6.6: Satellites and Kepler’s Laws- An Argument for Simplicity
Learning Objectives
By the end of this section, you will be able to:
- State Kepler’s laws of planetary motion.
- Derive the third Kepler’s law for circular orbits.
- Discuss the Ptolemaic model of the universe.
Examples of gravitational orbits abound. Hundreds of artificial satellites orbit Earth together with thousands of pieces of debris. The Moon’s orbit about Earth has intrigued humans from time immemorial. The orbits of planets, asteroids, meteors, and comets about the Sun are no less interesting. If we look further, we see almost unimaginable numbers of stars, galaxies, and other celestial objects orbiting one another and interacting through gravity.
All these motions are governed by gravitational force, and it is possible to describe them to various degrees of precision. Precise descriptions of complex systems must be made with large computers. However, we can describe an important class of orbits without the use of computers, and we shall find it instructive to study them. These orbits have the following characteristics:
- A small mass \(m\)orbits a much larger mass \(M\). This allows us to view the motion as if \(M\) were stationary—in fact, as if from an inertial frame of reference placed on \(M\) —without significant error. Mass \(m\) is the satellite of \(M\), if the orbit is gravitationally bound.
- The system is isolated from other masses . This allows us to neglect any small effects due to outside masses.
The conditions are satisfied, to good approximation, by Earth’s satellites (including the Moon), by objects orbiting the Sun, and by the satellites of other planets. Historically, planets were studied first, and there is a classical set of three laws, called Kepler’s laws of planetary motion, that describe the orbits of all bodies satisfying the two previous conditions (not just planets in our solar system). These descriptive laws are named for the German astronomer Johannes Kepler (1571–1630), who devised them after careful study (over some 20 years) of a large amount of meticulously recorded observations of planetary motion done by Tycho Brahe (1546–1601). Such careful collection and detailed recording of methods and data are hallmarks of good science. Data constitute the evidence from which new interpretations and meanings can be constructed.
Kepler’s Laws of Planetary Motion
Kepler's First Law
The orbit of each planet about the Sun is an ellipse with the Sun at one focus.
Kepler's Second Law
Each planet moves so that an imaginary line drawn from the Sun to the planet sweeps out equal areas in equal times (see Figure).
Kepler's Third Law
The ratio of the squares of the periods of any two planets about the Sun is equal to the ratio of the cubes of their average distances from the Sun. In equation form, this is \[ \dfrac{T_1^2}{T_2^2} =\dfrac{r_1^3}{r_2^3}\]
where \(T\) is the period (time for one orbit) and \(r\) is the average radius. This equation is valid only for comparing two small masses orbiting the same large one. Most importantly, this is a descriptive equation only, giving no information as to the cause of the equality.
Note again that while, for historical reasons, Kepler’s laws are stated for planets orbiting the Sun, they are actually valid for all bodies satisfying the two previously stated conditions.
Example \(\PageIndex{1}\): Find the Time for One Orbit of an Earth Satellite
Given that the Moon orbits Earth each 27.3 d and that it is an average distance of \(3.84 \times 10^8 \, m\) from the center of Earth, calculate the period of an artificial satellite orbiting at an average altitude of 1500 km above Earth’s surface.
Strategy
The period, or time for one orbit, is related to the radius of the orbit by Kepler’s third law, given in mathematical form in \(\frac{T_1^2}{T_2^2} =\frac{r_1^3}{r_2^3}\). Let us use the subscript 1 for the Moon and the subscript 2 for the satellite. We are asked to find \(T_2\). The given information tells us that the orbital radius of the Moon is \(r_1 = 3.84 \times 10^8 \, m\), and that the period of the Moon is \(T_1 = 27.3 \, d\).
The height of the artificial satellite above Earth’s surface is given, and so we must add the radius of Earth (6380 km) to get \(r_2 = (1500 +6380) km = 7880 \, km\). Now all quantities are known, and so \(T_2\) can be found.
Solution
Kepler’s third law is \[ \dfrac{T_1^2}{T_2^2} =\dfrac{r_1^3}{r_2^3}.\]
To solve for \(T_2\), we cross-multiply and take the square root, yielding
\[T_2^2 = T_1^2 \left(\dfrac{r_2}{r_1} \right)^3 \]
\[T_2 = T_1 \left(\dfrac{r_2}{r_1} \right)^{\frac{3}{2}}. \]
Substituting known values yields
\[T_2 = 27.3 \, d \times \dfrac {24 \, h}{d} \times \left( \dfrac {7880 \, km}{3.84 \times 10^5 km} \right )^{ \frac{3}{2}} \]
\[ = 1.93 \, h. \]
Discussion This is a reasonable period for a satellite in a fairly low orbit. It is interesting that any satellite at this altitude will orbit in the same amount of time. This fact is related to the condition that the satellite’s mass is small compared with that of Earth.
People immediately search for deeper meaning when broadly applicable laws, like Kepler’s, are discovered. It was Newton who took the next giant step when he proposed the law of universal gravitation. While Kepler was able to discover what was happening, Newton discovered that gravitational force was the cause.
Derivation of Kepler’s Third Law for Circular Orbits
We shall derive Kepler’s third law, starting with Newton’s laws of motion and his universal law of gravitation. The point is to demonstrate that the force of gravity is the cause for Kepler’s laws (although we will only derive the third one).
Let us consider a circular orbit of a small mass \(m\) around a large mass \(M\), satisfying the two conditions stated at the beginning of this section. Gravity supplies the centripetal force to mass \(m\). Starting with Newton’s second law applied to circular motion,
\[ F_{net} = ma_c = m\dfrac{v^2}{r}. \]
The net external force on mass \(m\) is gravity, and so we substitute the force of gravity for \(F_{net}\):
\[ G\dfrac{mM}{r^2} = m\dfrac{v^2}{r}. \]
The mass \(m\) cancels, yielding
\[G\dfrac{M}{r} = v^2. \]
The fact that \(m\) cancels out is another aspect of the oft-noted fact that at a given location all masses fall with the same acceleration. Here we see that at a given orbital radius \(r\), all masses orbit at the same speed. (This was implied by the result of the preceding worked example.) Now, to get at Kepler’s third law, we must get the period \(T\) into the equation. By definition, period \(T\) is the time for one complete orbit. Now the average speed \(v\) is the circumference divided by the period—that is,
\[v = \dfrac{2\pi r}{T}. \]
Substituting this into the previous equation gives \[G\dfrac{M}{r} = \dfrac{4\pi^2 r^2}{T^2}.\]
Solving for \(T^2\) yields \[T^2 = \dfrac{4\pi^2}{GM}r^3. \]
Using subscripts 1 and 2 to denote two different satellites, and taking the ratio of the last equation for satellite 1 to satellite 2 yields
\[\dfrac{T_1^2}{T_2^2} = \dfrac{r_1^3}{r_2^3}. \]
This is Kepler’s third law. Note that Kepler’s third law is valid only for comparing satellites of the same parent body, because only then does the mass of the parent body \(M\) cancel.
Now consider what we get if we solve \(T^2 = \frac{4\pi^2}{GM}r^3\) for the ratio \(r^3/T^2\). We obtain a relationship that can be used to determine the mass \(M\) of a parent body from the orbits of its satellites:
\[\dfrac{r^3}{T^2} = \dfrac{G}{4\pi^2}M.\]
If \(r\) and \(T\) are known for a satellite, then the mass \(M\) of the parent can be calculated. This principle has been used extensively to find the masses of heavenly bodies that have satellites. Furthermore, the ratio \(r^3/T^2\) should be a constant for all satellites of the same parent body (because \(r^3/T^2 = GM/4\pi\). (See Table).
It is clear from Table that the ratio of \(r^3/T^2\) is constant, at least to the third digit, for all listed satellites of the Sun, and for those of Jupiter. Small variations in that ratio have two causes—uncertainties in the \(r\) and \(T\) data, and perturbations of the orbits due to other bodies. Interestingly, those perturbations can be—and have been—used to predict the location of new planets and moons. This is another verification of Newton’s universal law of gravitation.
MAKING CONNECTIONS
Newton’s universal law of gravitation is modified by Einstein’s general theory of relativity, as we shall see in Particle Physics. Newton’s gravity is not seriously in error—it was and still is an extremely good approximation for most situations. Einstein’s modification is most noticeable in extremely large gravitational fields, such as near black holes. However, general relativity also explains such phenomena as small but long-known deviations of the orbit of the planet Mercury from classical predictions.
The Case for Simplicity
The development of the universal law of gravitation by Newton played a pivotal role in the history of ideas. While it is beyond the scope of this text to cover that history in any detail, we note some important points. The definition of planet set in 2006 by the International Astronomical Union (IAU) states that in the solar system, a planet is a celestial body that:
- is in orbit around the Sun,
- has sufficient mass to assume hydrostatic equilibrium and
- has cleared the neighborhood around its orbit.
A non-satellite body fulfilling only the first two of the above criteria is classified as “dwarf planet.”
In 2006, Pluto was demoted to a ‘dwarf planet’ after scientists revised their definition of what constitutes a “true” planet.
| Parent | Satellite | Average orbital radius r (km) | Period T(y) | \(r^3/T^2 (km^3/y^2)\) |
|---|---|---|---|---|
| Earth | Moon |
\(3.84 \times 10^5\)
×
10
5
|
0.07481 |
\(1.01 \times 10^{19}\)
.
01
×
10
19
|
| Sun | Mercury |
\(5.79 \times 10^7\) . 79 |
0.2409 |
\( 3,34 \times 10^{24}\)
.
34
×
10
24
|
| Venus |
\(1.082 \times 10^8\)
.
082
×
10
8
|
0.6150 |
\(3.35 \times 10^{24}\)
.
35
×
10
24
|
|
| Earth |
\(1.496 \times 10^8\)
.
496
×
10
8
|
1.000 |
\(3.35 \times 10^{24}\)
.
35
×
10
24
|
|
| Mars |
\(2.279 \times 10^8\)
.
279
×
10
8
|
1.881 |
\(3.35 \times 10^{24}\)
.
35
×
10
24
|
|
| Jupiter |
\(7.783 \times 10^8\)
.
783
×
10
8
|
11.86 |
\(3.35 \times 10^{24}\)
.
35
×
10
24
|
|
| Saturn |
\(1.427 \times 10^9\)
.
427
×
10
9
|
29.46 |
\(3.35 \times 10^{24}\)
.
35
×
10
24
|
|
| Neptune |
\(4.497 \times 10^9\)
.
497
×
10
9
|
164.8 |
\(3.35 \times 10^{24}\)
.
35
×
10
24
|
|
| Pluto |
\(5.90 \times 10^9\)
.
90
×
10
9
|
248.3 |
\(3.33 \times 10^{24}\)
.
33
×
10
24
|
|
| Jupiter | Io |
\(4.22 \times 10^5\)
.
22
×
10
5
|
0.00485 (1.77 d) |
\(3.19 \times 10^{21}\)
.
19
×
10
21
|
| Europa |
\(6.71 \times 10^5\)
.
71
×
10
5
|
0.00972 (3.55 d) |
\(3.20 \times 10^{21}\)
.
20
×
10
21
|
|
| Ganymede |
\(1.07 \times 10^6\)
.
07
×
10
6
|
0.0196 (7.16 d) |
\(3.19 \times 10^{21}\)
.
19
×
10
21
|
|
| Callisto |
\(1.88 \times 10^6\)
.
88
×
10
6
|
0.0457 (16.19 d) |
\(3.20 \times 10^{21}\)
.
20
×
10
21
|
The universal law of gravitation is a good example of a physical principle that is very broadly applicable. That single equation for the gravitational force describes all situations in which gravity acts. It gives a cause for a vast number of effects, such as the orbits of the planets and moons in the solar system. It epitomizes the underlying unity and simplicity of physics.
Before the discoveries of Kepler, Copernicus, Galileo, Newton, and others, the solar system was thought to revolve around Earth as shown in Figure (a). This is called the Ptolemaic view, for the Greek philosopher who lived in the second century AD. This model is characterized by a list of facts for the motions of planets with no cause and effect explanation. There tended to be a different rule for each heavenly body and a general lack of simplicity.
Figure (b) represents the modern or Copernican model. In this model, a small set of rules and a single underlying force explain not only all motions in the solar system, but all other situations involving gravity. The breadth and simplicity of the laws of physics are compelling. As our knowledge of nature has grown, the basic simplicity of its laws has become ever more evident.
Summary
- Kepler’s laws are stated for a small mass \(m\) orbiting a larger mass \(M\) in near-isolation. Kepler’s laws of planetary motion are then as follows:
Kepler’s first law
The orbit of each planet about the Sun is an ellipse with the Sun at one focus.
Kepler’s second law
Each planet moves so that an imaginary line drawn from the Sun to the planet sweeps out equal areas in equal times.
Kepler’s third law
The ratio of the squares of the periods of any two planets about the Sun is equal to the ratio of the cubes of their average distances from the Sun:
\[ \dfrac{T_1^2}{T_2^2} =\dfrac{r_1^3}{r_2^3},\]
where T is the period (time for one orbit) and \(r\) is the average radius of the orbit.
-
The period and radius of a satellite’s orbit about a larger body M are related by
\[T^2=\frac{4π^2}{GM}r^3\] or \[\dfrac{r^3}{T^2} = \dfrac{G}{4\pi^2}M.\]


