2.4: Descartes’ Meditations - Doubt
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5 Descartes’ Meditations: Doubt
Meditations 6
MEDITATION I.
OF THE THINGS OF WHICH WE MAY DOUBT.
1. SEVERAL years have now elapsed since I first became aware that I had accepted, even from my youth, many false opinions for true, and that consequently what I afterward based on such principles was highly doubtful; and from that time I was convinced of the necessity of undertaking once in my life to rid myself of all the opinions I had adopted, and of commencing anew the work of building from the foundation, if I desired to establish a firm and abiding superstructure in the sciences. But as this enterprise appeared to me to be one of great magnitude, I waited until I had attained an age so mature as to leave me no hope that at any stage of life more advanced I should be better able to execute my design. On this account, I have delayed so long that I should henceforth consider I was doing wrong were I still to consume in deliberation any of the time that now remains for action. To-day, then, since I have opportunely freed my mind from all cares [and am happily disturbed by no passions], and since I am in the secure possession of leisure in a peaceable retirement, I will at length apply myself earnestly and freely to the general overthrow of all my former opinions.
2. But, to this end, it will not be necessary for me to show that the whole of these are false--a point, perhaps, which I shall never reach; but as even now my reason convinces me that I ought not the less carefully to withhold belief from what is not entirely certain and indubitable, than from what is manifestly false, it will be sufficient to justify the rejection of the whole if I shall find in each some ground for doubt. Nor for this purpose will it be necessary even to deal with each belief individually, which would be truly an endless labor; but, as the removal from below of the foundation necessarily involves the downfall of the whole edifice, I will at once approach the criticism of the principles on which all my former beliefs rested.
3. All that I have, up to this moment, accepted as possessed of the highest truth and certainty, I received either from or through the senses. I observed, however, that these sometimes misled us; and it is the part of prudence not to place absolute confidence in that by which we have even once been deceived.
4. But it may be said, perhaps, that, although the senses occasionally mislead us respecting minute objects, and such as are so far removed from us as to be beyond the reach of close observation, there are yet many other of their informations (presentations), of the truth of which it is manifestly impossible to doubt; as for example, that I am in this place, seated by the fire, clothed in a winter dressing gown, that I hold in my hands this piece of paper, with other intimations of the same nature. But how could I deny that I possess these hands and this body, and withal escape being classed with persons in a state of insanity, whose brains are so disordered and clouded by dark bilious vapors as to cause them pertinaciously to assert that they are monarchs when they are in the greatest poverty; or clothed [in gold] and purple when destitute of any covering; or that their head is made of clay, their body of glass, or that they are gourds? I should certainly be not less insane than they, were I to regulate my procedure according to examples so extravagant.
5. Though this be true, I must nevertheless here consider that I am a man, and that, consequently, I am in the habit of sleeping, and representing to myself in dreams those same things, or even sometimes others less probable, which the insane think are presented to them in their waking moments. How often have I dreamt that I was in these familiar circumstances, that I was dressed, and occupied this place by the fire, when I was lying undressed in bed? At the present moment, however, I certainly look upon this paper with eyes wide awake; the head which I now move is not asleep; I extend this hand consciously and with express purpose, and I perceive it; the occurrences in sleep are not so distinct as all this. But I cannot forget that, at other times I have been deceived in sleep by similar illusions; and, attentively considering those cases, I perceive so clearly that there exist no certain marks by which the state of waking can ever be distinguished from sleep, that I feel greatly astonished; and in amazement I almost persuade myself that I am now dreaming.
6. Let us suppose, then, that we are dreaming, and that all these particulars--namely, the opening of the eyes, the motion of the head, the forth- putting of the hands--are merely illusions; and even that we really possess neither an entire body nor hands such as we see. Nevertheless it must be admitted at least that the objects which appear to us in sleep are, as it were, painted representations which could not have been formed unless in the likeness of realities; and, therefore, that those general objects, at all events, namely, eyes, a head, hands, and an entire body, are not simply imaginary, but really existent. For, in truth, painters themselves, even when they study to represent sirens and satyrs by forms the most fantastic and extraordinary, cannot bestow upon them natures absolutely new, but can only make a certain medley of the members of different animals; or if they chance to imagine something so novel that nothing at all similar has ever been seen before, and such as is, therefore, purely fictitious and absolutely false, it is at least certain that the colors of which this is composed are real. And on the same principle, although these general objects, viz. [a body], eyes, a head, hands, and the like, be imaginary, we are nevertheless absolutely necessitated to admit the reality at least of some other objects still more simple and universal than these, of which, just as of certain real colors, all those images of things, whether true and real, or false and fantastic, that are found in our consciousness (cogitatio), are formed.
7. To this class of objects seem to belong corporeal nature in general and its extension; the figure of extended things, their quantity or magnitude, and their number, as also the place in, and the time during, which they exist, and other things of the same sort.
8. We will not, therefore, perhaps reason illegitimately if we conclude from this that Physics, Astronomy, Medicine, and all the other sciences that have for their end the consideration of composite objects, are indeed of a doubtful character; but that Arithmetic, Geometry, and the other sciences of the same class, which regard merely the simplest and most general objects, and scarcely inquire whether or not these are really existent, contain somewhat that is certain and indubitable: for whether I am awake or dreaming, it remains true that two and three make five, and that a square has but four sides; nor does it seem possible that truths so apparent can ever fall under a suspicion of falsity [or incertitude].
9. Nevertheless, the belief that there is a God who is all powerful, and who created me, such as I am, has, for a long time, obtained steady possession of my mind. How, then, do I know that he has not arranged that there should be neither earth, nor sky, nor any extended thing, nor figure, nor magnitude, nor place, providing at the same time, however, for [the rise in me of the perceptions of all these objects, and] the persuasion that these do not exist otherwise than as I perceive them ? And further, as I sometimes think that others are in error respecting matters of which they believe themselves to possess a perfect knowledge, how do I know that I am not also deceived each time I add together two and three, or number the sides of a square, or form some judgment still more simple, if more simple indeed can be imagined? But perhaps Deity has not been willing that I should be thus deceived, for he is said to be supremely good. If, however, it were repugnant to the goodness of Deity to have created me subject to constant deception, it would seem likewise to be contrary to his goodness to allow me to be occasionally deceived; and yet it is clear that this is permitted.
10. Some, indeed, might perhaps be found who would be disposed rather to deny the existence of a Being so powerful than to believe that there is nothing certain. But let us for the present refrain from opposing this opinion, and grant that all which is here said of a Deity is fabulous: nevertheless, in whatever way it be supposed that I reach the state in which I exist, whether by fate, or chance, or by an endless series of antecedents and consequents, or by any other means, it is clear (since to be deceived and to err is a certain defect ) that the probability of my being so imperfect as to be the constant victim of deception, will be increased exactly in proportion as the power possessed by the cause, to which they assign my origin, is lessened. To these reasonings I have assuredly nothing to reply, but am constrained at last to avow that there is nothing of all that I formerly believed to be true of which it is impossible to doubt, and that not through thoughtlessness or levity, but from cogent and maturely considered reasons; so that henceforward, if I desire to discover anything certain, I ought not the less carefully to refrain from assenting to those same opinions than to what might be shown to be manifestly false.
11. But it is not sufficient to have made these observations; care must be taken likewise to keep them in remembrance. For those old and customary opinions perpetually recur-- long and familiar usage giving them the right of occupying my mind, even almost against my will, and subduing my belief; nor will I lose the habit of deferring to them and confiding in them so long as I shall consider them to be what in truth they are, viz, opinions to some extent doubtful, as I have already shown, but still highly probable, and such as it is much more reasonable to believe than deny. It is for this reason I am persuaded that I shall not be doing wrong, if, taking an opposite judgment of deliberate design, I become my own deceiver, by supposing, for a time, that all those opinions are entirely false and imaginary, until at length, having thus balanced my old by my new prejudices, my judgment shall no longer be turned aside by perverted usage from the path that may conduct to the perception of truth. For I am assured that, meanwhile, there will arise neither peril nor error from this course, and that I cannot for the present yield too much to distrust, since the end I now seek is not action but knowledge.
12. I will suppose, then, not that Deity, who is sovereignly good and the fountain of truth, but that some malignant demon, who is at once exceedingly potent and deceitful, has employed all his artifice to deceive me; I will suppose that the sky, the air, the earth, colors, figures, sounds, and all external things, are nothing better than the illusions of dreams, by means of which this being has laid snares for my credulity; I will consider myself as without hands, eyes, flesh, blood, or any of the senses, and as falsely believing that I am possessed of these; I will continue resolutely fixed in this belief, and if indeed by this means it be not in my power to arrive at the knowledge of truth, I shall at least do what is in my power, viz, [ suspend my judgment ], and guard with settled purpose against giving my assent to what is false, and being imposed upon by this deceiver, whatever be his power and artifice. But this undertaking is arduous, and a certain indolence insensibly leads me back to my ordinary course of life; and just as the captive, who, perchance, was enjoying in his dreams an imaginary liberty, when he begins to suspect that it is but a vision, dreads awakening, and conspires with the agreeable illusions that the deception may be prolonged; so I, of my own accord, fall back into the train of my former beliefs, and fear to arouse myself from my slumber, lest the time of laborious wakefulness that would succeed this quiet rest, in place of bringing any light of day, should prove inadequate to dispel the darkness that will arise from the difficulties that have now been raised.
MEDITATION II.
OF THE NATURE OF THE HUMAN MIND; AND THAT IT IS MORE EASILY KNOWN THAN THE BODY.
