3.4: Alkylation and Acylation of Aromatic Rings - The Friedel-Crafts Reaction
After completing this section, you should be able to
- write the equation and the detailed mechanism for a typical Friedel-Crafts alkylation, then identify the similarities between this reaction and the previous presented electrophilic aromatic substitution reactions.
- identify the product formed from the Friedel-Crafts alkylation of a given aromatic compound.
- identify the aromatic compound needed to prepare a given arene by a Friedel-Crafts alkylation.
- identify the alkyl halide and catalyst needed to form a specified arene from a given aromatic compound.
- show how alkyl halides and acyl-halides can be used as alkylating agents in Friedel-Crafts alkylation reactions.
- discuss the limitations of the Friedel-Crafts alkylation reaction, paying particular attention to the structure of the alkyl halide, the structure of the aromatic substrate, and the problem of polyalkylation.
- write an equation and the detailed mechanism for a typical Friedel-Crafts acylation.
- identify the product formed by the Friedel-Crafts acylation of a given aromatic compound.
- identify the aromatic compound, and the reagent and catalyst needed to prepare a given ketone through a Friedel-Crafts acylation reaction.
Friedel-Crafts Alkylation
Among the most useful electrophilic aromatic substitution reactions in the laboratory is alkylation —the introduction of an alkyl group onto the benzene ring. Called the Friedel–Crafts reaction after its founders in 1877, Charles Friedel and James Crafts, the reaction is carried out by treating an aromatic compound with an alkyl chloride, RCl, in the presence of AlCl 3 to generate a carbocation electrophile, R + .
Aluminum chloride catalyzes the reaction by helping the alkyl halide to generate a carbocation in much the same way that FeBr 3 catalyzes aromatic brominations by polarizing Br 2 .
The reactivity of haloalkanes increases as we move up the periodic table and increase polarity. This means that an RF haloalkane is most reactive followed by RCl then RBr and finally RI. This means that the Lewis acids used as catalysts in Friedel-Crafts Alkylation reactions tend to have similar halogen combinations, such as BF 3 , SbCl 5 , AlCl 3 , SbCl 5 , and AlBr 3 , all of which are commonly used in these reactions.
Mechanism
Step 1: An electron pair from the aromatic ring attacks the carbonation, forming a new C-C bond. The arenium ion intermediate results with stabilization from multiple resonance forms.
Steps 2: The loss of H + completes the reaction giving the neutral alkylated substitution product (Figure \(\PageIndex{1}\)).
Some limitations of Friedel-Crafts Alkylation
Despite its utility, the Friedel–Crafts alkylation has several limitations. For one thing, only alkyl halides can be used. Aromatic (aryl) halides and vinylic halides don’t react because aryl and vinylic carbocations are too high in energy to form under Friedel–Crafts conditions.
Another limitation is that Friedel–Crafts reactions don’t succeed on aromatic rings that are substituted by a strongly electron-withdrawing group, which will deactivate the ring, such as carbonyl (C=O) or nitro group (NO 2 ).
Adding to the list of substituents that will not allow the reaction to proceed are basic amino groups that can be protonated. The lone pair electrons on the amines react with the Lewis acid AlCl 3 . This places a positive charge next to the benzene ring, which is so strongly activating that the Friedel-Crafts reaction cannot occur.
We’ll see in the next section that the presence of a substituent group already on a ring can have a dramatic effect on that ring’s reactivity to further electrophilic substitution. Rings that contain any of the substituents listed in Figure \(\PageIndex{2}\) do not undergo Friedel–Crafts alkylation.
A third limitation to the Friedel–Crafts alkylation is that it’s often difficult to stop the reaction after a single substitution. Once the first alkyl group is on the ring, a second substitution reaction is facilitated for reasons we’ll discuss in the next section. Thus, we often observe polyalkylation. Reaction of benzene with 1 mol equivalent of 2-chloro-2-methylpropane, for example, yields p -di- tert -butylbenzene as the major product, along with small amounts of tert -butylbenzene and unreacted benzene. A high yield of mono-alkylation product is obtained only when a large excess of benzene is used.
A final limitation to the Friedel–Crafts reaction is that a skeletal rearrangement of the alkyl carbocation electrophile sometimes occurs during reaction, particularly when a primary alkyl halide is used. For instance, treating benzene with 1-chlorobutane at 0 °C gives an approximately 2 : 1 ratio of rearranged ( sec -butyl) to unrearranged (butyl) products.