16.5: Urinary Retention
Urinary retention is a condition when the patient cannot empty all of the urine from their bladder. Urinary retention can be acute (i.e., the sudden inability to urinate after receiving anesthesia during surgery) or chronic (i.e., a gradual inability to completely empty the bladder due to enlargement of the prostate gland in males). Urinary retention is caused by a blockage that partially or fully prevents the flow of urine or the bladder not being able to create a strong enough force to expel all the urine. In addition to causing discomfort, urinary retention increases the patient’s risk for developing a urinary tract infection (UTI). See Figure 16.5 [1] for an image of an enlarged prostate gland blocking the flow of urine from the bladder into the urethra.
Symptoms of urinary retention can range from none to severe abdominal pain. [2] Health care providers use a patient’s medical history, physical exam finding, and diagnostic tests to find the cause of urinary retention. Nurses typically receive orders to measure post-void residual amounts when urinary retention is suspected. Post-void residual measurements are taken after a patient has voided by using a bladder scanner or inserting a straight urinary catheter to determine how much urine is left in the bladder. See the following box regarding how to perform a bladder scan at the bedside. Read about other diagnostic tests related to urinary retention, such as urodynamic testing and cystoscopy, under the “ Applying the Nursing Process ” section of this chapter. [3]
A bladder scanner is a portable, noninvasive medical device that uses sound waves to calculate the amount of urine in a patient’s bladder. Nurses use bladder scanners at the bedside to determine post-void residual urine amounts in patients to avoid the need to perform an invasive urinary catheterization. Typically the use of a bladder scan does not require a physician order, but be sure to check agency policy.
After the patient voids and is lying in a supine position, turn on the device and indicate if the patient is male or female. (If the female has had a hysterectomy, then “male” is selected.) Apply warmed gel to the transducer head, and then place it approximately one inch above the symphysis pubis with the probe directed towards the bladder. Press the “scan” button, making sure to hold the scanner steady until you hear a beep. The bladder scanner will display the volume measured using a display with crosshairs. If the crosshairs are not centered on the urine displayed, adjust the probe and rescan until it is properly centered. If the post-void residual is greater than 300 mL, the provider should be notified and typically an order will be received for a straight urinary catheterization. Whenever possible, indwelling urinary catheterization (also referred to as a “Foley”) placement is avoided to reduce the patient’s risk of developing a catheter-associated urinary tract infection (CAUTI) . [4]
View this following YouTube video to see a bladder scanner in use [5] : How to Use BladderScan Prime Plus™ by Diane Newman
Interventions
Treatment for urinary retention depends on the cause. It may include urinary catheterization to drain the bladder, bladder training therapy, medications, or surgery. [6] Read more about bladder training therapy under the “ Urinary Incontinence ” section. Alpha blockers, such as tamsulosin (Flomax), are used to treat urinary retention caused by an enlarged prostate. A surgery called transurethral resection of the prostate (TURP) may be performed to treat urinary retention caused by an enlarged prostate that is not responsive to medication.
Read more about urinary catheterization and preventing catheter-associated urinary tract infection (CAUTI) in “ Facilitation of Elimination ” in Open RN Nursing Skills .
Read more about alpha-blocker medication (i.e., tamsulosin) in the “ Autonomic Nervous System ” chapter in Open RN Nursing Pharmacology .
- “ Normal-vs-enlarged-prostate.jpg ” by Akcmdu9 is licensed under CC BY-SA 3.0 ↵
- National Institute of Diabetes and Digestive and Kidney Diseases. (n.d.). Urinary retention. U.S. Department of Health and Human Services. https://www.niddk.nih.gov/health-information/urologic-diseases/urinary-retention ↵
- National Institute of Diabetes and Digestive and Kidney Diseases. (n.d.). Urinary retention. U.S. Department of Health and Human Services. https://www.niddk.nih.gov/health-information/urologic-diseases/urinary-retention ↵
- Agency for Healthcare Research and Quality. (2020, October). Toolkit for reducing catheter-associated urinary tract infections in hospital units: Implementation guide - Appendix C. Sample bladder scan policy . https://www.ahrq.gov/hai/cauti-tools/impl-guide/implementation-guide-appendix-c.html ↵
- BladderScanDevice. (2017, Novemeber 13). How to use BladderScan Prime PlusTM by Diane Newman . [Video]. YouTube. All rights reserved. https://youtu.be/Q-sQu0T2oUY . ↵
- National Institute of Diabetes and Digestive and Kidney Diseases. (n.d.). Urinary retention. U.S. Department of Health and Human Services. https://www.niddk.nih.gov/health-information/urologic-diseases/urinary-retention ↵