1. The Meditation of yesterday has filled my mind with so many doubts, that it is no longer in my power to forget them. Nor do I see, meanwhile, any principle on which they can be resolved; and, just as if I had fallen all of a sudden into very deep water, I am so greatly disconcerted as to be unable either to plant my feet firmly on the bottom or sustain myself by swimming on the surface. I will, nevertheless, make an effort, and try anew the same path on which I had entered yesterday, that is, proceed by casting aside all that admits of the slightest doubt, not less than if I had discovered it to be absolutely false; and I will continue always in this track until I shall find something that is certain, or at least, if I can do nothing more, until I shall know with certainty that there is nothing certain. Archimedes, that he might transport the entire globe from the place it occupied to another, demanded only a point that was firm and immovable; so, also, I shall be entitled to entertain the highest expectations, if I am fortunate enough to discover only one thing that is certain and indubitable.[
2. I suppose, accordingly, that all the things which I see are false (fictitious); I believe that none of those objects which my fallacious memory represents ever existed; I suppose that I possess no senses; I believe that body, figure, extension, motion, and place are merely fictions of my mind. What is there, then, that can be esteemed true ? Perhaps this only, that there is absolutely nothing certain.
3. But how do I know that there is not something different altogether from the objects I have now enumerated, of which it is impossible to entertain the slightest doubt? Is there not a God, or some being, by whatever name I may designate him, who causes these thoughts to arise in my mind ? But why suppose such a being, for it may be I myself am capable of producing them? Am I, then, at least not something? But I before denied that I possessed senses or a body; I hesitate, however, for what follows from that? Am I so dependent on the body and the senses that without these I cannot exist? But I had the persuasion that there was absolutely nothing in the world, that there was no sky and no earth, neither minds nor bodies; was I not, therefore, at the same time, persuaded that I did not exist? Far from it; I assuredly existed, since I was persuaded. But there is I know not what being, who is possessed at once of the highest power and the deepest cunning, who is constantly employing all his ingenuity in deceiving me. Doubtless, then, I exist, since I am deceived; and, let him deceive me as he may, he can never bring it about that I am nothing, so long as I shall be conscious that I am something. So that it must, in fine, be maintained, all things being maturely and carefully considered, that this proposition (pronunciatum ) I am, I exist , is necessarily true each time it is expressed by me, or conceived in my mind.


1.9: Chloramination and Nitrification
Learning Objectives
After reading this chapter you should be able to explain and identify the following topics:
- Why chloramine treatment has been employed, (hint) DBP formation
- The use of ammonia in water treatment
- Breakpoint chlorination
- Nitrification
The water treatment profession is constantly evolving and continuously in motion. Advancements in science and lab procedures help water professionals find constituents in drinking water that can be harmful to human beings. We take for granted the fact that clean drinking water will flow when we turn on the tap.
Currently, chlorine is the best disinfectant available. However, the use of chlorine causes any number of disinfectant by-products (DBPs). Many of the DBPs have not been researched and are not regularly monitored. As screening methods improve, it is likely that more of the possible DBPs will be regulated in the future. Trihalomethanes (THMs) are a DBP of chlorine disinfection and are classified as volatile organic chemicals. Rising THM levels in raw water are increasing the use of chloramine as the preferred disinfectant. In this chapter, we will discuss chloramine disinfection and the associated challenges that come with its implementation. Nothing is easy in water treatment.
Why Use Chloramines?
As discussed, chlorine has been used with great effectiveness in water treatment for many years in the United States. However recently, many water agencies have been switching to Chloramine treatment. Now here is the twist; they have done it because they must, not because they necessarily want to. When natural organic matter reacts with chlorine, it can result in the formation of DBPs, specifically trihalomethanes (THM) and haloacetic acids (HAA5). It is believed that long term exposure to these DBPs should be avoided as it may lead to cancer.
Because of the implementation of the Stage 2 DBP Rule in 2012, many water agencies could no longer meet the regulatory limits established, if they continued using chlorine as a primary disinfectant. To continue using chlorine alone, enhanced coagulation would have to be used but its effectiveness is limited. Carbon or GAC absorption is also a solution but it is very expensive, and thus not employed by larger water treatment facilities. The other option is to limit contact with the water and chlorine, but this is limited by the chlorine demand. The last and most effective option is to use chloramines.
Chloramine treatment is also effective in limiting taste and odor problems in finished water. When free chlorine reacts with phenol it can cause taste and odor issues. Chloramine disinfection is not as strong a disinfectant as free chlorine, chlorine dioxide, or ozone, but it does leave the longest-lasting residual in finished water. Because of chloramines weaker disinfecting power, it is often added just before the treated water enters the clearwell for storage. If implementing this method, the operator must ensure the THM levels are low enough as not to violate any regulations. Determining the most effective way to treat drinking water while remaining in compliance is a complex and difficult task.
Chloramine Chemistry
Chlorine reacts with a number of substances in water including dissolved organic matter, particulate organic matter, iron, nitrite, sulfide, and ammonia. It reacts by taking away an electron from them. Ammonia in the water supply is undesirable as it strips away the Chlorine residual which then requires the need to use more Chlorine. What makes chloramine treatment so effective is chlorine’s quick formation with ammonia (NH 3 ). (Ammonia is a compound of Nitrogen and Hydrogen.)
\[\ce{Cl2 + NH3 → NH2Cl + HCl}\]
Note that monochloramine is expressed as \(\ce{NH2Cl.}\)
The unit weight of Chlorine is 70 and the unit weight of Nitrogen is 14. When chloramine treatment is employed, the ratio of chlorine to ammonia will always be 5:1 because 5 mg/L of chlorine will always combine with 1.0 mg/L of ammonia. (70 ÷ 14 = 5) You will often hear monochloramines referred to as combined chlorine due to this reaction.
Below is a list of terms dealing with Chloramine disinfection:
- Free Chlorine: Cl 2 (Free chlorine can refer to gaseous, sodium hypochlorite, or calcium hypochlorite.) If using hypochlorite, remember you must dose the water to achieve 100 percent free chlorine.
- Monochloramine: NH 2 Cl or combined chlorine. Chloramine with the least taste and odor problems.
- Free Ammonia: NH 3 Ammonia is measured as nitrogen
- Chlorine: Ammonia ratio: Total chlorine to total ammonia nitrogen. You will always target a 5:1 ratio to avoid excess free chlorine or free nitrogen in the water supply.
Breakpoint Chlorination
The breakpoint chlorination curve is the visual representation of chlorine’s ability to react with a variety of compounds to form a combined chlorine residual or to completely react with compounds to form a free chlorine residual. When using free available chlorine as a disinfectant we want to stay out of the combined curve. Some water agencies buy imported water that has a combined residual. To treat the water more effectively or eliminate the possibility of nitrification, some of these agencies add chlorine to “break” over to a free chlorine residual. We will discuss Nitrification in greater detail later in this chapter.
A common problem that operators run into when using combined chlorine is “breaking over” to free chlorine by misdiagnosing problems on the treatment plant. For example, an operator is looking at a Supervisory Control And Data Acquisition (SCADA) screen. (SCADA is the computer system that monitors and controls the plant.) For the sake of ease, we will call this operator #1. He or she notices the chlorine effluent residual is beginning to drop. Operator #1 was not aware that the previous operator (Operator #2) had raised the chlorine dose a few hours before. With the added chlorine dose, the chlorine residual was “lowering” as it got closer to breaking over. If operator #1 panics and raises the chlorine dose again the combined chlorine residual will lower even further.
The important thing to remember is there are many things to look at if your chlorine dose is lowering. It may very well be an issue with your chlorine feed, but there are many other options to consider. If you check your chlorine feed system and everything is functioning normally, you may have an issue with the ammonia feed system. Many modern water treatment facilities have calculations built into the SCADA system that make chloramine dosing easier. Make sure to go to the screen with the chlorine: ammonia: ratio to make sure that it is set up correctly.
Lastly, free chlorine is always the preferred disinfectant as it is 25 times stronger a disinfectant than chloramines. However, chloramines last much longer in the distribution system. There are several factors that may lead to a lowered chlorine residual. An increase in biological growth is possible, but you should see other signs such as an increase in turbidity. The most common problem in chloramine treatment with a decreasing chlorine residual is operator error or an issue with the water treatment process.
Nitrification
Nitrification is a process in which bacteria reduces ammonia and organic nitrogen in treated water into nitrate and then nitrite. This condition usually occurs during the summer months when water temperatures are higher. It can also occur during the improper turnover of reservoirs, when there are high levels of ammonia, and in dark environments, a typical condition for most water reservoirs.
This condition is one of the drawbacks of chloramine treatment. Nitrate and Nitrite are inorganic chemicals regulated by primary drinking water standards. The MCL (Maximum Contaminate Level) for Nitrate is 10 ppm and the MCL for Nitrite is 1 ppm. High nitrate and nitrite levels in drinking water can lead to methemoglobinemia also known as “blue baby syndrome.” This condition affects infants under six months old that consume water contaminated with nitrates and nitrites. The infant's blood is not able to carry enough oxygen to blood cells and tissue within their body.
The condition of Nitrification is a reoccurring cycle that can lead to the total loss of chloramine residual in the distribution system and in water storage tanks. If this was to occur, all the water in the system would have to be dumped and it would be a total loss. As an operator, this is something that you never want to happen. Not only would it be a loss of money for your company, but it could lead to an outbreak of methemoglobinemia. The cycle is below:
Chloramine decay → Release of ammonia→ oxidation of ammonia to nitrite from oxidizing bacteria → Biomass increase → (back to chloramine decay until there is a possibility of total loss of residual)
Some early indicators of nitrification:
- Lowering chloramine residual
- Increase in bacterial heterotrophic plate counts
- Excess ammonia in the treated water
- Total coliform positive samples
In summation, nitrification can cause a variety of problems and violations within the water system. What can we as operators do to minimize any chance of nitrification from occurring? There are a few steps every operator can take to avoid nitrification. First, you want to minimize free ammonia in treated water. Sampling and equipment around the treatment plant will help operators determine if there is too much free ammonia in the treated water.
