easie to visualize than electic field stength. Just emembe that electic field stength, E, woks just like the displacement of a wave on a sting.) We can think of this as a snapshot of the wave. Fo simplicity, let s choose the phase angle,φ, to be zeo and take the snapshot at time t. What we plot then is just y ( x) Asin( kx) with A 5 mm and k ad/sec.
Figue 13.19. A snapshot of a wave on a sting at time t. The fist thing we can do is ask whee the peaks of the wave occu. In tems of angle we know those ae at kx π / and π / + π. In tems of x, the peaks ae then at x π π π, +. k k k Since one wavelength, λ, is the distance between peaks, this means that : π λ o k k π. λ This gives us a vey useful elationship between the wavelength and the wavenumbe. Now, let s take a snapshot of the wave just a little late. The function of the wave is now y( x, t) Asin( kx ωt) with ω ad/sec and t.1 sec. We know that the effect of adding this tem is to tanslate the wave to the ight by an angle ωt. ad. This is bone out in Fig. 13.. 1
Figue 13.. The sine wave of Fig. 13.19 a moment late. As can be seen fom these figues, the wave is moving to the ight. We can tell how fast the wave moves by detemining how fa the wave moves in a time t. Since ω kx ω t k x t k the wave is tanslated to the ight by a distance: ω t vt k ω v. k Fom this, we can also see that a wave taveling to the lest must have the fom y( x, t) Asin( kx + ωt). Since waves ae functions of both space and time, we can also choose a paticula point along the wave, x, fo example, and plot the wave at that point as a function of time. Since an oscilloscope shows a voltage as a function of time, we can think of this kind of gaph as an oscilloscope tace of the wave. Mathematically, fo a wave going to the ight, the oscilloscope tace at x is y(, t) Asin( ωt). This is illustated in Fig. 13.1.
Fig. 13.1. An oscilloscope tace of the wave on a sting. Now we can again ask whee the peaks of the wave occu. We know these ae at angles ω t π / and π / + π. In tems of t, the peaks ae then at t π π π, +. ω ω ω Since one peiod, T, is the time between peaks, this means that : T π ω o ω π π f. T Things to emembe: E( x, t) E sin( kx ± ωt) This wave moves in the E is the amplitude. m x diection. π k is the wavenumbe. k. λ π ω is the angula fequency ω π f. T 3
13.8. Maxwell s Equations and Radiation as: In Lesson 11, we found that Maxwell s Equations could be in witten in diffeential fom Gauss s Law of Electicity E Gauss s Law of Magnetism Ampèe s Law µ j + Faaday s Law E µ t ρ ε ε E t Let s conside a egion of space whee thee ae no chages and no cuents, just fields. Fom Ampèe s Law, we see that if thee is a changing electic field, thee must also be a magnetic field with cul. Fom Faaday s Law, we see that if thee is a changing magnetic field, thee must also be an electic field with cul. This suggests that changing electic fields with cul and changing magnetic fields with cul must go hand-in-hand. The math is a little messy, but if you can take a step o two on faith, we can see what the mathematical implications of this ae: E µ ε t ( ) µ ε ( E) ( ) µ ε t t µ ε t t The cul of a cul is a athe messy thing, but an identity fom vecto calculus (hee you have to execise the faith unless you want to wok though the details which you can do if you don t mind quite a bit of algeba) can simplify this elationship. This identity tells us ( ) ( ) That doesn t look a lot bette, but we do know fom Gauss s Law of Magnetism that. With that we can simplify the expession to 4