2.5: The Total Probability Sum at Constant N, V, and T
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In a collection of distinguishable independent molecules at constant \(N\), \(V\), and \(T\), the probability that a randomly selected molecule has energy \({\epsilon }_i\) is \(P_i\); we have \(1=P_1+P_2+\dots +P_i+\dots\). At any instant, every molecule in the \(N\)-molecule system has a specific energy, and the state of the system is described by a population set, \(\{N_1,\ N_2,\dots ,N_i,\dots .\}\), wherein \(N_i\) can have any value in the range \(0\le N_i\le N\), subject to the condition that
\[N=\sum^{\infty }_{i=1}{N_i} \nonumber \]
The probabilities that we assume for this system of molecules have the properties we assume in Chapter 19 where we find the total probability sum by raising the sum of the energy-level probabilities to the \(N^{th}\) power.
\[1={\left(P_1+P_2+\dots +P_i+\dots \right)}^N=\sum_{\{N_i\}}{\frac{N!}{N_1!N_2!\dots N_i!\dots }}P^{N_1}_1P^{N_2}_2\dots P^{N_i}_i\dots \nonumber \]
The total-probability sum is over all possible population sets, \(\{N_1,\ N_2,\dots ,N_i,\dots .\}\), which we abbreviate to \(\{N_i\}\), in indicating the range of the summation. Each term in this sum represents the probability of the corresponding population set \(\{N_1,\ N_2,\dots ,N_i,\dots .\}\),. At any given instant, one of the possible population sets describes the way that the molecules of the physical system are apportioned among the energy levels. The corresponding term in the total probability sum represents the probability of this apportionment. It is not necessary that all of the energy levels be occupied. We can have \(N_k=0\), in which case \(P^{N_k}_k=P^0_k=1\) and \(N_k!=1\). Energy levels that are not occupied have no effect on the probability of a population set. The unique population set
\[\{N^{\textrm{⦁}}_1,\ N^{\textrm{⦁}}_2,\dots ,N^{\textrm{⦁}}_i,\dots .\} \nonumber \]
that we conjecture to characterize the equilibrium state is represented by one of the terms in this total probability sum. We want to focus on the relationship between a term in the total probability sum and the corresponding state of the physical system.
Each term in the total probability sum includes a probability factor, \(P^{N_1}_1P^{N_2}_2\dots P^{N_i}_i\dots\) This factor is the probability that \(N_i\) molecules occupy each of the energy levels \({\epsilon }_i\). This term is not affected by our assumption that the molecules are distinguishable. The probability factor is multiplied by the polynomial coefficient
\[\frac{N!}{N_1!N_2!\dots N_i!\dots } \nonumber \]
This factor is the number of combinations of distinguishable molecules that arise from the population set \(\{N_1,\ N_2,\dots ,N_i,\dots \}\). It is the number of ways that the \(N\) distinguishable molecules can be assigned to the available energy levels so that \(N_1\) of them are in energy level, \({\epsilon }_1\), etc .
The combinations for the population set {3,2} are shown in Figure 2.
The expression for the number of combinations takes the form it does only because the molecules can be distinguished from one another. To emphasize this point, let us find the number of combinations using the method we develop in Chapter 19 . Briefly recapitulated, the argument is this:
- We can permute the \(N\) molecules in \(N!\) ways. If we were to distinguish (as different combinations) any two permutations of all of the molecules, this would also be the number of combinations.
- In fact, however, we do not distinguish between different permutations of those molecules that are assigned to the same energy level. If the \(N_1\) molecules assigned to the first energy level are \(B\), \(C\), \(Q\), … , \(X\), we do not distinguish the permutation \(BCQ\dots X\) from the permutation \(CBQ\dots X\) or from any other permutation of these \(N_1\) molecules. Then the complete set of \(N!\) permutations contains a subset of \(N_1!\) permutations, all of which are equivalent because they have the same molecules in the first energy level. So the total number of permutations, \(N!\), over-counts the number of combinations by a factor of \(N_1!\) We can correct for this over-count by dividing by \(N_1!\) That is, after correcting for the over-counting for the \(N_1\) molecules in the first energy level, the number of combinations is \({N!}/{N_1!}\) (If all \(N\) of the molecules were in the first energy level, there would be only one combination. We would have \(N=N_1\), and the number of combinations calculated from this formula would be \({N!}/{N!}=1\), as required.)
- The complete set of \(N!\) permutations also includes \(N_2!\) permutations of the \(N_2\) molecules in the second energy level. In finding the number of combinations, we want to include only one of these permutations, so correcting for the over-counting due to both the \(N_1\) molecules in the first energy level and the \(N_2\) molecules in the second energy level gives \[\frac{N!}{N_1!N_2!} \nonumber \]
- Continuing this argument through all of the occupied energy levels, we see that the total number of combinations is
\[C\left(N_1,N_2,\dots ,N_i,\dots \right)=\frac{N!}{N_1!N_2!\dots N_i!\dots } \nonumber \]
Because there are infinitely many energy levels and probabilities, \(P_i\), there are infinitely many terms in the total-probability sum. Every energy available to the macroscopic system is represented by one or more terms in this total-probability sum. Since there is no restriction on the energy levels that can be occupied, there are an infinite number of such system energies. There is an enormously large number of terms each of which corresponds to an enormously large system energy. Nevertheless, the sum of all of these terms must be one. The \(P_i\) form a convergent series, and the total probability sum must sum to unity.
Just as the \(P_i\) series can converge only if the probabilities of high molecular energies become very small, so the total probability sum can converge only if the probabilities of high system energies become very small. If a population set has \(N_i\) molecules in the \(i^{th}\) energy level, the probability of that population set is proportional to \(P^{N_i}_i\). We see therefore, that the probability of a population set in which there are many molecules in high energy levels must be very small. Terms in the total probability sum that correspond to population sets with many molecules in high energy levels must be negligible. Equivalently, at a particular temperature, macroscopic states in which the system energy is anomalously great must be exceedingly improbable.
What terms in the total probability sum do we need to consider? Evidently from among the infinitely many terms that occur, we can select a finite subset whose sum is very nearly one. If there are many terms that are small and nearly equal to one another, the number of terms in this finite subset could be large. Nevertheless, we can see that terms in this subset must involve the largest possible \(P_i\) values raised to the smallest possible powers, \(N_i\), consistent with the requirement that the \(N_i\) sum to \(N\).
If an equilibrium macroscopic system could have only one population set, the probability of that population set would be unity. Could an equilibrium system be characterized by two or more population sets for appreciable fractions of an observation period? Would this require that the macroscopic system change its properties with time as it jumps from one population set to another? Evidently, it would not, since our observations of macroscopic systems show that the equilibrium properties are unique. A system that wanders between two (or more) macroscopically distinguishable states cannot be at equilibrium. We are forced to the conclusion that, if a macroscopic equilibrium system has multiple population sets with non-negligible probabilities, the macroscopic properties associated with each of these population sets must be indistinguishably similar. (The alternative is to abandon the theory, which is useful only if its microscopic description of a system makes useful predictions about the system’s macroscopic behavior.)
To be a bit more precise about this, we recognize that our theory also rests on another premise: Any intensive macroscopic property of many independent molecules depends on the energy levels available to an individual molecule and the fraction of the molecules that populate each energy level. The average energy is a prime example. For the population set \(\{N_1,\ N_2,\dots ,N_i,\dots .\}\), the average molecular energy is
\[\overline{\epsilon }=\sum^{\infty }_{i=1}{\left(\frac{N_i}{N}\right)}{\epsilon }_i \nonumber \]
We recognize that many population sets may contribute to the total probability sum at equilibrium. If we calculate essentially the same \(\overline{\epsilon }\) from each of these contributing population sets, then all of the contributing population sets correspond to indistinguishably different macroscopic energies. We see in the next section that the central limit theorem guarantees that this happens whenever \(N\) is as large as the number of molecules in a macroscopic system.


4.1: Vectors in Rⁿ
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Outcomes
- Find the position vector of a point in \(\mathbb{R}^n\).
The notation \(\mathbb{R}^{n}\) refers to the collection of ordered lists of \(n\) real numbers, that is \[\mathbb{R}^{n} = \left\{ \left( x_{1}\cdots x_{n}\right) :x_{j}\in \mathbb{R}\text{ for }j=1,\cdots ,n\right\}\nonumber \] In this chapter, we take a closer look at vectors in \(\mathbb{R}^n\). First, we will consider what \(\mathbb{R}^n\) looks like in more detail. Recall that the point given by \(0=\left( 0, \cdots, 0 \right)\) is called the origin .
Now, consider the case of \(\mathbb{R}^n\) for \(n=1.\) Then from the definition we can identify \(\mathbb{R}\) with points in \(\mathbb{R}^{1}\) as follows: \[\mathbb{R} = \mathbb{R}^{1}= \left\{ \left( x_{1}\right) :x_{1}\in \mathbb{R} \right\}\nonumber \] Hence, \(\mathbb{R}\) is defined as the set of all real numbers and geometrically, we can describe this as all the points on a line.
Now suppose \(n=2\). Then, from the definition, \[\mathbb{R}^{2}= \left\{ \left(x_{1}, x_{2}\right) :x_{j}\in \mathbb{R}\text{ for }j=1,2 \right\}\nonumber \] Consider the familiar coordinate plane, with an \(x\) axis and a \(y\) axis. Any point within this coordinate plane is identified by where it is located along the \(x\) axis, and also where it is located along the \(y\) axis. Consider as an example the following diagram.
Hence, every element in \(\mathbb{R}^2\) is identified by two components, \(x\) and \(y\), in the usual manner. The coordinates \(x, y\) (or \(x_1\),\(x_2\)) uniquely determine a point in the plan. Note that while the definition uses \(x_1\) and \(x_2\) to label the coordinates and you may be used to \(x\) and \(y\), these notations are equivalent.
Now suppose \(n=3\). You may have previously encountered the \(3\)-dimensional coordinate system, given by \[\mathbb{R}^{3}= \left\{ \left( x_{1}, x_{2}, x_{3}\right) :x_{j}\in \mathbb{R}\text{ for }j=1,2,3 \right\}\nonumber \]
Points in \(\mathbb{R}^3\) will be determined by three coordinates, often written \(\left(x,y,z\right)\) which correspond to the \(x\), \(y\), and \(z\) axes. We can think as above that the first two coordinates determine a point in a plane. The third component determines the height above or below the plane, depending on whether this number is positive or negative, and all together this determines a point in space. You see that the ordered triples correspond to points in space just as the ordered pairs correspond to points in a plane and single real numbers correspond to points on a line.
The idea behind the more general \(\mathbb{R}^n\) is that we can extend these ideas beyond \(n = 3.\) This discussion regarding points in \(\mathbb{R}^n\) leads into a study of vectors in \(\mathbb{R}^n\). While we consider \(\mathbb{R}^n\) for all \(n\), we will largely focus on \(n=2,3\) in this section.
Consider the following definition.
Definition \(\PageIndex{1}\) THe Position Vector
Let \(P=\left( p_{1},\cdots ,p_{n}\right)\) be the coordinates of a point in \(\mathbb{R}^{n}.\) Then the vector \(\overrightarrow{0P}\) with its tail at \(0=\left( 0,\cdots ,0\right)\) and its tip at \(P\) is called the position vector of the point \(P\). We write \[\overrightarrow{0P} = \left [ \begin{array}{c} p_{1} \\ \vdots \\ p_{n} \end{array} \right ]\nonumber \]
For this reason we may write both \(P=\left( p_{1},\cdots ,p_{n}\right) \in \mathbb{R}^{n}\) and \(\overrightarrow{0P} = \left [ p_{1} \cdots p_{n} \right ]^T \in \mathbb{R}^{n}\).
This definition is illustrated in the following picture for the special case of \(\mathbb{R}^{3}\).
Thus every point \(P\) in \(\mathbb{R}^{n}\) determines its position vector \(\overrightarrow{0P}\). Conversely, every such position vector \(\overrightarrow{0P}\) which has its tail at \(0\) and point at \(P\) determines the point \(P\) of \(\mathbb{R}^{n}\).
Now suppose we are given two points, \(P,Q\) whose coordinates are \(\left( p_{1},\cdots ,p_{n}\right)\) and \(\left( q_{1},\cdots ,q_{n}\right)\) respectively. We can also determine the position vector from \(P\) to \(Q\) (also called the vector from \(P\) to \(Q\) ) defined as follows. \[\overrightarrow{PQ} = \left [ \begin{array}{c} q_{1}-p_{1}\\ \vdots \\ q_{n}-p_{n} \end{array} \right ] = \overrightarrow{0Q} - \overrightarrow{0P}\nonumber \]
Now, imagine taking a vector in \(\mathbb{R}^n\) and moving it around, always keeping it pointing in the same direction as shown in the following picture.
After moving it around, it is regarded as the same vector. Each vector, \(\overrightarrow{0P}\) and \(\overrightarrow{AB}\) has the same length (or magnitude) and direction. Therefore, they are equal.
Consider now the general definition for a vector in \(\mathbb{R}^n\).
Definition \(\PageIndex{2}\) Vectors in \(\mathbb{R}^n\)
Let \(\mathbb{R}^{n} = \left\{ \left( x_{1}, \cdots, x_{n}\right) :x_{j}\in \mathbb{R}\text{ for }j=1,\cdots ,n\right\} .\) Then, \[\vec{x} = \left [ \begin{array}{c} x_{1} \\ \vdots \\ x_{n} \end{array} \right ]\nonumber \] is called a vector . Vectors have both size (magnitude) and direction. The numbers \(x_{j}\) are called the components of \(\vec{x}\).
Using this notation, we may use \(\vec{p}\) to denote the position vector of point \(P\). Notice that in this context, \(\vec{p} = \overrightarrow{0P}\). These notations may be used interchangeably.
You can think of the components of a vector as directions for obtaining the vector. Consider \(n=3\). Draw a vector with its tail at the point \(\left( 0,0,0\right)\) and its tip at the point \(\left( a,b,c\right)\). This vector it is obtained by starting at \(\left( 0,0,0\right)\), moving parallel to the \(x\) axis to \(\left( a,0,0\right)\) and then from here, moving parallel to the \(y\) axis to \(\left( a,b,0\right)\) and finally parallel to the \(z\) axis to \(\left( a,b,c\right).\) Observe that the same vector would result if you began at the point \(\left( d,e,f \right)\), moved parallel to the \(x\) axis to \(\left( d+a,e,f\right) ,\) then parallel to the \(y\) axis to \(\left( d+a,e+b,f\right) ,\) and finally parallel to the \(z\) axis to \(\left( d+a,e+b,f+c\right)\). Here, the vector would have its tail sitting at the point determined by \(A= \left( d,e,f\right)\) and its point at \(B=\left( d+a,e+b,f+c\right) .\) It is the same vector because it will point in the same direction and have the same length. It is like you took an actual arrow, and moved it from one location to another keeping it pointing the same direction.
We conclude this section with a brief discussion regarding notation. In previous sections, we have written vectors as columns, or \(n \times 1\) matrices. For convenience in this chapter we may write vectors as the transpose of row vectors, or \(1 \times n\) matrices. These are of course equivalent and we may move between both notations. Therefore, recognize that \[\left [ \begin{array}{r} 2 \\ 3 \end{array} \right ] = \left [ \begin{array}{rr} 2 & 3 \end{array} \right ]^T\nonumber \]
Notice that two vectors \(\vec{u} = \left [ u_{1} \cdots u_{n}\right ]^T\) and \(\vec{v}=\left [ v_{1} \cdots v_{n}\right ]^T\) are equal if and only if all corresponding components are equal. Precisely, \[\begin{array}{c} \vec{u}=\vec{v} \; \mbox{if and only if}\\ u_{j}=v_{j} \; \mbox{for all}\; j=1,\cdots ,n \end{array}\nonumber \] Thus \(\left [ \begin{array}{rrr} 1 & 2 & 4 \end{array} \right ]^T \in \mathbb{R}^{3}\) and \(\left [ \begin{array}{rrr} 2 & 1 & 4 \end{array} \right ]^T \in \mathbb{R}^{3}\) but \(\left [ \begin{array}{rrr} 1 & 2 & 4 \end{array} \right ]^T \neq \left [ \begin{array}{rrr} 2 & 1 & 4 \end{array} \right ]^T\) because, even though the same numbers are involved, the order of the numbers is different.
For the specific case of \(\mathbb{R}^3\), there are three special vectors which we often use. They are given by \[\vec{i} = \left [ \begin{array}{rrr} 1 & 0 & 0 \end{array} \right ]^T\nonumber \] \[\vec{j} = \left [ \begin{array}{rrr} 0 & 1 & 0 \end{array} \right ]^T\nonumber \] \[\vec{k} = \left [ \begin{array}{rrr} 0 & 0 & 1 \end{array} \right ]^T\nonumber \] We can write any vector \(\vec{u} = \left [ \begin{array}{rrr} u_1 & u_2 & u_3 \end{array} \right ]^T\) as a linear combination of these vectors, written as \(\vec{u} = u_1 \vec{i} + u_2 \vec{j} + u_3 \vec{k}\). This notation will be used throughout this chapter.