Second, operators should always maintain a good chlorine residual. Isn’t that the name of the game? Sure, but it is easier said than done in some cases. Many water treatment companies will lower their target chlorine residual in the summer to save on costs because the disinfectant works more efficiently during the summer months. It is a good practice in theory, but not necessarily the best practice if you are working with chloramines and trying to combat nitrification.
Reducing water age is the third step operators can use to combat nitrification. This can be done by cycling reservoirs during the summer months and taking reservoirs out of service during the winter when water demands are lower. Distribution operators are continually balancing maintaining a good amount of water for fire flow and demand, but also cycling reservoirs to fight nitrification. A good practice is to take all the reservoirs in the distribution system and divide it into thirds. At any given point during the day 1/3 of the reservoirs are full, 1/3 are in the middle range, and 1/3 are in the lower range. This allows operators to cycle water while also keeping water in the system.
The last step is to keep the water system clean. This can be accomplished by flushing dead ends to prevent lower chlorine residuals at the end of the distribution system. Boosting chlorine by adding it directly to a reservoir is also an option but not always the best practice. The problem with this method is it could lead to short circuiting. Chlorine must mix adequately to be effective. Mixing can be accomplished by adding mechanical mixers in the reservoir to keep water moving and prevent short circuiting. Chlorine can also be added to the inlet of the reservoir so the water pressure acts as a mixer.
Hydrant flushing is a method that can be used to get rid of stagnant water at the ends of a distribution system. Stagnant water can lead to lower chlorine residuals and possible nitrification.
In summary, the use of chloramines as a disinfectant is being employed by many water agencies in the United States to prevent DBP formation. It is not the most ideal disinfectant available, but it does have some advantages. Chloramine disinfection limits DBP formation and has a longer-lasting disinfectant residual. This can be useful for larger systems where water must travel great distances to get to storage.
The drawbacks include the difficulty in effectively managing the breakpoint chlorination curve and managing nitrification. Because of advancements in water quality testing and more stringent regulations with DBPs, chloramine treatment is and will continue to be a more popular option in water treatment.
Chapter Review
-
What is the target chlorine to ammonia ratio?
- 2:1
- 3:1
- 4:1
- 5:1
-
What is the MCL for nitrates?
- 1 ppm
- 10 ppm
- 5 ppm
- None of the above
-
What is the atomic weight of chlorine?
- 70
- 14
- 65
- 20
-
What disinfectant has the longest-lasting residual?
- Ozone
- Chlorine
- Chloramine
- Chlorine dioxide
-
What are some of the early indicators of nitrification?
- Lowering chlorine residual
- Excess ammonia in treated water
- Raise in bacterial heterotrophic plate counts
- All of the above
-
What are THMs classified as?
- Turbidity
- Radiological
- Volatile organic chemicals
- Salts
-
What method can operators employ to combat nitrification?
- Lower residual chlorine target
- Keep reservoir levels static
- Minimize free ammonia in treated water
- Increase water age
-
How many times stronger is Chlorine compared to monochloramine?
- 25 times
- 20 times
- 15 times
- 5 times


12.1D: Vaccine Safety
- Describe the possible side effects of vaccine administration
Vaccines are biological products with biological effects. Vaccines are made with a variety of ingredients including antigens, stabilizers, adjuvants, and preservatives; they may also contain residual by-products from the production process. These might be the cause of allergic reactions and side effects.
Vaccines carry risks, ranging from rashes or tenderness at the site of injection to fever-associated seizures called febrile convulsions and dangerous infections in those with compromised immune systems. Technological advances have made modern vaccines purer and safer than their historical counterparts. Most developed countries have switched to the inactivated polio vaccine and stopped using whole-cell pertussis (whooping cough) vaccines, which are made from killed bacteria and cause relatively high rates of arm swelling, febrile convulsions and periods of limpness or unresponsiveness.
Researchers have long known that some individuals are more susceptible to vaccine risks than others. Immunocompromised individuals have generally been discouraged from receiving live-virus vaccines. Some speculate that children with metabolic disorders might be prone to vaccine side effects. Safer vaccines and manufacturing processes are also in the works. New influenza vaccine doses are produced in cell culture, rather than the industry-standard chicken eggs. This process will improve reliability and reduce allergic reactions to egg proteins. Researchers are also developing replacements for vaccines that can be risky for vulnerable groups. These include current smallpox vaccines that cannot safely be given to immunocompromised people; the tuberculosis vaccine, which is not recommended for HIV-positive infants; and the yellow-fever vaccine, which puts elderly people at particular risk of a yellow-fever-like illness. Researchers are quick to emphasize that the benefits of vaccines still greatly outweigh the risks.
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Key Points
- Vaccines can cause side effects in immunocompromised people, and allergic reactions due to the ingredients used to make them stable.
- Side effects from vaccines are rare and most reported ailments after vaccination have not been proven to be caused by vaccination (e.g. autism, asthma).
- Researchers continue to ameliorate vaccines composition making them safer and effective.
Key Terms
- convulsion : An intense, paroxysmal, involuntary muscular contraction.
- subunit vaccine : a vaccine that presents an antigen to the immune system without introducing viral particles, whole or otherwise
- limpness : Property of being limp.


18.4: Diameters of Stars
By the end of this section, you will be able to:
- Describe the methods used to determine star diameter s
- Identify the parts of an eclipsing binary star light curve that correspond to the diameters of the individual components
It is easy to measure the diameter of the Sun. Its angular diameter—that is, its apparent size on the sky—is about 1/2°. If we know the angle the Sun takes up in the sky and how far away it is, we can calculate its true (linear) diameter, which is 1.39 million kilometers, or about 109 times the diameter of Earth.
Unfortunately, the Sun is the only star whose angular diameter is easily measured. All the other stars are so far away that they look like pinpoints of light through even the largest ground-based telescopes. (They often seem to be bigger, but that is merely distortion introduced by turbulence in Earth’s atmosphere.) Luckily, there are several techniques that astronomers can use to estimate the sizes of stars.
Stars Blocked by the Moon
One technique, which gives very precise diameters but can be used for only a few stars, is to observe the dimming of light that occurs when the Moon passes in front of a star. What astronomers measure (with great precision) is the time required for the star’s brightness to drop to zero as the edge of the Moon moves across the star’s disk. Since we know how rapidly the Moon moves in its orbit around Earth, it is possible to calculate the angular diameter of the star. If the distance to the star is also known, we can calculate its diameter in kilometers. This method works only for fairly bright stars that happen to lie along the zodiac, where the Moon (or, much more rarely, a planet) can pass in front of them as seen from Earth.
Eclipsing Binary Stars
Accurate sizes for a large number of stars come from measurements of eclipsing binary star systems, and so we must make a brief detour from our main story to examine this type of star system. Some binary stars are lined up in such a way that, when viewed from Earth, each star passes in front of the other during every revolution (Figure 18.10). When one star blocks the light of the other, preventing it from reaching Earth, the brightness of the system decreases, and astronomers say that an eclipse has occurred.
The discovery of the first eclipsing binary helped solve a long-standing puzzle in astronomy. The star Algol, in the constellation of Perseus, changes its brightness in an odd but regular way. Normally, Algol is a fairly bright star, but at intervals of 2 days, 20 hours, 49 minutes, it fades to one-third of its regular brightness. After a few hours, it brightens to normal again. This effect is easily seen, even without a telescope, if you know what to look for.
In 1783, a young English astronomer named John Goodricke (1764–1786) made a careful study of Algol (see the feature on John Goodricke for a discussion of his life and work). Even though Goodricke could neither hear nor speak, he made a number of major discoveries in the 21 years of his brief life. He suggested that Algol’s unusual brightness variations might be due to an invisible companion that regularly passes in front of the brighter star and blocks its light. Unfortunately, Goodricke had no way to test this idea, since it was not until about a century later that equipment became good enough to measure Algol’s spectrum.
In 1889, the German astronomer Hermann Vogel (1841–1907) demonstrated that, like Mizar, Algol is a spectroscopic binary. The spectral lines of Algol were not observed to be double because the fainter star of the pair gives off too-little light compared with the brighter star for its lines to be conspicuous in the composite spectrum. Nevertheless, the periodic shifting back and forth of the brighter star’s lines gave evidence that it was revolving about an unseen companion. (The lines of both components need not be visible for a star to be recognized as a spectroscopic binary.)
The discovery that Algol is a spectroscopic binary verified Goodricke’s hypothesis. The plane in which the stars revolve is turned nearly edgewise to our line of sight, and each star passes in front of the other during every revolution. (The eclipse of the fainter star in the Algol system is not very noticeable because the part of it that is covered contributes little to the total light of the system. This second eclipse can, however, be detected by careful measurements.)
Any binary star produces eclipses if viewed from the proper direction, near the plane of its orbit, so that one star passes in front of the other (see Figure 18.10). But from our vantage point on Earth, only a few binary star systems are oriented in this way.
The name Algol comes from the Arabic Ras al Ghul , meaning “the demon’s head.” 3 The word “ghoul” in English has the same derivation. As discussed in Observing the Sky: The Birth of Astronomy, many of the bright stars have Arabic names because during the long dark ages in medieval Europe, it was Arabic astronomers who preserved and expanded the Greek and Roman knowledge of the skies. The reference to the demon is part of the ancient Greek legend of the hero Perseus, who is commemorated by the constellation in which we find Algol and whose adventures involve many of the characters associated with the northern constellations.
Perseus was one of the many half-god heroes fathered by Zeus (Jupiter in the Roman version), the king of the gods in Greek mythology. Zeus had, to put it delicately, a roving eye and was always fathering somebody or other with a human maiden who caught his fancy. (Perseus derives from Per Zeus , meaning “fathered by Zeus.”)  The king there, taking an interest in Perseus’ mother, tried to get rid of the young man by assigning him an extremely difficult task.