5 t ε µ. While this may not look vey pomising, it s a beautiful thing to a mathematician it s the wave equation. This is the equation satisfied by a wave with a velocity of 1 ε µ v. In about 1864, James Clek Maxwell had just discoveed the Maxwell tem of Ampèe s Law, t E ε µ. Soon afte he did, Maxwell went though this little execise. He then plugged known values in fo the constants and discoveed that v c, the speed of light. Maxwell believed he had discoveed the secet of light. He was almost ight; he discoveed one secet of light: light is an electomagnetic wave. Since most of us ae not quite the mathematical genius that Maxwell was, let s go back and wok though some of the details, making use of some facts we know about electomagnetic waves. Fom ou study of the adiation of acceleating chages, we leaned that fa away fom the souce of adiation, the electic and magnetic fields ae pependicula and that E is in the diection the wave tavels. We also found that the amplitude of the magnetic field is 1/c the amplitude of the electic field. This suggest that a solution to Maxwell s equations would be: z t kx c E t x y t kx E t x E ) ˆ sin( ), ( ˆ ) sin( ), ( ω ω fo a wave taveling in the +x diection. Using the diffeential-opeato foms of divegence and cul, we can see if these esults do indeed satisfy Maxwell s equations. z y x x y z x y z z y x z y x z y x y x z x z y z y x cul z E y E x E E div E + + + + ˆ ˆ ˆ ˆ ˆ ˆ
E y E y z z ρ away z z E xˆ y y y + ˆ x ˆ c E ye ˆ ω sin( kx ω t) t k E E + µ ε ω c t t fom the k sin( kx ω t) souce E y E y E xˆ + zˆ ze ˆ k sin( kx ω t) z x E zˆ ω sin( kx ω t) t c kc ω 1 E as c ω t t k µ ε Theefoe, these functions do satisfy Maxwell s equations. Things to emembe: e sue you undestand the tems amplitude, wavenumbe, angula fequency, fequency, and peiod. The electic and magnetic fields ae pependicula to each othe and pependicula to the diection the wave tavels. The vecto E points in the diection of the velocity. The amplitude of the magnetic field is 1/c times the amplitude of the electic field. The electic and magnetic fields ae in phase. That is, if the electic field is lage and a given point in space and time, the magnetic field is also lage at that point. 1 Know that c. µ ε 13.9. Electomagnetic Radiation and Radio Waves Electomagnetic adiation can be poduced with any abitay fequency. We can poduce adiation at low fequencies by letting chages undego oscillatoy motion. Highe fequency adiation can be poduced by oscillating electic cicuits. eyond what we can poduce with cicuits, we can poduce even highe fequency adiation in atomic and nuclea tansitions o by acceleating vey high-enegy paticle beams. Whateve the fequency, we can deduce the wavelength by the impotant elation: 6
(13.4 wavelength-fequency elationship) c λ f whee: c is the speed of light. λ is the wavelength. f is the fequency. This elationship is moe o less intuitive. If twenty full wavelengths go past you in a second and each wave is 3 metes long, the velocity is 6 m/s. We can, howeve, deive the elationship fom esults of the pevious section. ω λ c πf λ f. k π Electomagnetic waves ae often classified by thei wavelength o fequency. The divisions between the types of adiation ae athe hazy, but the following is a useful table, nonetheless. Table 13.3. Electomagnetic Radiation Name Typical Souce Appoximate Wavelength Radio Oscillating cicuits >1 cm Micowave Electonic devices 1 m 1 cm Infaed Atoms, molecules 7 nm 1 m Visible Light Atoms 4 7 nm Ultaviolet Atoms 1 4 nm X-ays Inne shells of atoms 1pm 1nm Gamma-ays Nuclei < 1pm While each egion of the electomagnetic spectum is inteesting fo diffeent easons, we ae going to spend some time consideing adio waves as a special case. Radio waves can be ceated and detected by simple cicuits that we can constuct with the knowledge we have gained in this couse. (Of couse, many adio cicuits ae much moe complicated that the ones we will study in this section.) The basic pinciple of adio communications is to combine a signal with a caie wave, boadcast the wave, eceive the wave, and sepaate the caie and signal once again. Theefoe, we want to supeimpose an audio wave o a digital signal on a adio wave fo the wave to be useful fo communication. The simplest way we can do this is to tun the wave on and off and send a digital signal. The ealiest electomagnetic communications used Mose code in this 7