19.6: Stool Specimen Collection
Stool samples are collected from patients to test for cancer, parasites, or for occult blood (i.e., hidden blood). Follow specific instructions from the laboratory for collecting the sample.
The Guaiac-Based Fecal Occult Blood Test (gFOBT) is a commonly used test to find hidden blood in the stool that is not visibly apparent. As a screening test for colon cancer, it is typically obtained by the patient in their home using samples from three different bowel movements. Nurses may assist in gFOBT specimen collection during inpatient care.
Before the test, the patient should avoid red meat for three days and should not take aspirin or nonsteroidal anti-inflammatory drugs (NSAIDs), such as ibuprofen, for seven days prior to the test. (Blood from the meat can cause a false positive test, and aspirin and NSAIDS can cause bleeding, leading to a false positive result.) Vitamin C (more than 250 mg a day) from supplements, citrus fruits, or citrus juices should be avoided for 3 to 7 days before testing because it can affect the chemicals in the test and make the result negative, even if blood is present.
To perform a gFOBT in an inpatient setting, perform the following steps.
- Verify the patient has not consumed red meat for three days, has not taken aspirin or NSAIDs for seven days prior to the test, and has not had vitamin C greater than 250 mg daily for the past 3-7 days because these substances can affect the results.
- Explain the procedure to the patient. Instruct them to flush the toilet before defecating to remove any potential chemicals and to not place toilet paper in the toilet after defecating. Request they notify you when they have had a bowel movement.
- Review the manufacturer’s instructions because different test kits may have different instructions. Contact the laboratory with any questions.
- Label the card with the patient’s name and medical information per agency policy. Open the flap of the guaiac test card.
- Apply nonsterile gloves. Use the applicator stick to apply a thin smear of the stool specimen to one of the squares of filter paper on the card. Obtain a second specimen from a different part of the stool and apply it to the second square of filter paper on the card. (Occult blood isn’t typically equally dispersed throughout the stool.)
- Place the labeled test card in a transport bag and send it to the laboratory for analysis.
- If you are working in an agency where nurses apply the guaiac developer solution to the card, allow the specimen to dry for 3 to 5 minutes. Open the reverse side of the card and apply two drops of guaiac developer solution to each square. A blue reaction will occur within 60 seconds if the test is positive. The absence of a blue color after 60 seconds is considered a negative test.
- Document the date and time of the test and any unusual characteristics of the stool sample. [1] , [2]
- American Cancer Society (2020, June 29). Colorectal cancer screening tests . www.cancer.org/cancer/colon-rectal-cancer/detection-diagnosis-staging/screening-tests-used.htm↵
- A.D.A.M. Medical Encyclopedia [Internet]. Atlanta (GA): A.D.A.M., Inc.; c1997-2021. Stool guaiac test; [updated 2021, February 26]. https://medlineplus.gov/ency/article/003393.htm ↵


2.5.3: Quantifier Rules
Rules for \(\lforall{}{}\)
In the rules for \(\lforall{}{}\) , \(t\) is a ground term (a term that does not contain any variables), and \(a\) is a constant symbol which does not occur in the conclusion \(\lforall{x}{A(x)}\) , or in any assumption which is undischarged in the derivation ending with the premise \(A(a)\) . We call \(a\) the eigenvariable of the \(\Intro{\lforall{}{}}\) inference.
Rules for \(\lexists{}{}\)
Again, \(t\) is a ground term, and \(a\) is a constant which does not occur in the premise \(\lexists{x}{A(x)}\) , in the conclusion \(C\) , or any assumption which is undischarged in the derivations ending with the two premises (other than the assumptions \(A(a)\) ). We call \(a\) the eigenvariable of the \(\Elim{\lexists{}{}}\) inference.
The condition that an eigenvariable neither occur in the premises nor in any assumption that is undischarged in the derivations leading to the premises for the \(\Intro{\lforall{}{}}\) or \(\Elim{\lexists{}{}}\) inference is called the eigenvariable condition .
We use the term “eigenvariable” even though \(a\) in the above rules is a constant. This has historical reasons.
In \(\Intro{\lexists{}{}}\) and \(\Elim{\lforall{}{}}\) there are no restrictions, and the term \(t\) can be anything, so we do not have to worry about any conditions. On the other hand, in the \(\Elim{\lexists{}{}}\) and \(\Intro{\lforall{}{}}\) rules, the eigenvariable condition requires that the constant symbol \(a\) does not occur anywhere in the conclusion or in an undischarged assumption. The condition is necessary to ensure that the system is sound, i.e., only derives sentences from undischarged assumptions from which they follow. Without this condition, the following would be allowed:
However, \(\lexists{x}{A(x)} \EntailsN \lforall{x}{A(x)}\) .