In a moment of overarching pride, a beautiful young woman named Medusa had compared her golden hair to that of the goddess Athena (Minerva for the Romans). The Greek gods did not take kindly to being compared to mere mortals, and Athena turned Medusa into a gorgon: a hideous, evil creature with writhing snakes for hair and a face that turned anyone who looked at it into stone. Perseus was given the task of slaying this demon, which seemed like a pretty sure way to get him out of the way forever.
But because Perseus had a god for a father, some of the other gods gave him tools for the job, including Athena’s reflective shield and the winged sandals of Hermes (Mercury in the Roman story). By flying over her and looking only at her reflection, Perseus was able to cut off Medusa’s head without ever looking at her directly. Taking her head (which, conveniently, could still turn onlookers to stone even without being attached to her body) with him, Perseus continued on to other adventures.
He next came to a rocky seashore, where boasting had gotten another family into serious trouble with the gods. Queen Cassiopeia had dared to compare her own beauty to that of the Nereids, sea nymphs who were daughters of Poseidon (Neptune in Roman mythology), the god of the sea. Poseidon was so offended that he created a sea-monster named Cetus to devastate the kingdom. King Cepheus, Cassiopeia’s beleaguered husband, consulted the oracle, who told him that he must sacrifice his beautiful daughter Andromeda to the monster.
When Perseus came along and found Andromeda chained to a rock near the sea, awaiting her fate, he rescued her by turning the monster to stone. (Scholars of mythology actually trace the essence of this story back to far-older legends from ancient Mesopotamia, in which the god-hero Marduk vanquishes a monster named Tiamat. Symbolically, a hero like Perseus or Marduk is usually associated with the Sun, the monster with the power of night, and the beautiful maiden with the fragile beauty of dawn, which the Sun releases after its nightly struggle with darkness.)
Many of the characters in these Greek legends can be found as constellations in the sky, not necessarily resembling their namesakes but serving as reminders of the story. For example, vain Cassiopeia is sentenced to be very close to the celestial pole, rotating perpetually around the sky and hanging upside down every winter. The ancients imagined Andromeda still chained to her rock (it is much easier to see the chain of stars than to recognize the beautiful maiden in this star grouping). Perseus is next to her with the head of Medusa swinging from his belt. Algol represents this gorgon head and has long been associated with evil and bad fortune in such tales. Some commentators have speculated that the star’s change in brightness (which can be observed with the unaided eye) may have contributed to its unpleasant reputation, with the ancients regarding such a change as a sort of evil “wink.”
Diameters of Eclipsing Binary Stars
We now turn back to the main thread of our story to discuss how all this can be used to measure the sizes of stars. The technique involves making a light curve of an eclipsing binary, a graph that plots how the brightness changes with time. Let us consider a hypothetical binary system in which the stars are very different in size, like those illustrated in Figure 18.11. To make life easy, we will assume that the orbit is viewed exactly edge-on.
Even though we cannot see the two stars separately in such a system, the light curve can tell us what is happening. When the smaller star just starts to pass behind the larger star (a point we call first contact ), the brightness begins to drop. The eclipse becomes total (the smaller star is completely hidden) at the point called second contact . At the end of the total eclipse ( third contact ) , the smaller star begins to emerge. When the smaller star has reached last contact , the eclipse is completely over.
To see how this allows us to measure diameters, look carefully at Figure 18.11. During the time interval between the first and second contacts, the smaller star has moved a distance equal to its own diameter. During the time interval from the first to third contacts, the smaller star has moved a distance equal to the diameter of the larger star. If the spectral lines of both stars are visible in the spectrum of the binary, then the speed of the smaller star with respect to the larger one can be measured from the Doppler shift. But knowing the speed with which the smaller star is moving and how long it took to cover some distance can tell the span of that distance—in this case, the diameters of the stars. The speed multiplied by the time interval from the first to second contact gives the diameter of the smaller star. We multiply the speed by the time between the first and third contacts to get the diameter of the larger star.
In actuality, the situation with eclipsing binaries is often a bit more complicated: orbits are generally not seen exactly edge-on, and the light from each star may be only partially blocked by the other. Furthermore, binary star orbits, just like the orbits of the planets, are ellipses, not circles. However, all these effects can be sorted out from very careful measurements of the light curve.
The Eclipsing Binary Simulator allows you to explore how the eclipse timing can be used to determine the size of stars in a binary pair. Other traits can be explored as well, such as their mass, separation, and surface temperatures.
Using the Radiation Law to Get the Diameter
Another method for measuring star diameters makes use of the Stefan-Boltzmann law for the relationship between energy radiated and temperature (see Radiation and Spectra). In this method, the energy flux (energy emitted per second per square meter by a blackbody, like the Sun) is given by
where σ is a constant and T is the temperature. The surface area of a sphere (like a star) is given by
The luminosity ( L ) of a star is then given by its surface area in square meters times the energy flux:
Previously, we determined the masses of the two stars in the Sirius binary system. Sirius gives off 8200 times more energy than its fainter companion star, although both stars have nearly identical temperatures. The extremely large difference in luminosity is due to the difference in radius, since the temperatures and hence the energy fluxes for the two stars are nearly the same. To determine the relative sizes of the two stars, we take the ratio of the corresponding luminosities:
Therefore, the relative sizes of the two stars can be found by taking the square root of the relative luminosity. Since , the radius of Sirius is 91 times larger than the radius of its faint companion.
The method for determining the radius shown here requires both stars be visible, which is not always the case.
Use the Stellar Luminosity Simulato to explore the relationship between a star’s surface temperature, luminosity, and radius. Move the sliders to see what happens. Try to make two stars with the same luminosity but different surface temperatures.
Stellar Diameters
The results of many stellar size measurements over the years have shown that most nearby stars are roughly the size of the Sun, with typical diameters of a million kilometers or so. Faint stars, as we might have expected, are generally smaller than more luminous stars. However, there are some dramatic exceptions to this simple generalization.
A few of the very luminous stars, those that are also red (indicating relatively low surface temperatures), turn out to be truly enormous. These stars are called, appropriately enough, giant stars or supergiant stars . An example is Betelgeuse , the second brightest star in the constellation of Orion and one of the dozen brightest stars in our sky. Its diameter, remarkably, is greater than 10 AU (1.5 billion kilometers!), large enough to fill the entire inner solar system almost as far out as Jupiter. In Stars from Adolescence to Old Age, we will look in detail at the evolutionary process that leads to the formation of such giant and supergiant stars.
Watch this star size comparison video for a striking visual that highlights the size of stars versus planets and the range of sizes among stars.
Footnotes
- 3 Fans of Batman comic books and movies will recognize that this name was given to an archvillain in the series.


13.1: Part-Part-Whole Ratios
Lesson
Let's look at situations where you can add the quantities in a ratio together.
Exercise \(\PageIndex{1}\): True or False: Mulltiplying by a Unit Fraction
True or false?
\(\frac{1}{5}\cdot 45=\frac{45}{5}\)
\(\frac{1}{5}\cdot 20=\frac{1}{4}\cdot 24\)
\(42\cdot\frac{1}{6}=\frac{1}{6}\cdot 42\)
\(486\cdot\frac{1}{12}=\frac{480}{12}+\frac{6}{12}\)
Exercise \(\PageIndex{2}\): Cubes of Paint
A recipe for maroon paint says, “Mix 5 ml of red paint with 3 ml of blue paint.”
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Use snap cubes to represent the amounts of red and blue paint in the recipe. Then, draw a sketch of your snap-cube representation of the maroon paint.
- What amount does each cube represent?
- How many milliliters of maroon paint will there be?
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-
Suppose each cube represents 2 ml. How much of each color paint is there?
Red: _______ ml
Blue: _______ ml
Maroon: _______ ml -
Suppose each cube represents 5 ml. How much of each color paint is there?
Red: _______ ml
Blue: _______ ml
Maroon: _______ ml
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Suppose each cube represents 2 ml. How much of each color paint is there?
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-
Suppose you need 80 ml of maroon paint. How much red and blue paint would you mix? Be prepared to explain your reasoning.
Red: _______ ml
Blue: _______ ml
Maroon: 80 ml - If the original recipe is for one batch of maroon paint, how many batches are in 80 ml of maroon paint?
-
Suppose you need 80 ml of maroon paint. How much red and blue paint would you mix? Be prepared to explain your reasoning.
Exercise \(\PageIndex{3}\): Sneakers, Chicken, and Fruit Juice
Solve each of the following problems and show your thinking. If you get stuck, consider drawing a tape diagram to represent the situation.
- The ratio of students wearing sneakers to those wearing boots is 5 to 6. If there are 33 students in the class, and all of them are wearing either sneakers or boots, how many of them are wearing sneakers?
- A recipe for chicken marinade says, “Mix 3 parts oil with 2 parts soy sauce and 1 part orange juice.” If you need 42 cups of marinade in all, how much of each ingredient should you use?
- Elena makes fruit punch by mixing 4 parts cranberry juice to 3 parts apple juice to 2 parts grape juice. If one batch of fruit punch includes 30 cups of apple juice, how large is this batch of fruit punch?
Are you ready for more?
Using the recipe from earlier, how much fruit punch can you make if you have 50 cups of cranberry juice, 40 cups of apple juice, and 30 cups of grape juice?
Exercise \(\PageIndex{4}\): Invent Your Own Ratio Problem
- Invent another ratio problem that can be solved with a tape diagram and solve it. If you get stuck, consider looking back at the problems you solved in the earlier activity.
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Create a visual display that includes:
- The new problem that you wrote, without the solution.
- Enough work space for someone to show a solution.
- Trade your display with another group, and solve each other’s problem. Include a tape diagram as part of your solution. Be prepared to share the solution with the class.
- When the solution to the problem you invented is being shared by another group, check their answer for accuracy.