fashion. Moe fequently, howeve, we want to change some chaacteistic of the wave at a signal fequency, ωs. Usually the signal fequency is the fequency of a sound wave in speech o music. This fequency vaies in time, but much moe slowly that the adio wave vaies in time. The pocess of changing the adio wave at the signal fequency is called modulation. Thee ae thee pinciple types of modulation: amplitude modulation (AM), fequency modulation (FM), and phase modulation (PM). Let us begin by taking a high-fequency sine wave as ou basic caie wave. That is, when thee is a plain sine wave, no infomation is caied on the signal. Such a wave is shown in Fig. 13.. A typical caie wave has a fequency in the MHz ange. Figue 13.. A high fequency caie wave. The infomation we wish to convey is contained in a wave of much lowe fequency. Audio signals ae in the Hz khz ange. Figue 13.3. A lowe fequency audio signal. As the name implies, amplitude modulation changes the amplitude of the caie wave as a function of time. Mathematically, we may wite the modulated wave as: E( x, t) Asin( ω t)sin( k x ω t) whee k c ω / c. E(, t) Asin( ω t)sin c s s c ( ω t) c c ωc Asin( ωst)sin ( x ct) c A plot of this as a function of time fo x is shown in Fig. 13.4 below. 8
Figue 13.4. An AM wave with the audio signal supeimposed fo compaison. The second method of modulating the wave is to change the phase as a function of time. In amateu adio, phase modulation is fequently used. PM waves ae poduced by a device called a eactance modulato. The mathematical and gaphical epesentations of phase modulated wave ae shown below. ωc E( x, t) sin ( x ct) + Asin( ωst) c E(, t) sin ω t + Asin( ω t) [ ] c s Figue 13.5. A PM wave with the audio signal supeimposed fo compaison. The last kind of modulation, fequency modulation, is vey simila to phase modulation because wheneve we change the phase, we momentaily change the fequency as well. A fequency modulated wave has a highe fequency when the signal wave is lage, and a lowe fequency when the signal wave is small. It is illustated below: Figue 13.5. A PM wave with the audio signal supeimposed fo compaison. 9
This section is a little mathematical divesion. You can skip it, if you don t feel inclined to ead it. Mathematically, fequency modulation is a little moe subtle. It seems that all we would need to do is wite the fequency as the caie fequency modulated by the souce fequency:? ωc ( 1 + Asin( ωst)) E( x, t) sin ( x ct) c φ ut this does not wok. The poblem is that the wave at a specified time is given as the wave that would have been poduced had the fequency emained constant ove the entie inteval [,t]. y gaphing the function above, you can demonstate to youself that the fequency actually gets highe and highe in the couse of time. Instead, let s assume that we know the entie agument of the sine function we ll call it at some time. Since ϕ kx ωt, ϕ( t + t) ϕ( t) ω ( t) t ϕ( t + t) ϕ( t) d ϕ ω c ( t) t d t We haven t done anything to modulate the caie fequency yet. ut now we can simply conside the caie fequency to be a base fequency, ω, modulated by the signal fequency. That is, ω ( t) ω (1 + Asinω t) c d ϕ ω (1 + Asinωst) d t Aω ϕ( t) ω t + ω c s s cos( ω t) s To poduce a adio wave, all we have to do is acceleate electons back and foth along wies. The simplest fom of tansmission antenna is called a cente-fed dipole antenna, as illustated in Fig. 13.6. This type of antenna is typically constucted fom two equal lengths of bae wie connected to a coaxial cable fom an oscillating cicuit. (A coaxial cable has a hollow cylindical wie suound a nomal cylindical wie. Why doesn t the coaxial cable emit adiation?) The antenna adiates well if the length of each segment is chosen to be one-half of the wavelength of the adio wave, as measued in ai. 3
Fig. 13.6. A cente-fed dipole antenna. If the cicuit is aanged popely, we can set up a standing wave on the antenna, in much the same way that we can set up a standing wave on a sting. Note that, since cuent can not pass in o out of the ends of the antenna, the cuent at the ends must be zeo, in analogy to the amplitude of the sting in Fig. 13.7. Figue 13.7. A standing wave on a sting. If we take x to be the cente of the antenna and x ± L/ to be the ends, then we may wite fo the cuent in the antenna ω πx i( x, t) i cos ( cosω t) L π whee i is the maximum cuent in the antenna and f is the angula fequency of the oscillating cicuit. Futhemoe we choose L to be one-half wavelength, so c L λ f 31.