8.1: Introduction to Welding and Cutting
Welding in Construction
Welding and cutting on construction sites is a common task which is often performed by several different trades. OSHA investigations into accidents related to welding and cutting discovered that the most frequent accidents resulted from the ignition of fumes/vapors or other explosive materials in the vicinity of the welding or cutting operation. These types of fire related accidents may be attributed to improper material handling and storage.
Another hazard that workers encounter is the long-term effects of welding and cutting. These effects include damage to the eyes, lungs, and skin. Other hazards include falls from elevations, electrocutions, caught between, and explosions as welding activity can occur during all phases of construction.
To address these safety issues Subpart J of the 1926 OSHA standards covers welding and cutting requirements for construction sites. This subpart includes requirements for gas welding and cutting, Arc welding and cutting, fire prevention, ventilation and health concerns when heating treated metals.
Welding produces toxic vapors and fumes such as beryllium, chromium, and fuel gases for welding are considered hazardous materials. Because the production of toxic vapors during welding may create confined space conditions or if the welding activity is occurring in a confined space it is important to ensure proper ventilation and when appropriate air supplied respirators.
Gas Welding and Cutting
General
When transporting, moving, and storing compressed gas cylinders, valve protection caps shall be in place and secured. When cylinders are hoisted, they shall be secured on a cradle, sling board or pallet. They shall not be hoisted or transported by means of magnets or choker slings. A suitable cylinder truck, chain, or other steadying device shall be used to keep cylinders from being knocked over while in use.
Transporting
When cylinders are transported by powered vehicles, they shall be secured in a vertical position.
Moving
Unless cylinders are firmly secured on a special carrier intended for this purpose, regulators shall be removed and valve protection caps put in place before cylinders are moved.
Compressed gas cylinders shall be secured in an upright position at all times except, if necessary, for short periods of time while cylinders are actually being hoisted or carried. This is the most frequently cited welding and cutting violation.
Cylinders shall be moved by tilting and rolling them on their bottom edges. They shall not be intentionally dropped, struck, or permitted to strike each other violently. Valve protection caps shall not be used for lifting cylinders from one vertical position to another. Bars shall not be used under valves or valve protection caps to pry cylinders loose when frozen. Warm, not boiling, water shall be used to thaw cylinders loose.
Storage
When work is finished, when cylinders are empty, or when cylinders are moved at any time, the cylinder valve shall be closed. Oxygen cylinders in storage shall be separated from fuel- gas cylinders or combustible materials (especially oil or grease), a minimum distance of 20 feet (6.1 m) or by a noncombustible barrier at least 5 feet (1.5 m) high having a fire- resistance rating of at least one-half hour.
Inside of buildings, cylinders shall be stored in a well-protected well-ventilated, dry location, at least 20 feet (6.1 m) from highly combustible materials such as oil or excelsior. Cylinders should be stored in definitely assigned places away from elevators, stairs, or gangways.
Assigned storage places shall be located where cylinders will not be knocked over or damaged by passing or falling objects, or subject to tampering by unauthorized persons. Cylinders shall not be kept in unventilated enclosures such as lockers and cupboards.
Cylinder Placement
Cylinders shall be kept far enough away from the actual welding or cutting operation so that sparks, hot slag, or flame will not reach them. When this is impractical, fire resistant shields shall be provided. Cylinders shall be placed where they cannot become part of an electrical circuit. Electrodes shall not be struck against a cylinder to strike an arc.
Fuel gas cylinders shall be placed with valve end up whenever they are in use. They shall not be placed in a location where they would be subject to open flame, hot metal, or other sources of artificial heat. Cylinders containing oxygen or acetylene or other fuel gas shall not be taken into confined spaces.
Cylinders, whether full or empty, shall not be used as rollers or supports. No damaged or defective cylinder shall be used.
Use of Fuel Gas
Training
Employers shall instruct employees in the safe use of fuel gas, as follows:
- Before a regulator is connected to a cylinder valve, the valve shall be opened slightly and closed immediately. (This action is generally termed "cracking" and is intended to clear the valve of dust or dirt that might otherwise enter the regulator.) The person cracking the valve shall stand to one side of the outlet, not in front of it. The valve of a fuel gas cylinder shall not be cracked where the gas would reach welding work, sparks, flame, or other possible sources of ignition.
- The cylinder valve shall always be opened slowly to prevent damage to the regulator. For quick closing, valves on fuel gas cylinders shall not be opened more than 1 ½ turns. When a special wrench is required, it shall be left in position on the stem of the valve while the cylinder is in use so that the fuel gas flow can be shut off quickly in case of an emergency. In the case of manifolded or coupled cylinders, at least one such wrench shall always be available for immediate use. Nothing shall be placed on top of a fuel gas cylinder when in use, which may damage the safety device or interfere with the quick closing of the valve.
- Fuel gas shall not be released from cylinders through torches or other devices, which are equipped with shutoff valves without reducing the pressure through a suitable regulator, attached to the cylinder valve or manifold.
- Before a regulator is removed from a cylinder valve, the cylinder valve shall always be closed and the gas released from the regulator.
- If, when the valve on a fuel gas cylinder is opened, and a leak is found around the valve stem, the valve shall be closed and the gland nut tightened. If this action does not stop the leak, the use of the cylinder shall be discontinued, and it shall be properly tagged and removed from the work area. In the event that fuel gas should leak from the cylinder valve, rather than from the valve stem, and the gas cannot be shut off, the cylinder shall be properly tagged and removed from the work area. If a regulator attached to a cylinder valve will effectively stop a leak through the valve seat, the cylinder need not be removed from the work area.
- If a leak should develop at a fuse plug or other safety device, the cylinder shall be removed from the work area.
Fuel Gas and Oxygen Manifold
Marking
Fuel gas and oxygen manifolds shall bear the name of the substance they contain in letters at least 1-inch high which shall be either painted on the manifold or on a sign permanently attached to it.
Location
Fuel gas and oxygen manifolds shall be placed in safe, well ventilated, and accessible locations. They shall not be located within enclosed spaces.
Connections
Manifold hose connections, including both ends of the supply hose that lead to the manifold, shall be such that the hose cannot be interchanged between fuel gas and oxygen manifolds and supply header connections. Adapters shall not be used to permit the interchange of hose. Hose connections shall be kept free of grease and oil.
Storage
When not in use, manifold and header hose connections shall be capped. Nothing shall be placed on top of a manifold, when in use, which will damage the manifold or interfere with the quick closing of the valves.
Fuel Hoses
Identification
Fuel gas hoses and oxygen hoses shall be easily distinguishable from each other. The contrast may be made by different colors or by surface characteristics readily distinguishable by the sense of touch. Oxygen and fuel gas hoses shall not be interchangeable. A single hose having more than one gas passage shall not be used.
Maintenance, inspection, and testing
All hoses in use, carrying acetylene, oxygen, natural or manufactured fuel gas, or any gas or substance which may ignite or enter into combustion or be in any way harmful to employees, shall be inspected at the beginning of each working shift. Defective hoses shall be removed from service.
Hose which has been subject to flashback, or which shows evidence of severe wear or damage, shall be tested to twice the normal pressure to which it is subject, but in no case less than 300 psi. Defective hose, or hose in doubtful condition, shall not be used.
When parallel sections of oxygen and fuel gas hose are taped together, not more than 4 inches out of 12 inches shall be covered by tape.
Couplings
Hose couplings shall be of the type that cannot be unlocked or disconnected by means of a straight pull without rotary motion. Boxes used for the storage of gas hose shall be ventilated. Hoses, cables, and other equipment shall be kept clear of passageways, ladders and stairs.
Torches
Clogged torch tip openings shall be cleaned with suitable cleaning wires drills, or other devices designed for such purpose. Torches in use shall be inspected at the beginning of each working shift for leaking shutoff valves, hose couplings, and tip connections. Defective torches shall not be used. Torches shall be lighted by friction lighters or other approved devices, and not by matches or from hot work.
Oil and grease hazards
Oxygen cylinders and fittings shall be kept away from oil or grease. Cylinders, cylinder caps and valves, couplings, regulators, hose, and apparatus shall be kept free from oil or greasy substances and shall not be handled with oily hands or gloves. Oxygen shall not be directed at oily surfaces, greasy clothes, or within a fuel oil or other storage tank or vessel.
Arc Welding and Cutting
Manual Electrode Holders
Only manual electrode holders which are specifically designed for arc welding and cutting, and are of a capacity capable of safely handling the maximum rated current required by the electrodes, shall be used.
Any current-carrying parts passing through the portion of the holder, which the arc welder or cutter grips in his hand, and the outer surfaces of the jaws of the holder, shall be fully insulated against the maximum voltage, encountered to ground.
Welding cables and connectors
All arc welding and cutting cables shall be of the completely insulated flexible type, capable of handling the maximum current requirements of the work in progress, taking into account the duty cycle under which the arc welder or cutter is working.
Use cable free from repair or splices for a minimum distance of 10 feet from the cable end to which the electrode holder is connected, except that cables with standard insulated connectors or with splices whose insulating quality is equal to that of the cable are permitted.
When it becomes necessary to connect or splice lengths of cable one to another, substantial insulated connectors of a capacity at least equivalent to that of the cable shall be used. If connections are effected by means of cable lugs, they shall be securely fastened together to give good electrical contact and the exposed metal parts of the lugs shall be completely insulated.
Cables in need of repair shall not be used. When a cable, other than those with acceptable splices, becomes worn to the extent of exposing bare conductors, the portion thus exposed shall be protected by means of rubber and friction tape or other equivalent insulation.
Ground returns and machine grounding
A ground return cable shall have a safe current carrying capacity equal to or exceeding the specified maximum output capacity of the arc welding or cutting unit, which it services. When a single ground return cable services more than one unit, it’s safe current-carrying capacity shall equal or exceed the total specified maximum output capacities of all the units, which it services. Pipelines containing gases or flammable liquids, or conduits containing electrical circuits, shall not be used as a ground return.
When a structure or pipeline is employed as a ground return circuit, it shall be determined that the required electrical contact exists at all joints. The generation of an arc, sparks, or heat at any point shall cause rejection of the structures as a ground circuit.
When a structure or pipeline is continuously employed as a ground return circuit, all joints shall be bonded, and periodic inspections shall be conducted to ensure that no condition of electrolysis or fire hazard exists by virtue of such use.
The frames of all arc welding and cutting machines shall be grounded either through a third wire in the cable containing the circuit conductor or through a separate wire which is grounded at the source of the current. Grounding circuits, other than by means of the structure, shall be checked to ensure that the circuit between the ground and the grounded power conductor has resistance low enough to permit sufficient current to flow to cause the fuse or circuit breaker to interrupt the current.
All ground connections shall be inspected to ensure that they are mechanically strong and electrically adequate for the required current.
Training
Employers shall instruct employees in the safe means of arc welding and cutting as follows:
- When electrode holders are to be left unattended, the electrodes shall be removed and the holders shall be so placed or protected that they cannot make electrical contact with employees or conducting objects.
- Hot electrode holders shall not be dipped in water; to do so may expose the arc welder or cutter to electric shock.
- When the arc welder or cutter has occasion to leave his work or to stop work for any appreciable length of time, or when the arc welding or cutting machine is to be moved, the power supply switch to the equipment shall be opened.
- Any faulty or defective equipment shall be reported to the supervisor.
Shielding
Whenever practicable, all arc welding and cutting operations shall be shielded by noncombustible or flameproof screens which will protect employees and other persons working in the vicinity from the direct rays of the arc.


22.1: Muscle Strength and Endurance Reading
Purpose
This section will focus on the last two components of fitness: muscular strength and muscular endurance.
Directions
The rest of this chapter contains your reading for this section.
Additionally, information on these external sources will be useful to deepen your understanding:
- Read Musculoskeletal Fitness (Kravitz, Len. “Musculoskeletal Fitness.” Encyclopedia of Lifestyle Medicine & Health . Ed. James M. Rippe. Vol. 2. Thousand Oaks, CA: SAGE Reference, 2012. 734-737. Gale Virtual Reference Library . Retrieved from http://go.galegroup.com/ps/i.do?id=G...9e21e93a586875 )
This article explains in detail the benefits of improving musculoskeletal fitness.
- Review sections one, two, and four in this document: Health-Related Fitness Overview
You have already read this document before. Therefore, this time you can focus specifically on the information about muscular strength, muscular endurance, and resistance training.
- Lifetime Fitness and Wellness. Provided by : Extended Learning Institute of Northern Virginia Community College. Located at : https://eli.nvcc.edu/ . License : CC BY: Attribution