Summary
A tape diagram is another way to represent a ratio. All the parts of the diagram that are the same size have the same value.
For example, this tape diagram represents the ratio of ducks to swans in a pond, which is \(4:5\).
The first tape represents the number of ducks. It has 4 parts.
The second tape represents the number of swans. It has 5 parts.
There are 9 parts in all, because \(4+5=9\).
Suppose we know there are 18 of these birds in the pond, and we want to know how many are ducks.
The 9 equal parts on the diagram need to represent 18 birds in all. This means that each part of the tape diagram represents 2 birds, because \(18\div 9=2\).
There are 4 parts of the tape representing ducks, and \(4\cdot 2=8\), so there are 8 ducks in the pond.
Glossary Entries
Definition: Tape Diagram
A tape diagram is a group of rectangles put together to represent a relationship between quantities.
For example, this tape diagram shows a ratio of 30 gallons of yellow paint to 50 gallons of blue paint.
If each rectangle were labeled 5, instead of 10, then the same picture could represent the equivalent ratio of 15 gallons of yellow paint to 25 gallons of blue paint.
Practice
Exercise \(\PageIndex{5}\)
Here is a tape diagram representing the ratio of red paint to yellow paint in a mixture of orange paint.
- What is the ratio of yellow paint to red paint?
- How many total cups of orange paint will this mixture yield?
Exercise \(\PageIndex{6}\)
At the kennel, the ratio of cats to dogs is \(4:5\). There are 27 animals in all. Here is a tape diagram representing this ratio.
- What is the value of each small rectangle?
- How many dogs are at the kennel?
- How many cats are at the kennel?
Exercise \(\PageIndex{7}\)
Last month, there were 4 sunny days for every rainy day. If there were 30 days in the month, how many days were rainy? Explain your reasoning. If you get stuck, consider using a tape diagram.
Exercise \(\PageIndex{8}\)
Noah entered a 100-mile bike race. He knows he can ride 32 miles in 160 minutes. At this rate, how long will it take him to finish the race? Use each table to find the answer. Next, explain which table you think works better in finding the answer.
Table A:
| distance (miles) | elapsed time (minutes) |
|---|---|
| \(32\) | \(160\) |
| \(1\) | |
| \(100\) |
Table B:
| distance (miles) | elapsed time (minutes) |
|---|---|
| \(32\) | \(160\) |
| \(96\) | |
| \(4\) | |
| \(100\) |
(From Unit 2.4.2)
Exercise \(\PageIndex{9}\)
A cashier worked an 8-hour day, and earned $58.00. The double number line shows the amount she earned for working different numbers of hours. For each question, explain your reasoning.
- How much does the cashier earn per hour?
- How much does the cashier earn if she works 3 hours?
(From Unit 2.4.3)
Exercise \(\PageIndex{10}\)
A grocery store sells bags of oranges in two different sizes.
- The 3-pound bags of oranges cost $4.
- The 8-pound bags of oranges for $9.
Which oranges cost less per pound? Explain or show your reasoning.
(From Unit 2.3.5)


6.6: Regional Scale Winds - The Monsoon
Monsoon means wind that changes direction with season. Monsoon conditions are found in a variety of regions around the world, the most noted is the monsoon of Asia. During the summer, the continent of Asia heats up more than the surrounding ocean due to the differences in the way land and water heat. The warm surface creates a large area of low pressure over north-central Asia and a smaller one over India. This creates an onshore wind bringing the moisture laden maritime air from the Pacific and Indian oceans onto land. As the air streams across the land, convection induced by the warm surface, convergence into the areas of low pressure, and uplift along major mountain systems like the Himalayas force the air to rise creating the abundant rainfall of the wet monsoon season . During the winter, the flow of air reverses. The continent cools rapidly forming a large area of high pressure over north central Asia, known as the Siberian High, and a smaller area over India. Now the drier, colder air of the continent blows offshore creating the dry monsoon season .
Monsoon conditions are also found in east-central Africa. This region experiences a seasonality of precipitation induced by alternating influence of global pressure belts. During the high sun season, the ITCZ moves in bringing warm, moist unstable air that induces precipitation. During the low sun season the ITCZ shifts out and the influence of the descending motions of the subtropical high dominate the region. During this period the subtropical high inhibits the development of precipitation.
The southwest desert of the United States experiences a pseudo-monsoon during the summer. The intense heating of the surface creates a thermal low over the desert at this time of year. The low induces moist Gulf air to move into the region where it is uplift by convergence or orographic uplift to produce precipitation. During the winter the desert surface cools, the low dissipates, and dry conditions prevail.


6.7: Music of Franz Liszt
Franz Liszt (b. 1811-1886) was born in Doborján, Hungary (now Raiding, Austria). His father, employed as a steward for a wealthy family, was an amateur musician who recognized his son’s talent. A group of Hungarian noblemen sponsored him with a stipend that enabled Franz to pursue his musical interest in Paris. There, he became the friend of Mendelssohn, Hugo, Chopin, Delacroix, George Sand, and Berlioz; these friends influenced him to become part of the French Romanticism movement.
Also in Paris in 1831, Liszt attended a performance of virtuoso violinist Paganini, who was touring. Paganini’s style and success helped make Liszt aware of the demand for a solo artist who performed with showmanship. The ever growing mass public audience desired gifted virtuoso soloists performers at the time. Liszt, one of the best pianists of his time, became a great showman who knew how to energize an audience. Up until Liszt, the standard practice of performing piano solos was with the solo artist’s back to the audience. This limited—and actually blocked—the audience from viewing the artist’s hands, facial expression, and musical nuance. Liszt changed the entire presentation by turning the piano sideways so the audience could view his facial expressions and the manner in which his fingers interacted with the keys, from playing loud and thunderously to gracefully light and legato. Liszt possessed great charisma and performance appeal; indeed, he had a following of young ladies that idolized his performances. During his career of music stardom, Liszt never married and was considered one of the most eligible bachelors of the time. But he did have several “relationships” with different women, one of whom was the novelist Countess Marie d’Agoult who wrote under the pen name of Daniel Stern. She and Liszt travelled to Switzerland for a few years and they had three children, including Cosima who ultimately married Wagner.
While at the height of his performance career, Liszt retreated from his piano soloist career to devote all his energy to composition. He moved to Weimer in 1948 and assumed the post of court musician for the Grand Duke, remaining in Weimer until 1861. There, he produced his greatest orchestral works. His position in Weimer included the responsibility as director to the Grand Duke’s opera house. In this position, Liszt could influence the public’s taste in music and construct musical expectations for future com- positions. And he used his influential position to program what Wagner called “Music of the Future.” Liszt and Wagner both advocated and promoted highly dramatic music in Weimer, with Liszt conducting the first performances of Wagner’s Lohengrin, Be- lioz’s Benevenuto Cellini, as well as many other contemporary compositions.
While in Weimer, Liszt began a relationship with a woman who had a tremendous influence on his life and music. A wife of a nobleman in the court of the Tsar, Princess Carolyne Sayn-Wittenstein met and fell in love with Liszt on his final per- formance tour of Russia. Later she left her husband and moved to Weimer to be with Liszt. She assisted Liszt in writing literary works, among which included a fabricated biography by Liszt on the Life of Chopin and a book on “Gypsy,” a book also considered eccentric and inaccurate.
While Liszt had an eventful romantic life, he remained a Roman Catholic, and he eventually sought solitude in the Catholic Church. His association with the church led to the writing of his major religious works. He also joined the Oratory of the Madonna del Rosario and studied the preliminary stage for priesthood, tak- ing his minor orders and becoming known as the Abbé Liszt. He dressed as a priest and composed Masses, oratorios, and religious music for the church.
Still active at the age of seventy-five, he earned respect from England as a com- poser and was awarded an honor in person by Queen Victoria. Returning from this celebration, he met Claud Debussy in Paris then journeyed to visit his widowed daughter Cosima in Bayreuth and attended a Wagnerian Festival. He died during that festival, and even on his death bed, dying of pneumonia, Liszt named one of the “Music of the Future” masterpieces: Wagner’s Tristan.
Liszt’s primary goal in music composition was pure expression through the idiom of tone. His freedom of expression necessitated his creation of the symphonic poem , sometimes called a tone poem--a one movement program piece written for orchestra that portrays images of a place, story, novel, landscape or non-musical source or image. This form utilizes transformations of a few themes through the en- tire work for continuity. The themes are varied by adjusting the rhythm, harmony, dynamics, tempos, instrumental registers, instrumentation in the orchestra, timbre, and melodic outline, or shape. By making these slight-to-major adjustments, Liszt found it possible to convey the extremes of emotion—from love to hate, war to peace, triumph to defeat—within a thematic piece. His thirteen symphonic poems greatly influenced the nineteenth century, an influence that continues through to- day. Liszt’s most famous piece for orchestra is the three portrait work Symphony after Goethe’s Faust ( the portraits include Faust, Gretchen, and Mephistopheles) . A similar work, his Symphony of Dante’s Divine Comedy, has three movements: Inferno , Purgatory , and Vision of Paradise. His most famous of the symphonic poems is Les Preludes (The Preludes) written in 1854.
His best known works include nineteen Hungarian Rhapsodies (Liszt’s Hungarian Rhapsody No. 2 may be heard at the following link: www.52compos- ers.com/liszt.html), Piano concertos (Piano concerto No. 1, Part 1 may be heard at the following link: www.52composers.com/liszt.html), Mephisto Waltzes, Faust Symphony ( Mephisto from Faust Symphony Part 1 may be heard at follow- ing link: www.52composers.com/liszt.html), and Lieberstaumes (may be heard at the following link: www.52composers.com/liszt.html).