f. Hee we have made use of the elation c λ To eceive a adio wave, we can use a vey simila antenna, called a half-wavelength dipole antenna. The electic field of the adio wave causes electons to oscillate back and foth along the antenna. The antenna in the eceive cicuit is just like an AC powe supply. ut thee is one poblem: a adio antenna is constantly being bombaded with thousands of signals fom many diffeent souces. We need to be able to tune a adio to a given fequency. To do this, we just use a seies LRC cicuit. The capacito in the LRC cicuit has a high impedance fo lowfequency oscillations, and vey little cuent will flow. The inducto has a high impedance fo high-fequency oscillations. It s only when the caie wave is at the esonance fequency of a cicuit that much cuent can flow. The esonance fequency of the cicuit is usually adjusted by changing the capacitance of a vaiable capacito. antenna C R L Figue 13.8. An LRC cicuit fo eceiving adio signals. The aow though the capacito indicates that it s a vaiable capacito. Of couse, a esonating cicuit by itself isn t sufficient to poduce a sound on a speake. The voltage, say acoss the esisto, must be amplified and the caie wave sepaated fom the signal in ode to make a wokable adio. Things to emembe: Know the electomagnetic spectum. You don t need to emembe the wavelengths listed in the table, but you should emembe the names of the diffeent types of electomagnetic adiation and thei ode in the spectum. c λ f. Radio communication equies us to modulate a caie wave with a signal wave, tansmit the wave, eceive the wave, and then sepaate the signal fom the caie once again. e qualitatively familia with the thee methods of modulating caie waves: AM, FM, and PM. Cente-fed dipole antennas ae often used fo tansmission. Similaly, half-wavelength dipole antennas ae often used fo eception. The length of an antenna is geneally on the ode of one wavelength. A seies LRC with a vaiable capacito is used to tune the adio. The capacito adjusts the esonant fequency of the cicuit to match the tansmission fequency. 3
13.1. Caying Infomation on Electomagnetic Waves With moden demands on sending and eceiving moe and moe infomation in shote and shote times, it is vey impotant to undestand some of the limitations of electomagnetic tansmission. Two impotant paametes in detemining how much infomation can be put on waves ae fequency and bandwidth. 1. Fequency: In ode fo a caie wave to maintain its basic fequency so that it can be detected and tuned, the modulation fequency needs to be less than the caie fequency. In othe wods, it is difficult to put moe than about one bit of data on a full wavelength. Theefoe, the highe the fequency and hence the shote the peiod and the shote the wavelength the moe quickly data can be tansmitted.. andwidth: Tuning cicuits ae not pefect. If a second signal with nealy the same signal as the caie signal is picked up by an antenna, the tuning cicuit will not be able to filte it completely out and the signal will be muddled. You have pobably expeienced two adio stations coming in to you adio at the same time. If a wide ange of fequencies is available eithe by egulation o fo technical easons then multiple signals can be tansmitted simultaneously. The diffeence between the maximum and minimum fequencies that can be used is called the bandwidth. The numbe of signals that can tansmitted at the same time without intefeence depends on the chaacteistics of the tansmission and eceive cicuits as well as upon the bandwidth. In geneal FM signals equie moe bandwidth than AM signals because the fequency of the FM signal is modulated. Note that by extension of this meaning, the tem bandwidth is also used fo the numbe of bits pe second that can be tansmitted by any means. Things to emembe: The maximum ate at which infomation can be sent on a wave is about the same as the fequency. bandwidth is liteally the fequency ange ove which signals can be tansmitted. andwidth detemines the numbe of signals that can be tansmitted simultaneously. 13.11. Polaization If we think of the adiation of a single point chage oscillating sinusoidally, we can easily deduce the diection of the electic and magnetic fields in the wave by using the methods of Lesson 1. The one thing that makes the analysis a little complicated is that we have to think of the motion of the souce when the thead is emitted, not the motion of the souce when the thead aives at a field point. In Fig. 13.9 we wish to find the fields at point P die to a chage (we choose the neaest chage fo convenience) oscillating in the antenna. Let s assume that the theads aiving at P wee emitted fom the souce when the souce acceleation was in the +x diection. The vecto fom the souce to P is R. We know that the diection of the electic field is then Rˆ ( Rˆ aˆ ), which is to the left. The diection of the magnetic field is R ˆ E ˆ, out of the. (Wok these out youself to be sue you agee.) 33