28.6: Popular Culture and Mass Media
With a greater generational consciousness than previous generations, the baby boomers sought to define and redefine their identities in numerous ways. Music, especially rock and roll, reflected their desire to rebel against adult authority. Other forms of popular culture, such as movies and television, sought to entertain, while reinforcing values such as religious faith, patriotism, and conformity to societal norms.
 This experimentation led to the creation of a new musical form known as rockabilly, and by the 1950s, rockabilly had developed into rock and roll. Rock and roll music celebrated themes such as young love and freedom from the oppression of middle-class society. It quickly grew in favor among American teens, thanks largely to the efforts of disc jockey Alan Freed, who named and popularized the music by playing it on the radio in Cleveland, where he also organized the first rock and roll concert, and later in New York.
The theme of rebellion against authority, present in many rock and roll songs, appealed to teens. In 1954, Bill Haley and His Comets provided youth with an anthem for their rebellion—”Rock Around the Clock” (Figure 28.4.1). The song, used in the 1955 movie Blackboard Jungle about a white teacher at a troubled inner-city high school, seemed to be calling for teens to declare their independence from adult control.
Haley illustrated how white artists could take musical motifs from the African American community and achieve mainstream success. Teen heartthrob Elvis Presley rose to stardom doing the same. Thus, besides encouraging a feeling of youthful rebellion, rock and roll also began to tear down color barriers, as white youths sought out African American musicians such as Chuck Berry and Little Richard (Figure 28.4.1).
While youth had found an outlet for their feelings and concerns, parents were much less enthused about rock and roll and the values it seemed to promote. Many regarded the music as a threat to American values. When Elvis Presley appeared on The Ed Sullivan Show , a popular television variety program, the camera deliberately focused on his torso and did not show his swiveling hips or legs shaking in time to the music. Despite adults’ dislike of the genre, or perhaps because of it, more than 68 percent of the music played on the radio in 1956 was rock and roll.
HOLLYWOOD ON THE DEFENSIVE
At first, Hollywood encountered difficulties in adjusting to the post-World War II environment. Although domestic audiences reached a record high in 1946 and the war’s end meant expanding international markets too, the groundwork for the eventual dismantling of the traditional studio system was laid in 1948, with a landmark decision by the U.S. Supreme Court. Previously, film studios had owned their own movie theater chains in which they exhibited the films they produced; however, in United States v. Paramount Pictures, Inc. , this vertical integration of the industry—the complete control by one firm of the production, distribution, and exhibition of motion pictures—was deemed a violation of antitrust laws.
The HUAC hearings also targeted Hollywood. When Senator McCarthy called eleven “unfriendly witnesses” to testify before Congress about Communism in the film industry in October 1947, only playwright Bertolt Brecht answered questions. The other ten, who refused to testify, were cited for contempt of Congress on November 24. The next day, film executives declared that the so-called “Hollywood Ten” would no longer be employed in the industry until they had sworn they were not Communists (Figure 28.4.2). Eventually, more than three hundred actors, screenwriters, directors, musicians, and other entertainment professionals were placed on the industry blacklist. Some never worked in Hollywood again; others directed films or wrote screenplays under assumed names.
Watch a 1953 episode of a popular television show from the 1950s, I Led Three Lives , the highly fictionalized story of a member of a Communist organization who is also an FBI informant.
Hollywood reacted aggressively to these various challenges. Filmmakers tried new techniques, like CinemaScope and Cinerama, which allowed movies to be shown on large screens and in 3-D. Audiences were drawn to movies not because of gimmicks, however, but because of the stories they told. Dramas and romantic comedies continued to be popular fare for adults, and, to appeal to teens, studios produced large numbers of horror films and movies starring music idols such as Elvis. Many films took espionage, a timely topic, as their subject matter, and science fiction hits such as Invasion of the Body Snatchers , about a small town whose inhabitants fall prey to space aliens, played on audience fears of both Communist invasion and nuclear technology.
THE TRIUMPH OF TELEVISION
By far the greatest challenge to Hollywood, however, came from the relatively new medium of television. Although the technology had been developed in the late 1920s, through much of the 1940s, only a fairly small audience of the wealthy had access to it. As a result, programming was limited. With the post-World War II economic boom, all this changed. Where there had been only 178,000 televisions in homes in 1948, by 1955, over three-quarters of a million U.S. households, about half of all homes, had television (Figure 28.4.3).
Various types of programs were broadcast on the handful of major networks: situation comedies, variety programs, game shows, soap operas, talk shows, medical dramas, adventure series, cartoons, and police procedurals. Many comedies presented an idealized image of white suburban family life: Happy housewife mothers, wise fathers, and mischievous but not dangerously rebellious children were constants on shows like Leave It to Beaver and Father Knows Best in the late 1950s. These shows also reinforced certain perspectives on the values of individualism and family—values that came to be redefined as “American” in opposition to alleged Communist collectivism. Westerns, which stressed unity in the face of danger and the ability to survive in hostile environments, were popular too. Programming designed specifically for children began to emerge with shows such as Captain Kangaroo , Romper Room , and The Mickey Mouse Club designed to appeal to members of the baby boom.
Section Summary
Young Americans in the postwar period had more disposable income and enjoyed greater material comfort than their forebears. These factors allowed them to devote more time and money to leisure activities and the consumption of popular culture. Rock and roll, which drew from African American roots in the blues, embraced themes popular among teenagers, such as young love and rebellion against authority. At the same time, traditional forms of entertainment, such as motion pictures, came under increasing competition from a relatively new technology, television.
Review Questions
The disc jockey who popularized rock and roll was ________.
Bill Haley
Elvis Presley
Alan Freed
Ed Sullivan
C
What challenges did Hollywood face in the 1950s?
Antitrust lawsuits deprived studios of their theaters, and the careers of many actors, directors, and screenwriters were destroyed by Senator McCarthy’s blacklist of suspected Communists. Meanwhile, the new technology of television drew audiences away from the movies by providing convenient at-home entertainment.
Glossary
- rock and roll
- a musical form popular among the baby boomers that encompassed styles ranging from county to blues, and embraced themes such as youthful rebellion and love


13.2: Net and Gross Cash Flows
Learning Outcomes
- Differentiate between net and gross cash flows
Gross cash flows don’t exist in the operating portion of the cash flow statement. GAAP (generally accepted accounting principles) and IFRS (international financial reporting standards) for foreign companies, require us to disclose the gross cash flows for the investing and financing sections of the cash flow statement.
Gross cash flows essentially include the purchase price in cash of a new piece of property or equipment, and the cash gain of the sale of a piece of property or equipment.
So if a company purchased $25,000 of new equipment and sold $10,000 of equipment, the net cash flow would be $15,000. But in the investing and financing sections, we need to keep those separate. It is still important to note the difference in balance netted, then backing it out to the gross calculation!
Here is an outline of how noncash balance sheet accounts affect the investing and financing sections of your cash flow statement:
| If the account balance increases | If the account balance decreases | |
|---|---|---|
| Non-Current Assets (Investing Activities) | ||
| Property, Plant, and Equipment (P,P,&E) | Subtract | Add |
| Long-term investments | Subtract | Add |
| Loans to others | Subtract | Add |
| Liabilities and Equity (Financing Activities) | ||
| Bonds payable | Add | Subtract |
| Common stock (our own company) | Add | Subtract |
| Retained earnings | * | * |
*Retained earnings will need more analysis
So you will notice, this chart looks at the overall change in the balance, but to calculate the gross cash flow by looking at the detail in all of the each of the accounts to fill out the cash flow analysis to meet the requirements of GAAP and IFRS reporting requirements.
Practice Questions


2.2: Annotated Bibliographies - What? Why? How?
Your final essay for the class will be a Toulmin Analysis/Research Essay (I’ll teach you how to do this!) of an article you will choose about the effects of social media on society (from the “Social Media Arti- cles for Final Essay” tab on the left side navigation bar). Although the article you choose will be the primary source of your analysis, you will also need to do a good deal of external research to help you develop an objective, academic and analytical approach to your critique. Two weeks before your final essay is due, you will turn in an annotated bibliography of at least ten sources related to your research topic. These should not be the only ten sources you examine but rather most useful or influential ones. Your grade will be based on the variety and relevance of the sources and on the insight and clarity of your annotations.
Definition
A bibliography is a list of sources (books, journals, websites, periodicals, etc.) one has used for researching a topic. A bibliography usually just includes the bibliographic in- formation (i.e., the author, title, publisher; etc.).
An annotation is a summary and/or evaluation. Therefore, an annotated bibliography in- cludes a summary and/or evaluation of each of the sources.
For your assignment, your annotations must do all of the following:
Summarize: What are the main arguments? What is the point of this book or article? What topics are covered? If someone asked what this article/book is about, what would you say?
Assess : After summarizing a source, it is helpful to evaluate it. How does it compare with other sources in your bibliography? Is the information reliable, relevant and author- itative? How can you tell? Is it this source inappropriately biased or relatively objective? What is the goal of this source?
Reflect: Once you’ve summarized and assessed a source, you need to ask how it fits specifically into your research. Was this source helpful to you? How does it help you shape your argument? How can you use this source in your research project? Has it changed how you think about your topic?
Purpose
To learn about your topic: Writing an annotated bibliography is excellent preparation for a research project. Just collecting sources for a bibliography is useful, but when you have to write annotations for each source, you’re forced to read each source more carefully. You be- gin to read more critically instead of just collecting information.
To help you formulate a thesis: Every good research paper is an argument. The purpose of research is to state and support a thesis. So a very important part of research is developing a thesis that is debatable, interesting , and current. Writing an annotated bibliography can help you gain a good perspective on what is being said about your topic. By reading and respond- ing to a variety of sources on a topic, you’ll start to see what the issues are, what people are arguing about, and you’ll then be able to develop your own point of view.
To develop enough authoritative, relevant and reliable support for your topic: Most top- ics that you will be writing about in college classes are very complex and multi-faceted. Usu- ally just finding a single source or two on a topic will not be sufficient to really understand all of the angles and nuances of the topic under consideration. A well researched and well written annotated bibliography will help you to be thorough and exhaustive in your research and guard against oversimplification and other forms of research-related blind spots.
Format
The bibliographic information: The bibliographic information of the source (the title, au- thor, publisher, date, etc.) is written in MLA format.
The annotations: The annotations for each source are written in paragraph form. The lengths of the annotations can vary significantly from a couple of sentences to a couple of pages. The length will depend on the purpose and the particular complexity of each source.
Annotated Bibliography Checklist
Before your final essay is due, you will turn in an Annotated Bibliography of at least TEN sources related to your topic. These should not be the only sources you have examined but rather most useful ones . Your grade will be based on the variety and relevance of the sources as well as on the insight and clarity of your annotations.
- Consult the handout on Annotated Bibliographies for information on the purpose, scope and content of the assignment.
- Your annotations should be listed alphabetically, by the last name of the primary author of the selection and in proper MLA format.
- At least TWO of your sources should be books (if possible).
- At least THREE of your sources must be academic, scholarly or peer-reviewed journals.
- At least TWO of your annotations must be longer and more thorough (at least a page each).
- Be as specific as possible with your summaries, analyses and reflections on the relevancy of each source. Don’t just tell me it is a good source or that it is very useful. I am assuming this since it is one of the ones you have selected to annotate. Tell me exactly why it is a good source and exactly ho w it is useful to your topic.
- Be sure to use the citation builder websites to help you build your entries in proper MLA format. They can be found on the Library website and on the our WR 122 Library Guide in the “Citation Style Guides” section.
- Sources found in the MHCC library databases are already in MLA format. Scroll down to the end of the article and the entry will be posted in full MLA format. Just copy and past it into your Bibliography. (Make sure to make it double-spaced and in “hanging” format as well.)