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Listening Guide For audio, go to: |
| Composer: Franz Liszt |
| Composition: Hungarian Rhapsody No. 2 |
| Date: 1847 |
| Genre: The second of a set of 19 Hungarian Rhapsodies |
| Performing Forces: Piano solo |
|
What we want you to remember about this composition:
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|
Other things to listen for:
|
| Timing | Performing Forces, Melody, and Texture |
| 00:00-04:26 | The lasson opens at a slow tempo |
| 04:26- to the end | The friska follows and builds feverishly. Dance rhythms with heavy pulse. |
For more information and listening opportunities for Liszt selections, go to:
www.classicfm.com/composers/l...szts-piano-mu- sic-where-start/#qhmCIMkIdTaD2o88.97
We shift now from smaller compositions for small forces to larger-scale com-
positions written for entire orchestras.


3.5: The Aggregates of Culture
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- Last updated
- Save as PDF
- Jennifer Hasty, David G. Lewis, & Marjorie M. Snipes
- OpenStax
Learning Objectives
By the end of this section, you will be able to:
- Explain how elements of culture combine in aggregates.
- Give three detailed examples of cultural symbols.
- Explain how symbols are embedded in rituals.
- Describe how social structures organize important cultural processes.
An aggregate is a combination of elements. What we make, what we do, and what we think all combine in larger aggregates of culture. For instance, it’s pretty clear that toothbrushes, moon rocks, restaurants, and Mother’s Day cards must be understood as aggregates of material objects, practices, and ideas. In order to fully understand the toothbrush as a cultural object, we must examine not only its design and production but also how people use toothbrushes and why they use them. A set of routine practices surround our cultural objects (brushing), and those practices are supported by cultural ideas (hygiene).
Symbols
A symbol is an object, image, gesture, vocalization, or event conventionally associated with a particular meaning. anthropologist Jennifer Hasty was conducting fieldwork in Ghana during an election year, she noticed that the posters and pamphlets of one politician featured a broom. Confused, she asked a friend why a male politician would choose a humble domestic tool associated with women’s work as his political motif. Making a sweeping motion with her hands, she explained, “Because he is promising to sweep away all the corruption.” Turns out, he was the not the first to use this symbol. Over time, the broom has come to acquire political meaning as a symbolic anti-corruption tool in Ghana.
Colors, shapes, gestures, animals, plants—all of these commonly acquire specific cultural meaning. For a Hindu wedding, a bride typically wears a bright red sari, as red is an auspicious color associated with change, passion, and prosperity. White, on the other hand, is typically worn to Indian funerals.
Symbols are useful cultural aggregates because they provide a kind of shorthand for expressing complex ideas. Consider the American bumper sticker shown in Figure 3.9.
Combining symbols from Islam, Judaism, Taoism, Christianity, paganism, women, men, and the peace movement, this sticker aims to promote multicultural diversity. Rather than listing the various religions, identities, and ideologies and describing the conflicts among them, the message simply incorporates their symbols into a word urging mutual tolerance.
Although symbols have conventional meanings, they can mean different things in different contexts or to different people. Although the intended meaning of the above bumper sticker is diversity, some people interpret it as an emblem of radical atheism. In the wake of the 2016 presidential election, some Americans started wearing safety pins to show their solidarity with LGBTQIA+ people, people of color, and others who had become targets of post-election harassment. For some, however, the safety pin symbolized pretentiousness and hypocrisy.
Ritual
Combining objects, actions, and meanings, ritual is a special kind of repeated, patterned action conventionally associated with a particular meaning. Rituals incorporate symbols and roles along with routinized activities such as gestures, music, and movement. Many rituals are performed by specialists in group settings to accomplish specific group or individual goals. Rituals bring together symbols, practices, and worldviews.
Consider this popular American ritual. On the first Sunday in February, many Americans gather in each other’s homes to watch the annual championship game of the National Football League (NFL) on television. So widespread is this practice that stores are nearly empty and many Christian churches cancel afternoon and evening activities. As a whole, the ritual consists of many roles and relationships as well as patterned actions and conventional meanings. At the heart of the action are the two teams competing against one another in a chaotic game featuring an oddly shaped ball carried forward in campaigns of full-frontal assault across a carefully marked field. The players are surrounded by referees, coaches, camera people, and cheerleaders, each group having a strategic role in the action. Surrounding the field are commentators who interpret and contextualize, giving meaning to the actions of the game. At home, some people watch the game closely, exclaiming with joy or disappointment and commenting on the comments of the commentators. Other people socialize with one another, watching the game intermittently. Vast amounts of food and drink are consumed by Americans on Super Bowl Sunday. Typical foods include potato chips, dips, barbecued chicken wings, and pizza. Beer is the beverage of choice for this occasion. An event celebrating competition, spectatorship, and consumption, Super Bowl Sunday is an effective ritual for reinforcing dominant values in a society structured by corporate capitalism. Notions of gender, race, and class are threaded through the various levels of play and consumption as well.
In the Akan communities of central and southern Ghana, in West Africa, leaders perform a ritual called Adae that uses important cultural symbols and reinforces cultural commitments to authority, ancestors, and shared prosperity. In the Akan society, people are given special wooden stools to mark certain stages in life, such as puberty and marriage. A person’s stool is said to contain the personal power of the owner, symbolizing the life essence of that person.
When an eminent person dies, that person’s stool is enshrined in a special shed called a stool house, or nkonuafieso . Twice every 42 days (once on a Wednesday and once on a Sunday), a community leader makes a procession to the stool house of the ancestral leaders of the community. Entering the stool house, the leader must remove their sandals and lower the cloth worn draped around their shoulders, symbolizing their humility and respect for the ancestors. Then the leader greets the ancestral leaders one by one, making offerings of drink and food and asking for blessings and prosperity for the community.
Special rituals called rites of passage are used to mark the movement of a person from one social status to another. Naming ceremonies, puberty rites, weddings, and funerals are all common rites of passage. Anthropologist Arnold van Gennep (1960) identified three stages in rites of passage: separation, transition, and incorporation. In the first phase, separation, individuals, or groups are taken out of their everyday social context, leaving their original social status. In the second phase, transition, people exist in an in-between state outside of conventional norms of dress and action. In this phase, people are often dressed in special costume, made to engage in unusual behaviors, and taught special forms of secret knowledge. In the third phase, people are brought back into society in a formal ceremony and introduced as subjects in a new social category.
Initiation rituals are a common rite of passage in many societies. In many African societies that practice initiation, young people are gathered together in a group and taken to a special camp outside the town or village. This constitutes the separation phase. In the next phase, transition, members of the group are often dressed alike and made to follow a common set of rules and schedule of activities. They may be required to perform unusual feats, such as eating strange foods. Their bodies may be scarified or tattooed. Elders give them special knowledge essential to performing their future roles as women or men. For instance, girls may learn explicit lessons about conception and childbirth. Finally, when the transition is complete, initiates are returned to the town or village and presented as women or men. Often, the completion of initiation marks a young woman as formally eligible for courtship and marriage.
Social Structure
The way a society is formally organized is called social structure . Typically, a society organizes a set of routine activities and objects in space and time to accomplish a particular function, such as community decision-making, the production and circulation of goods, or religious observance. Social structure is the framework for those realms, designating when, where, how, and by whom these functions are accomplished. Social structures combine material culture (such as buildings) with practices (such as meetings) and ideas (such as the rules and procedures of those meetings).
Consider the social structure of community decision-making, or the political realm. In some societies, community decisions are routinely made under the authority of a person inhabiting an inherited political office, called a chief or king (such as the Akans, discussed in the last section). Chiefs often have a council of community elders, the heads of local extended families. A chief is expected to consult with this council in all community matters. Other groups in society may represent the interests of youth, women, farmers, or traders. Each group will have its own leader who communicates directly with the local chief. Regular procedures govern how issues are raised and discussed and how decisions are taken. Together, the groups, roles, relationships, and procedures all constitute the social structure of the political system.
Rather than seeing social structure as fixed and immobile, some anthropologists emphasize that people continually make and alter their social structures through everyday forms of interpretation, participation, and resistance. These processes mean that social structures are always subject to a variety of forces in a constant state of change.


7.1: Electromagnetic Energy
- Explain the basic behavior of waves, including traveling waves and standing waves
- Describe the wave nature of light
- Use appropriate equations to calculate related light-wave properties such as period, frequency, wavelength, and energy
- Distinguish between line and continuous emission spectra
- Describe the particle nature of light
The nature of light has been a subject of inquiry since antiquity. In the seventeenth century, Isaac Newton performed experiments with lenses and prisms and was able to demonstrate that white light consists of the individual colors of the rainbow combined together. Newton explained his optics findings in terms of a "corpuscular" view of light, in which light was composed of streams of extremely tiny particles traveling at high speeds according to Newton's laws of motion. Others in the seventeenth century, such as Christiaan Huygens , had shown that optical phenomena such as reflection and refraction could be equally well explained in terms of light as waves traveling at high speed through a medium called "luminiferous aether" that was thought to permeate all space. Early in the nineteenth century, Thomas Young demonstrated that light passing through narrow, closely spaced slits produced interference patterns that could not be explained in terms of Newtonian particles but could be easily explained in terms of waves. Later in the nineteenth century, after James Clerk Maxwell developed his theory of electromagnetic radiation and showed that light was the visible part of a vast spectrum of electromagnetic waves, the particle view of light became thoroughly discredited. By the end of the nineteenth century, scientists viewed the physical universe as roughly comprising two separate domains: matter composed of particles moving according to Newton's laws of motion, and electromagnetic radiation consisting of waves governed by Maxwell's equations. Today, these domains are referred to as classical mechanics and classical electrodynamics (or classical electromagnetism). Although there were a few physical phenomena that could not be explained within this framework, scientists at that time were so confident of the overall soundness of this framework that they viewed these aberrations as puzzling paradoxes that would ultimately be resolved somehow within this framework. As we shall see, these paradoxes led to a contemporary framework that intimately connects particles and waves at a fundamental level called wave-particle duality, which has superseded the classical view.