E R P P P Figue 13.9. The fields of an oscillating souce. If we look at a point P a little fathe fom the wie, so that the theads eaching that point wee emitted when the oscillating souce was acceleating to the left, both the electic and magnetic fields ae evesed in diection. At a thid point a little fathe fom the wie still, the field diections evese once again. Fom this figue, we see that the fequency of the electomagnetic wave is the same as the souce s oscillation fequency and that the wavelength is λ ct c / f as we aleady have obseved. ut most impotantly, we note that the electic field is always oiented sideways. It gets lage o smalle, depending on position and time, but its diection is always sideways. Similaly the magnetic field is always diected in o out of the sceen. We define polaization to be the diection of the electic field in an electomagnetic wave. Unlike a nomal vecto diection, howeve, the polaization has two diections, such as up and down, sideways, in and out. This, of couse is due to the fact that the diection of the electic field is oscillating in time. Now what happens when we have many oscillatos moving in diffeent diections. This is what we have in light bulbs o light fom the sun. In Fig. 13.3, we see light coming towad us fom the sun. We know that the atoms in the sun that emitted the light oscillate andomly. Howeve, we also know that the electic field is pependicula to the diection of motion, so that thee can be no component of the electic field in o out of the sceen. Anothe way to say the same thing is that all the theads emitted by chages in the sun must lie in the plane of the sceen. This is illustated by the aows dawn in all diections. We say that light fom the sun is unpolaized. 34
Figue 13.3. Unpolaized light fom the sun. Now let s think of sunlight eflecting off a lake. Light comes fom the sun along the ay i. When the light stikes the lake, the electic field of the light causes electons in the wate to oscillate, just as adio waves cause electons in a eceiving antenna to oscillate. The electons in the wate effectively become little tansmitting antennas that e-adiate light to you eyes. These antennas emit adiation pefeentially in the plane pependicula to the line along which the wate molecules oscillate. Let s fist conside light that is polaized in the hoizontal diection, as shown in Fig. 13.31. The electic field of the sunlight causes electons on the lake s suface to oscillate in and out of the sceen, as indicated by the ed cicle on the wate s suface. These electons then adiate pimaily in the plane pependicula to thei oscillation; that is, in the plane of the sceen. Light fom the wate suface then stikes ou eyes with polaization in the hoizontal diection. i Figue 13.31. Reflected light polaized hoizontally. If light is polaized in the opposite diection we ll call it non-hoizontally then the electons in the wate oscillate in the diection of the ed aow in Fig. 13.3. These electons pimaily emit light in the plane pependicula to this aow. This means that vey little of the light polaized in this diection will get to ou eyes. 35
i Figue 13.3. Vey little light eflects when the polaization is in the opposite diection. The bottom line is that the light that eaches ou eyes is pefeentially polaized in the hoizontal diection. A moe detailed analysis tells us that the amount of polaization is lagest when the angle between in the incident and eflected ays is 9. Although it is believed that some animals ae sensitive to polaization diection, we don t eally notice the diffeence at all. If you have a pai of Polaoid sunglasses; howeve, you can detect polaization by otating you lenses and see if the intensity of the light vaies as you otate. Light can also be polaized by scatteing, such as when light is scatteed in the atmosphee. Light that comes fom the sky at an angle of 9 fom the sun is somewhat polaized. A athe simple way to detemine the diection of polaization in eflection and scatteing pocesses is to conside the polaization planes of the incident and eflected (o scatteed) ays. The polaization plane of a ay is the plane pependicula to the ay; that is, the plane in which it is possible fo the electic field of the light to point. The diection in which light is polaized is the intesection of the two polaization planes. This is depicted in Fig. 13.33. i polaization planes Figue 13.33. The intesection of the polaization planes on the suface tells us the pefeed diection of polaization. 36
Although light can be polaized by eflection o scatteing, a moe effective means of polaizing light is by passing it though cetain special mateials. These mateials come in two distinct types. The fist type is a biefingent cystal, such as calcite. ecause of peculiaities in the inteaction of light with the cystal lattice, light that passes though calcite sepaates into two diffeent ays with opposite polaization. Replace with a photogaph This This is is an an example of of some witten mateial seen seen though a calcite cystal. Figue 13.34. Witing viewed though a calcite cystal. The two sets of witing ae polaized in diffeent diections. The second way mateial can polaize light is though selective absoption. Some mateials with long molecules allow electons to oscillate up and down along the molecules, but allow vey little sideways motion of the electons. The pat of the wave that is polaized along the length of the molecules is absobed, as the light enegy is tansfeed to kinetic enegy of the electons. Light that is polaized in the opposite diection is not significantly affected. Such mateials ae called, polaizes, polaizing filtes, o Polaoid filtes afte the company that developed them. To undestand the physics of polaizes, we need a few facts: 1. Polaizes have an axis. The axis is defined to be the diection of polaization that can pass though the filte (as opposed to the diection that is absobed).. The electic field vecto of light enteing a polaize can be boken down into two components, the component along the axis and the component pependicula to the axis. 3. The component of the electic field that is paallel to the axis is all that can pass though the filte. 4. The intensity of light is popotional to the enegy in the electic field. As we leaned in Lesson 6, the enegy is in tun popotional to E. Hence, intensity is popotional to E. 5. When unpolaized light passes though a polaize, its intensity is cut in half. What happens if we have unpolaized light pass though two polaizes with thei axes 9 apat? This situation is illustated in Fig. 13.35. 37