3.2.3: Trends in Electron Affinity and Metallic Character
Learning Objectives
- Describe and explain the observed trends in electron affinity and metallic character of the elements
- To recognize the inverse relationship of ionization energies and electron affinities
Variation in Electron Affinities
Video \(\PageIndex{1}\) : A brief overview of electron affinity.
The electron affinity [EA] is the energy change for the process of adding an electron to a gaseous atom to form an anion (negative ion).
This process can be either endothermic or exothermic, depending on the element. The EA of some of the elements is given in Figure \(\PageIndex{6}\). You can see that many of these elements have negative values of EA, which means that energy is released when the gaseous atom accepts an electron. However, for some elements, energy is required for the atom to become negatively charged and the value of their EA is positive. Just as with ionization energy, subsequent EA values are associated with forming ions with more charge. The second EA is the energy associated with adding an electron to an anion to form a –2 ion, and so on.
Figure \(\PageIndex{1}\): This version of the periodic table displays the electron affinity values (in kJ/mol) for selected elements.:
As we might predict, it becomes easier to add an electron across a series of atoms as the effective nuclear charge of the atoms increases. We find, as we go from left to right across a period, EAs tend to become more negative. The exceptions found among the elements of group 2 (2A), group 15 (5A), and group 18 (8A) can be understood based on the electronic structure of these groups. The noble gases, group 18 (8A), have a completely filled shell and the incoming electron must be added to a higher n level, which is more difficult to do. Group 2 (2A) has a filled ns subshell, and so the next electron added goes into the higher energy np , so, again, the observed EA value is not as the trend would predict. Finally, group 15 (5A) has a half-filled np subshell and the next electron must be paired with an existing np electron. In all of these cases, the initial relative stability of the electron configuration disrupts the trend in EA.
We also might expect the atom at the top of each group to have the largest EA; their first ionization potentials suggest that these atoms have the largest effective nuclear charges. However, as we move down a group, we see that the second element in the group most often has the greatest EA. The reduction of the EA of the first member can be attributed to the small size of the n = 2 shell and the resulting large electron–electron repulsions. For example, chlorine, with an EA value of –348 kJ/mol, has the highest value of any element in the periodic table. The EA of fluorine is –322 kJ/mol. When we add an electron to a fluorine atom to form a fluoride anion (F – ), we add an electron to the n = 2 shell. The electron is attracted to the nucleus, but there is also significant repulsion from the other electrons already present in this small valence shell. The chlorine atom has the same electron configuration in the valence shell, but because the entering electron is going into the n = 3 shell, it occupies a considerably larger region of space and the electron–electron repulsions are reduced. The entering electron does not experience as much repulsion and the chlorine atom accepts an additional electron more readily.
The properties discussed in this unit (size of atoms and ions, effective nuclear charge, ionization energies, and electron affinities) are central to understanding chemical reactivity. For example, because fluorine has an energetically favorable EA and a large energy barrier to ionization (IE), it is much easier to form fluorine anions than cations.
Video \(\PageIndex{2}\) : An overview of electron effinity.
Trends in Metallic and Nonmetallic Character
In this course thusfar (and in the previous course), we have seen that metals are on the left side of the periodic table while nonmetals are generally on the right (exception: Hydrogen). We can classify the metallic character of all the elements with trends similar to those we've previous examined in this unit. Metallic character has the opposite trend of ionization energy and electronegativity, but is similar to the trend we saw for atomic radius: it INCREASES as you move from right to left across rows of the periodic table and INCREASES as you move down columns. This trend does not necessarily apply to transition metals. Figure \(\PageIndex{2}\) shows this trend as well as the others we've discussed in Unit 3.
Figure \(\PageIndex{2}\):
Metallic character increases as you go from right to left on periods and move down groups on the periodic table. The gray squares represent metals, the orange squares are "metalloids" and the yellow squares are nonmetals.
Credit: Wikimedia Commons, licensed under the
Creative Commons
Attribution-Share Alike 4.0 International
license.
Metallic properties including conductivity and malleability (the ability to be formed into sheets) depend on having electrons that can be removed easily. Thus, metallic character increases as we move down a group and decreases across a period in the same trend observed for atomic size because it is easier to remove an electron that is farther away from the nucleus.
Summary
Electron affinity (the energy associated with forming an anion) is more favorable (exothermic) when electrons are placed into lower energy orbitals, closer to the nucleus. Therefore, electron affinity becomes increasingly negative as we move left to right across the periodic table and decreases as we move down a group. For both IE and electron affinity data, there are exceptions to the trends when dealing with completely filled or half-filled subshells. Metallic character increases from right to left across periods on the periodic table and down columns.
Glossary
- electron affinity
- energy required to add an electron to a gaseous atom to form an anion
Contributors and Attributions
-
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>).
- Adelaide Clark, Oregon Institute of Technology
- Crash Course Chemistry: Crash Course is a division of Complexly and videos are free to stream for educational purposes.
- Guillotined Chemistry is a production of Mark Anticole and available for free on Youtube.


10.6: Discussion- Sun City Boards’ Self-Leadership Audit
Post 1: Initial Post
Instructions
This posting should be a minimum of one short paragraph and a maximum of two paragraphs. Word totals for this post should be in the 100–200-word range. Whether you agree or disagree, explain why with supporting evidence and concepts from the readings or a related experience. Include a reference, link, or citation when appropriate.
Scenario
Despite cultural and economic turbulence, Sun City Boards CEO Tom Wilson is doubling down on his plans to grow the business and to grow as a leader. In the Sun City Boards Assignment for Module 10, he asked you to develop a composite profile of a successful entrepreneurial leader. His follow-up request: help him develop a leadership audit framework and associated plan for his development as a leader.
For Discussion
- As preparation for this discussion, review the Leadership module and specifically the What Makes an Effective Leader section.
- Conduct additional research on how to conduct a leadership audit.
Distill your findings into an email to Tom that outlines how to conduct a self-leadership audit with specific steps and explanations. For perspective, refer to OnPurpose Consulting’s How to Perform a Self-Leadership Audit post. Note: Your email recommendations must reflect substantial original thinking and not be a simple copy of Sylvia’s (the linked) post.
Posts 2 and 3: Respond to Classmates’ Posts
Instructions
After you have created your initial post, look over the discussion posts of your classmates and give at least two thoughtful responses to two different classmates (one per classmate).
A response posting should be a minimum of one short paragraph. Word totals for these posts should be in the 75–100-word range. Whether you agree or disagree, explain why with supporting evidence and concepts from the readings or a related experience. Include a reference, link, or citation when appropriate. The goal of your response posts is to extend discussions already taking place or pose new possibilities or opinions not previously voiced. Your goal should be to motivate the group discussion and present a creative approach to the topic.
Contributors and Attributions
- Discussion: Sun City Boards' Self-Leadership Audit. Authored by : Nina Burokas. Provided by : Lumen Learning. License : CC BY: Attribution