Visible light and other forms of electromagnetic radiation play important roles in chemistry, since they can be used to infer the energies of electrons within atoms and molecules. Much of modern technology is based on electromagnetic radiation. For example, radio waves from a mobile phone, X-rays used by dentists, the energy used to cook food in your microwave, the radiant heat from red-hot objects, and the light from your television screen are forms of electromagnetic radiation that all exhibit wavelike behavior.
Waves
A wave is an oscillation or periodic movement that can transport energy from one point in space to another. Common examples of waves are all around us. Shaking the end of a rope transfers energy from your hand to the other end of the rope, dropping a pebble into a pond causes waves to ripple outward along the water's surface, and the expansion of air that accompanies a lightning strike generates sound waves (thunder) that can travel outward for several miles. In each of these cases, kinetic energy is transferred through matter (the rope, water, or air) while the matter remains essentially in place. An insightful example of a wave occurs in sports stadiums when fans in a narrow region of seats rise simultaneously and stand with their arms raised up for a few seconds before sitting down again while the fans in neighboring sections likewise stand up and sit down in sequence. While this wave can quickly encircle a large stadium in a few seconds, none of the fans actually travel with the wave-they all stay in or above their seats.
Waves need not be restricted to travel through matter. As Maxwell showed, electromagnetic waves consist of an electric field oscillating in step with a perpendicular magnetic field, both of which are perpendicular to the direction of travel. These waves can travel through a vacuum at a constant speed of 2.998 × 10 8 m/s, the speed of light (denoted by c ).
All waves, including forms of electromagnetic radiation, are characterized by, a wavelength (denoted by λ , the lowercase Greek letter lambda), a frequency (denoted by \(\nu\), the lowercase Greek letter nu), and an amplitude . As can be seen in Figure \(\PageIndex{1}\), the wavelength is the distance between two consecutive peaks or troughs in a wave (measured in meters in the SI system). Electromagnetic waves have wavelengths that fall within an enormous range-wavelengths of kilometers (10 3 m) to picometers (10 −12 m) have been observed. The frequency is the number of wave cycles that pass a specified point in space in a specified amount of time (in the SI system, this is measured in seconds). A cycle corresponds to one complete wavelength. The unit for frequency, expressed as cycles per second [s −1 ], is the hertz (Hz) . Common multiples of this unit are megahertz, (1 MHz = 1 × 10 6 Hz) and gigahertz (1 GHz = 1 × 10 9 Hz). The amplitude corresponds to the magnitude of the wave's displacement and so, in Figure, this corresponds to one-half the height between the peaks and troughs. The amplitude is related to the intensity of the wave, which for sound is the loudness.
The product of a wave's wavelength ( λ ) and its frequency ( \(\nu\) ), λ \(\nu\) , is the speed of the wave. Thus, for electromagnetic radiation in a vacuum:
\[c=\mathrm{2.998×10^8\,ms^{−1}}=λ \nu \]
Wavelength and frequency are inversely proportional: As the wavelength increases, the frequency decreases. The inverse proportionality is illustrated in Figure \(\PageIndex{2}\). This figure also shows the electromagnetic spectrum , the range of all types of electromagnetic radiation. Each of the various colors of visible light has specific frequencies and wavelengths associated with them, and you can see that visible light makes up only a small portion of the electromagnetic spectrum. Because the technologies developed to work in various parts of the electromagnetic spectrum are different, for reasons of convenience and historical legacies, different units are typically used for different parts of the spectrum. For example, radio waves are usually specified as frequencies (typically in units of MHz), while the visible region is usually specified in wavelengths (typically in units of nm or angstroms).
A sodium streetlight gives off yellow light that has a wavelength of 589 nm (1 nm = 1 × 10 −9 m). What is the frequency of this light?
Solution
We can rearrange the Equation \(\PageIndex{1}\) to solve for the frequency:
\[ \nu=\dfrac{c}{λ} \nonumber\]
Since c is expressed in meters per second, we must also convert 589 nm to meters.
\[ \nu=\mathrm{\left(\dfrac{2.998×10^8\:\cancel{m}s^{−1}}{589\cancel{nm}}\right)\left(\dfrac{1×10^9\cancel{nm}}{1\cancel{m}}\right)=5.09×10^{14}\,s^{−1}} \nonumber\]
One of the frequencies used to transmit and receive cellular telephone signals in the United States is 850 MHz. What is the wavelength in meters of these radio waves?
- Answer
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0.353 m = 35.3 cm
One particularly characteristic phenomenon of waves results when two or more waves come into contact: They interfere with each other. Figure \(\PageIndex{5}\) shows the interference patterns that arise when light passes through narrow slits closely spaced about a wavelength apart. The fringe patterns produced depend on the wavelength, with the fringes being more closely spaced for shorter wavelength light passing through a given set of slits. When the light passes through the two slits, each slit effectively acts as a new source, resulting in two closely spaced waves coming into contact at the detector (the camera in this case). The dark regions in Figure \(\PageIndex{5}\) correspond to regions where the peaks for the wave from one slit happen to coincide with the troughs for the wave from the other slit (destructive interference), while the brightest regions correspond to the regions where the peaks for the two waves (or their two troughs) happen to coincide (constructive interference). Likewise, when two stones are tossed close together into a pond, interference patterns are visible in the interactions between the waves produced by the stones. Such interference patterns cannot be explained by particles moving according to the laws of classical mechanics.
Because the wavelengths of X-rays (10-10,000 picometers [pm]) are comparable to the size of atoms, X-rays can be used to determine the structure of molecules. When a beam of X-rays is passed through molecules packed together in a crystal, the X-rays collide with the electrons and scatter. Constructive and destructive interference of these scattered X-rays creates a specific diffraction pattern. Calculating backward from this pattern, the positions of each of the atoms in the molecule can be determined very precisely. One of the pioneers who helped create this technology was Dorothy Crowfoot Hodgkin.
She was born in Cairo, Egypt, in 1910, where her British parents were studying archeology. Even as a young girl, she was fascinated with minerals and crystals. When she was a student at Oxford University, she began researching how X-ray crystallography could be used to determine the structure of biomolecules. She invented new techniques that allowed her and her students to determine the structures of vitamin B 12 , penicillin, and many other important molecules. Diabetes, a disease that affects 382 million people worldwide, involves the hormone insulin. Hodgkin began studying the structure of insulin in 1934, but it required several decades of advances in the field before she finally reported the structure in 1969. Understanding the structure has led to better understanding of the disease and treatment options.
Not all waves are traveling waves. Standing waves (also known as stationary waves ) remain constrained within some region of space. As we shall see, standing waves play an important role in our understanding of the electronic structure of atoms and molecules. The simplest example of a standing wave is a one-dimensional wave associated with a vibrating string that is held fixed at its two end points. Figure \(\PageIndex{6}\) shows the four lowest-energy standing waves (the fundamental wave and the lowest three harmonics) for a vibrating string at a particular amplitude. Although the string's motion lies mostly within a plane, the wave itself is considered to be one dimensional, since it lies along the length of the string. The motion of string segments in a direction perpendicular to the string length generates the waves and so the amplitude of the waves is visible as the maximum displacement of the curves seen in Figure \(\PageIndex{6}\). The key observation from the figure is that only those waves having an integer number, n, of half-wavelengths between the end points can form. A system with fixed end points such as this restricts the number and type of the possible waveforms. This is an example of quantization , in which only discrete values from a more general set of continuous values of some property are observed. Another important observation is that the harmonic waves (those waves displaying more than one-half wavelength) all have one or more points between the two end points that are not in motion. These special points are nodes . The energies of the standing waves with a given amplitude in a vibrating string increase with the number of half-wavelengths n . Since the number of nodes is n – 1, the energy can also be said to depend on the number of nodes, generally increasing as the number of nodes increases.
An example of two-dimensional standing waves is shown in Figure \(\PageIndex{7}\) which shows the vibrational patterns on a flat surface. Although the vibrational amplitudes cannot be seen like they could in the vibrating string, the nodes have been made visible by sprinkling the drum surface with a powder that collects on the areas of the surface that have minimal displacement. For one-dimensional standing waves, the nodes were points on the line, but for two-dimensional standing waves, the nodes are lines on the surface. Because of the circular symmetry of the drum surface, its boundary conditions (the drum surface being tightly constrained to the circumference of the drum) result in two types of nodes: radial nodes that sweep out all angles at constant radii and, thus, are seen as circles about the center, and angular nodes that sweep out all radii at constant angles and, thus, are seen as lines passing through the center. The upper left image in Figure \(\PageIndex{7}\) shows two radial nodes, while the image in the lower right shows the vibrational pattern associated with three radial nodes and two angular nodes.
The Photoelectric Effect
The next paradox in the classical theory to be resolved concerned the photoelectric effect (Figure \(\PageIndex{8}\)). It had been observed that electrons could be ejected from the clean surface of a metal when light having a frequency greater than some threshold frequency was shone on it. Surprisingly, the kinetic energy of the ejected electrons did not depend on the brightness of the light, but increased with increasing frequency of the light. Since the electrons in the metal had a certain amount of binding energy keeping them there, the incident light needed to have more energy to free the electrons. According to classical wave theory, a wave's energy depends on its amplitude, not its frequency. One part of these observations was that the number of electrons ejected within in a given time period was seen to increase as the brightness increased. In 1905, Albert Einstein was able to resolve the paradox by incorporating Planck's quantization findings into the discredited particle view of light. Einstein won his Nobel prize for this work, and not for his theories of relativity for which he is most famous.
Einstein argued that the quantized energies that Planck had postulated in his treatment of blackbody radiation could be applied to the light in the photoelectric effect so that the light striking the metal surface should not be viewed as a wave, but instead as a stream of particles (later called photons ) whose energy depended on their frequency, not amplitude. Electrons were ejected when hit by photons having sufficient energy (a frequency greater than the threshold). The greater the frequency, the greater the kinetic energy imparted to the escaping electrons by the collisions. Einstein also argued that the light intensity did not depend on the amplitude of the incoming wave, but instead corresponded to the number of photons striking the surface within a given time period. This explains why the number of ejected electrons increased with increasing brightness, since the greater the number of incoming photons, the greater the likelihood that they would collide with some of the electrons. This shows the brightness or intensity of light depends on the number of photos, not the amplitude of the wave.
According to Planck's formula where E is the energy of a single photon, h is planks constant (6.62607015 x 10 -34 J sec) and \(\nu\) is the frequency.
\[E=\mathrm{h \nu} \]
With Einstein's findings, the nature of light took on a new air of mystery. Although many light phenomena could be explained either in terms of waves or particles, certain phenomena, such as the interference patterns obtained when light passed through a double slit, were completely contrary to a particle view of light, while other phenomena, such as the photoelectric effect, were completely contrary to a wave view of light. Somehow, at a deep fundamental level still not fully understood, light is both wavelike and particle-like. This is known as wave-particle duality .
When we see light from a neon sign, we are observing radiation from excited neon atoms. If this radiation has a wavelength of 640 nm, what is the energy of the photon being emitted?
Solution
We use the part of Planck's equation that includes the wavelength, λ , and convert units of nanometers to meters so that the units of λ and c are the same.
\[\begin{align*} E&=\dfrac{hc}{λ} \\[4pt] &=\mathrm{\dfrac{(6.626×10^{−34}\:J\cancel{s})(2.998×10^{8}\:m\cancel{s}^{−1})}{(640\cancel{nm})\left(\dfrac{1\:m}{10^9\cancel{nm}}\right)}}\\[4pt] &=\mathrm{3.10×10^{−19}\:J} \end{align*}\]
The microwaves in an oven are of a specific frequency that will heat the water molecules contained in food. (This is why most plastics and glass do not become hot in a microwave oven-they do not contain water molecules.) This frequency is about 3 × 10 9 Hz. What is the energy of one photon in these microwaves?
- Answer
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2 × 10 −24 J
Line Spectra
Another paradox within the classical electromagnetic theory that scientists in the late nineteenth century struggled with concerned the light emitted from atoms and molecules. When solids, liquids, or condensed gases are heated sufficiently, they radiate some of the excess energy as light. Photons produced in this manner have a range of energies, and thereby produce a continuous spectrum in which an unbroken series of wavelengths is present. Most of the light generated from stars (including our sun) is produced in this fashion. You can see all the visible wavelengths of light present in sunlight by using a prism to separate them. Sunlight also contains UV light (shorter wavelengths) and IR light (longer wavelengths) that can be detected using instruments but that are invisible to the human eye. Incandescent (glowing) solids such as tungsten filaments in incandescent lights also give off light that contains all wavelengths of visible light. These continuous spectra can often be approximated by blackbody radiation curves at some appropriate temperature.
In contrast to continuous spectra, light can also occur as discrete or line spectra having very narrow line widths interspersed throughout the spectral regions. Exciting a gas at low partial pressure using an electrical current, or heating it, will produce line spectra. Fluorescent light bulbs and neon signs operate in this way (Figure \(\PageIndex{9}\)). Each element displays its own characteristic set of lines, as do molecules, although their spectra are generally much more complicated.
Each emission line consists of a single wavelength of light, which implies that the light emitted by a gas consists of a set of discrete energies. For example, when an electric discharge passes through a tube containing hydrogen gas at low pressure, the H 2 molecules are broken apart into separate H atoms, and we see a blue-pink color. Passing the light through a prism produces a line spectrum, indicating that this light is composed of photons of four visible wavelengths, as shown in Figure \(\PageIndex{10}\).
The origin of discrete spectra in atoms and molecules was extremely puzzling to scientists in the late nineteenth century, since according to classical electromagnetic theory, only continuous spectra should be observed. Even more puzzling, in 1885, Johann Balmer was able to derive an empirical equation that related the four visible wavelengths of light emitted by hydrogen atoms to whole integers. That equation is the following one, in which k is a constant:
\[\dfrac{1}{λ}=k\left(\dfrac{1}{4}−\dfrac{1}{n^2}\right),\:n=3,\:4,\:5,\:6\]
Other discrete lines for the hydrogen atom were found in the UV and IR regions. Johannes Rydberg generalized Balmer's work and developed an empirical formula that predicted all of hydrogen's emission lines, not just those restricted to the visible range, where, n 1 and n 2 are integers, n 1 < n 2 , and \(R_∞\) is the Rydberg constant (1.097 × 10 7 m −1 ).
\[\dfrac{1}{λ}=R_∞\left(\dfrac{1}{n^2_1}−\dfrac{1}{n^2_2}\right)\]
Even in the late nineteenth century, spectroscopy was a very precise science, and so the wavelengths of hydrogen were measured to very high accuracy, which implied that the Rydberg constant could be determined very precisely as well. That such a simple formula as the Rydberg formula could account for such precise measurements seemed astounding at the time, but it was the eventual explanation for emission spectra by Neils Bohr in 1913 that ultimately convinced scientists to abandon classical physics and spurred the development of modern quantum mechanics.
Summary
Light and other forms of electromagnetic radiation move through a vacuum with a constant speed, c , of 2.998 × 10 8 m s −1 . This radiation shows wavelike behavior, which can be characterized by a frequency, ν , and a wavelength, λ , such that c = λν . Light is an example of a travelling wave. Other important wave phenomena include standing waves, periodic oscillations, and vibrations. Standing waves exhibit quantization, since their wavelengths are limited to discrete integer multiples of some characteristic lengths. Electromagnetic radiation that passes through two closely spaced narrow slits having dimensions roughly similar to the wavelength will show an interference pattern that is a result of constructive and destructive interference of the waves. Electromagnetic radiation also demonstrates properties of particles called photons. The energy of a photon is related to the frequency (or alternatively, the wavelength) of the radiation as E = hν (or \(E=\dfrac{hc}{λ}\)), where h is Planck's constant. That light demonstrates both wavelike and particle-like behavior is known as wave-particle duality. All forms of electromagnetic radiation share these properties, although various forms including X-rays, visible light, microwaves, and radio waves interact differently with matter and have very different practical applications. Electromagnetic radiation can be generated by exciting matter to higher energies, such as by heating it. The emitted light can be either continuous (incandescent sources like the sun) or discrete (from specific types of excited atoms). Continuous spectra often have distributions that can be approximated as blackbody radiation at some appropriate temperature. The line spectrum of hydrogen can be obtained by passing the light from an electrified tube of hydrogen gas through a prism. This line spectrum was simple enough that an empirical formula called the Rydberg formula could be derived from the spectrum. Three historically important paradoxes from the late 19th and early 20th centuries that could not be explained within the existing framework of classical mechanics and classical electromagnetism were the blackbody problem, the photoelectric effect, and the discrete spectra of atoms. The resolution of these paradoxes ultimately led to quantum theories that superseded the classical theories.
Key Equations
- c = λν
- \(E=hν=\dfrac{hc}{λ}\), where h = 6.626 × 10 −34 J s
- \(\dfrac{1}{λ}=R_∞\left(\dfrac{1}{n^2_1}−\dfrac{1}{n^2_2}\right)\)
Glossary
- amplitude
- extent of the displacement caused by a wave (for sinusoidal waves, it is one-half the difference from the peak height to the trough depth, and the intensity is proportional to the square of the amplitude)
- blackbody
- idealized perfect absorber of all incident electromagnetic radiation; such bodies emit electromagnetic radiation in characteristic continuous spectra called blackbody radiation
- continuous spectrum
- electromagnetic radiation given off in an unbroken series of wavelengths (e.g., white light from the sun)
- electromagnetic radiation
- energy transmitted by waves that have an electric-field component and a magnetic-field component
- electromagnetic spectrum
- range of energies that electromagnetic radiation can comprise, including radio, microwaves, infrared, visible, ultraviolet, X-rays, and gamma rays; since electromagnetic radiation energy is proportional to the frequency and inversely proportional to the wavelength, the spectrum can also be specified by ranges of frequencies or wavelengths
- frequency (\(\nu\))
- number of wave cycles (peaks or troughs) that pass a specified point in space per unit time
- hertz (Hz)
- the unit of frequency, which is the number of cycles per second, s −1
- intensity
- property of wave-propagated energy related to the amplitude of the wave, such as brightness of light or loudness of sound
- interference pattern
- pattern typically consisting of alternating bright and dark fringes; it results from constructive and destructive interference of waves
- line spectrum
- electromagnetic radiation emitted at discrete wavelengths by a specific atom (or atoms) in an excited state
- node
- any point of a standing wave with zero amplitude
- photon
- smallest possible packet of electromagnetic radiation, a particle of light
- quantization
- occurring only in specific discrete values, not continuous
- standing wave
- (also, stationary wave) localized wave phenomenon characterized by discrete wavelengths determined by the boundary conditions used to generate the waves; standing waves are inherently quantized
- wave
- oscillation that can transport energy from one point to another in space
- wavelength ( λ )
- distance between two consecutive peaks or troughs in a wave
- wave-particle duality
- term used to describe the fact that elementary particles including matter exhibit properties of both particles (including localized position, momentum) and waves (including nonlocalization, wavelength, frequency)
Contributors and Attributions
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Paul Flowers (University of North Carolina - Pembroke), Klaus Theopold (University of Delaware) and Richard Langley (Stephen F. Austin State University) with contributing authors. Textbook content produced by OpenStax College is licensed under a Creative Commons Attribution License 4.0 license. Download for free at<EMAIL_ADDRESS>).