4.3: Intensity in Single-Slit Diffraction
-
- Last updated
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Learning Objectives
By the end of this section, you will be able to:
- Calculate the intensity relative to the central maximum of the single-slit diffraction peaks
- Calculate the intensity relative to the central maximum of an arbitrary point on the screen
To calculate the intensity of the diffraction pattern, we follow the phasor method used for calculations with ac circuits in Alternating-Current Circuits . If we consider that there are \(N\) Huygens sources across the slit shown previously , with each source separated by a distance a/N from its adjacent neighbors, the path difference between waves from adjacent sources reaching the arbitrary point \(P\) on the screen is \((a/N) \, \sin \theta\). This distance is equivalent to a phase difference of \((2\pi a/\lambda N) \, \sin \, \theta\). The phasor diagram for the waves arriving at the point whose angular position is \(\theta\) is shown in Figure \(\PageIndex{1}\). The amplitude of the phasor for each Huygens wavelet is \(\Delta E_0\), the amplitude of the resultant phasor is \(E\), and the phase difference between the wavelets from the first and the last sources is
\[\phi = \left(\dfrac{2\pi}{\lambda}\right) \, a \, \sin \theta. \nonumber \]
With \(N → ∞\), the phasor diagram approaches a circular arc of length \(N \Delta E_0\) and radius \(r\). Since the length of the arc is \(N \Delta E_0\) for any \(ϕ\), the radius \(r\) of the arc must decrease as \(ϕ\) increases (or equivalently, as the phasors form tighter spirals).
The phasor diagram for ϕ = 0 (the center of the diffraction pattern) is shown in Figure \(\PageIndex{1a}\) using N=30. In this case, the phasors are laid end to end in a straight line of length \(N \Delta E_0\), the radius r goes to infinity, and the resultant has its maximum value \(E = N\Delta E_0\). The intensity of the light can be obtained using the relation \(I = \dfrac{1}{2} c \epsilon_0 E^2\) from Electromagnetic Waves . The intensity of the maximum is then
\[I_0 = \dfrac{1}{2} c\epsilon_0 (N \Delta E_0)^2 = \dfrac{1}{2\mu_0 c}(N\Delta E_0)^2, \nonumber \]
where \(\epsilon_0 = 1/\mu_0 c^2\). The phasor diagrams for the first two zeros of the diffraction pattern are shown in Figure \(\PageIndex{1b}\) and Figure \(\PageIndex{1d}\). In both cases, the phasors add to zero, after rotating through \(\phi = 2\pi\) rad for m = 1 and \(4 \pi\) rad for m = 2.
The next two maxima beyond the central maxima are represented by the phasor diagrams of parts (c) and (e). In part (c), the phasors have rotated through \(\phi = 3\pi\) rad and have formed a resultant phasor of magnitude \(E_1\). The length of the arc formed by the phasors is \(N\Delta E_0\). Since this corresponds to 1.5 rotations around a circle of diameter \(E_1\), we have
\[\dfrac{3}{2} \pi E_1 = N \Delta E_0, \nonumber \]
so
\[E_1 = \dfrac{2N\Delta E_0}{3\pi} \nonumber \]
and
\[I_1 = \dfrac{1}{2\mu_0 c}E_1^2 = \dfrac{4(N\Delta E_0)^2}{(9\pi^2)(2\mu_0c)} = 0.045 I_0, \nonumber \]
where
\[I_0 = \dfrac{(N\Delta E_0)^2}{2\mu_0 c}. \nonumber \]
In part (e), the phasors have rotated through \(\phi = 5\pi\) rad, corresponding to 2.5 rotations around a circle of diameter \(E_2\) and arc length \(N\Delta E_0\). This results in \(I_2 = 0.016 I_0\). The proof is left as an exercise for the student (Exercise 4.119).
These two maxima actually correspond to values of ϕ slightly less than \(3\pi\) rad and \(5\pi\) rad. Since the total length of the arc of the phasor diagram is always \(N \Delta E_0\), the radius of the arc decreases as \(ϕ\) increases. As a result, \(E_1\) and \(E_2\) turn out to be slightly larger for arcs that have not quite curled through \(3\pi\) rad and \(5\pi\) rad, respectively. The exact values of \(ϕ\) for the maxima are investigated in Exercise 4.120. In solving that problem, you will find that they are less than, but very close to, \(\phi = 3\pi, \, 5\pi, \, 7\pi,\)… rad.
To calculate the intensity at an arbitrary point \(P\) on the screen, we return to the phasor diagram of Figure \(\PageIndex{1}\). Since the arc subtends an angle ϕ at the center of the circle,
\[N\Delta E_0 = r\phi \label{eq10} \]
and
\[\sin \left(\dfrac{\phi}{2}\right) = \dfrac{E}{2r}. \label{eq11} \]
where \(E\) is the amplitude of the resultant field. Solving the Equation \ref{eq11} for \(E\) and then substituting \(r\) from Equation \ref{eq10}, we find
\[\begin{align*} E &= 2r \, \sin \, \dfrac{\phi}{2} \\[5pt] &= 2\dfrac{N\Delta E_0}{\phi} \sin \, \dfrac{\phi}{2}. \end{align*} \nonumber \]
Now defining
\[\beta = \dfrac{\phi}{2} = \dfrac{\pi a \, \sin \, \theta}{\lambda} \label{4.2} \]
we obtain
\[E = N\Delta E_0 \dfrac{\sin \, \beta}{\beta} \label{eq15} \]
Equation \ref{eq15} relates the amplitude of the resultant field at any point in the diffraction pattern to the amplitude \(N \Delta E_0\) at the central maximum. The intensity is proportional to the square of the amplitude, so
\[I = I_0 \left(\dfrac{\sin \, \beta}{\beta}\right)^2 \label{eq20} \]
where \(I_0 = (N\delta E_0)^2/2\mu_0 c\) is the intensity at the center of the pattern.
For the central maximum, ϕ = 0, β is also zero and we see from l’Hôpital’s rule that \(\lim_{\beta \rightarrow 0}(sin \, \beta/\beta) = 1\), so that \(lim_{\phi \rightarrow 0}I = I_0\). For the next maximum, \(\phi = 3\pi\) rad, we have \(\beta = 3\pi/2\) rad and when substituted into Equation \ref{eq20}, it yields
\[I_1 = I_0 \left(\dfrac{\sin \, 3\pi/2}{3\pi/2}\right)^2 = 0.045 I_0, \nonumber \]
in agreement with what we found earlier in this section using the diameters and circumferences of phasor diagrams. Substituting \(\phi = 5\pi\) rad into Equation \ref{eq20} yields a similar result for \(I_2\).
A plot of Equation \ref{eq20} is shown in Figure \(\PageIndex{3}\) and directly below it is a photograph of an actual diffraction pattern. Notice that the central peak is much brighter than the others, and that the zeros of the pattern are located at those points where \(sin \, \beta = 0\), which occurs when \(\beta = m\pi\) rad. This corresponds to
\[\dfrac{\pi a \, \sin \theta}{\lambda} = m\pi, \nonumber \]
or
\[a \, \sin \, \theta = m \lambda, \nonumber \]
which we derived for the destructive interference ro a single slit previously .
Example \(\PageIndex{1}\): Intensity in Single-Slit Diffraction
Light of wavelength 550 nm passes through a slit of width 2.00 μm and produces a diffraction pattern similar to that shown in Figure \(\PageIndex{3a}\).
- Find the locations of the first two minima in terms of the angle from the central maximum.
- Determine the intensity relative to the central maximum at a point halfway between these two minima.
Strategy
The minima are given by Equation 4.2.1 , \(a \, sin \, \theta = m\lambda\). The first two minima are for m = 1 and m = 2. Equation \ref{eq20} and Equation \ref{4.2} can be used to determine the intensity once the angle has been worked out.
Solution
-
Solving Equation
4.2.1
for θ gives us \(\theta_m = \sin^{-1}(m\lambda/a)\), so that
\[\theta_1 = \sin^{-1} \left(\dfrac{(+1)(550 \times 10^{-9} m)}{2.00 \times 10^{-6}m}\right) = +16.0° \nonumber \]
and\[\theta_2 = \sin^{-1} \left(\dfrac{(+2)(550 \times 10^{-9}m)}{2.00 \times 10^{-6}m}\right) = +33.4°. \nonumber \]
-
The halfway point between \(\theta_1\) and \(\theta_2\) is
\[\theta = (\theta_1 + \theta_2) /2 = (16.0° + 33.4°)/2 = 24.7°. \nonumber \]
Equation \ref{4.2} gives
\[\beta = \dfrac{\pi a \, sin \, \theta}{\lambda} = \dfrac{\pi (2.00 \times 10^{-6}m) \, \sin \, (24.7°)}{(550 \times 10^{-9}m)} = 1.52\pi \, or \, 4.77 \, rad. \nonumber \]
From Equation \ref{eq20}, we can calculate
\[\dfrac{I}{I_0} = \left(\dfrac{\sin \, \beta}{\beta}\right)^2 = \left(\dfrac{\sin \, (4.77)}{4.77}\right)^2 = \left(\dfrac{-0.9985}{4.77}\right)^2 = 0.044. \nonumber \]
Significance
This position, halfway between two minima, is very close to the location of the maximum, expected near \(\beta = 3\pi/2\), or \(1.5\pi\).
Exercise \(\PageIndex{1}\)
For the experiment in Example \(\PageIndex{1}\), at what angle from the center is the third maximum and what is its intensity relative to the central maximum?
- Answer
-
\(74.3^o\), \(0.0083 I_0\)
If the slit width \(a\) is varied, the intensity distribution changes, as illustrated in Figure \(\PageIndex{4}\). The central peak is distributed over the region from \(sin \, \theta = -\lambda/a\) to \(sin \, \theta = +\lambda/a\). For small θ, this corresponds to an angular width \(\Delta \theta \approx 2\lambda /a\). Hence, an increase in the slit width results in a decrease in the width of the central peak . For a slit with a ≫ λ, the central peak is very sharp, whereas if a ≈ λ, it becomes quite broad.
Diffraction Simulation
A diffraction experiment in optics can require a lot of preparation but this simulation by Andrew Duffy offers not only a quick set up but also the ability to change the slit width instantly. Run the simulation and select “Single slit.” You can adjust the slit width and see the effect on the diffraction pattern on a screen and as a graph.


4: Molecular Geometry and Bonding Theories
-
- 4.1: Molecular Shapes
- The Lewis electron-pair approach described previously can be used to predict the number and types of bonds between the atoms in a substance, and it indicates which atoms have lone pairs of electrons. This approach gives no information about the actual arrangement of atoms in space, however.
-
- 4.2: The VSEPR Model
- The VSEPR model can predict the structure of nearly any molecule or polyatomic ion in which the central atom is a nonmetal, as well as the structures of many molecules and polyatomic ions with a central metal atom. The VSEPR model is not a theory; it does not attempt to explain observations. Instead, it is a counting procedure that accurately predicts the three-dimensional structures of a large number of compounds, which cannot be predicted using the Lewis electron-pair approach.
-
- 4.3: Molecular Shape and Molecular Polarity
- Compounds with polar covalent bonds have electrons that are shared unequally between the bonded atoms. The polarity of such a bond is determined largely by the relative electronegativites of the bonded atoms. The asymmetrical charge distribution in a polar substance produces a dipole moment, which is the product of the partial charges on the bonded atoms and the distance between them.
-
- 4.4: Covalent Bonding and Orbital Overlap
- A more sophisticated treatment of bonding is needed for systems such as these. In this section, we present a quantum mechanical description of bonding, in which bonding electrons are viewed as being localized between the nuclei of the bonded atoms. The overlap of bonding orbitals is substantially increased through a process called hybridization, which results in the formation of stronger bonds.
-
- 4.5: Hybrid Orbitals
- The localized valence bonding theory uses a process called hybridization, in which atomic orbitals that are similar in energy but not equivalent are combined mathematically to produce sets of equivalent orbitals that are properly oriented to form bonds. These new combinations are called hybrid atomic orbitals because they are produced by combining (hybridizing) two or more atomic orbitals from the same atom.
-
- 4.6: Multiple Bonds
- To describe the bonding in more complex molecules with multiple bonds, we can use an approach that uses hybrid atomic orbitals to describe the \(\sigma\) bonding and molecular orbitals to describe the \(\pi\) bonding. In this approach, unhybridized np orbitals on atoms bonded to one another are allowed to interact to produce bonding, antibonding, or nonbonding combinations.
-
- 4.7: Molecular Orbitals
- A molecular orbital is an allowed spatial distribution of electrons in a molecule that is associated with a particular orbital energy. Unlike an atomic orbital, which is centered on a single atom, a molecular orbital extends over all the atoms in a molecule or ion. Hence the molecular orbital theory of bonding is a delocalized approach. Molecular orbitals are constructed using linear combinations of atomic orbitals (LCAOs), which describe overlapping atomic orbitals.
-
- 4.8: Second-Row Diatomic Molecules
- Molecular orbital energy-level diagrams for diatomic molecules can be created if the electron configuration of the parent atoms is known, following a few simple rules. Most important, the number of molecular orbitals in a molecule is the same as the number of atomic orbitals that interact. The difference between bonding and antibonding molecular orbital combinations is proportional to the overlap of the parent orbitals and decreases as the energy difference between the atomic orbitals increases.
Thumbnail: Covalent bonding in ethane (CC BY-SA-NC 3.0; Anonymous)


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HGTC BIO 225: Microbiology Lab Manual
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HGTC BIO 225: Microbiology Lab Manual
4: Simple and Negative Stains
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4: Simple and Negative Stains
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6.14: Introduction to Informal Reports
What you’ll learn to do: Create an informal report
Informal reports in the business setting are usually shorter in length and have fewer sections than a formal report. Employees in most organizations create and use informal reports. Almost all informal reports are for internal use. Some institutions have prescribed formats and others do not. As we’ve previously discussed, an informal report fits in one of two large categories:
- informational report
- analytical report
An informational report provides background and information without reaching an evaluation. These include simple reports like meeting minutes, expense reports, and progress, or status updates. An analytical report provides much the same information as the informational report along with evaluation or recommendation. These reports may include feasibility studies, justification reports, and proposals.
Most organizations have specific forms and policies for the simplest reports, such as a mileage reimbursement report. Other simple reports are not as restricted.
Contributors and Attributions
- Introduction to Informal Reports. Authored by : Susan Kendall. Provided by : Lumen Learning. License : CC BY: Attribution
- Report Icon. Authored by : Korokoro. Provided by : Noun Project. Located at : https://thenounproject.com/term/report/131419/ . License : CC BY: Attribution