The carbocation rearrangements that accompany Friedel–Crafts reactions are like those that accompany electrophilic additions to alkenes and occur either by hydride shift or alkyl shift. For example, the relatively unstable primary butyl carbocation produced by reaction of 1-chlorobutane with AlCl 3 rearranges to the more stable secondary butyl carbocation by the shift of a hydrogen atom and its electron pair (a hydride ion, H : – ) from C2 to C1. Similarly, alkylation of benzene with 1-chloro-2,2-dimethylpropane yields (1,1-dimethylpropyl)benzene. The initially formed primary carbocation rearranges to a tertiary carbocation by shift of a methyl group and its electron pair from C2 to C1.
The Friedel–Crafts reaction of benzene with 2-chloro-3-methylbutane in the presence of AlCl 3 occurs with a carbocation rearrangement. What is the structure of the product?
Strategy
A Friedel–Crafts reaction involves initial formation of a carbocation, which can rearrange by either a hydride shift or an alkyl shift to give a more stable carbocation. Draw the initial carbocation, assess its stability, and see if the shift of a hydride ion or an alkyl group from a neighboring carbon will result in increased stability. In the present instance, the initial carbocation is a secondary one that can rearrange to a more stable tertiary one by a hydride shift.
Use this more stable tertiary carbocation to complete the Friedel–Crafts reaction.
Solution
Friedel-Crafts Acylation
Just as an aromatic ring can be alkylated by reaction with an alkyl chloride, it can be acylated by reaction with a carboxylic acid chloride, RCOCl, in the presence of AlCl 3 . That is, an acyl group (–COR; pronounced a -sil) is substituted onto the aromatic ring. For example, reaction of benzene with acetyl chloride yields the ketone acetophenone.
Mechanism
The mechanism of Friedel–Crafts acylation is similar to that of Friedel–Crafts alkylation, and the same limitations on the aromatic substrate noted previously in Figure \(\PageIndex{2}\) for alkylation also apply to acylation. The reactive electrophile is a resonance-stabilized acyl cation, generated by reaction between the acyl chloride and AlCl 3 (Figure \(\PageIndex{3}\)). As the resonance structures in the figure indicate, an acyl cation is stabilized by interaction of the vacant orbital on carbon with lone-pair electrons on the neighboring oxygen. Because of this stabilization, no carbocation rearrangement occurs during acylation.
Unlike the multiple substitutions that often occur in Friedel–Crafts alkylations, acylations never occur more than once on a ring because the product acylbenzene is less reactive than the starting material.
Aromatic alkylations occur in numerous biological pathways, although there is of course no AlCl 3 present in living systems to catalyze the reaction. Instead, the carbocation electrophile is typically formed by dissociation of an organodiphosphate. The dissociation is usually assisted by complexation to a divalent metal cation such as Mg 2 + , just as the dissociation of an alkyl chloride is assisted by AlCl 3 .
An example of a biological Friedel–Crafts reaction occurs during the biosynthesis of phylloquinone, or vitamin K 1 , the human blood-clotting factor. Phylloquinone is formed by reaction of 1,4-dihydroxynaphthoic acid with phytyl diphosphate. Phytyl diphosphate first dissociates to a resonance-stabilized allylic carbocation, which then substitutes onto the aromatic ring in the typical way. Several further transformations lead to phylloquinone (Figure \(\PageIndex{4}\)).
Exercises
Which of the following will NOT undergo a rearrangement in a Friedel-Crafts reaction?
- Answer
-
A, B, and E will not undergo a rearrangement.
What is the major monosubstitution product from the Friedel–Crafts reaction of benzene with 1-chloro-2-methylpropane in the presence of AlCl 3 ?
- Answer
-
tert -Butylbenzene
Suggest an acyl chloride that was used to make the following compounds:
- Answer
-
Make certain that you can define, and use in context, the key terms below.
- acyl group
- Friedel-Crafts acylation reaction
- Friedel-Crafts alkylation reaction
- polyalkylation
A Friedel-Crafts alkylation reaction is an electrophilic aromatic substitution reaction in which a carbocation is attacked by a pi bond from an aromatic ring with the net result that one of the aromatic protons is replaced by an alkyl group. If you prefer, you may regard these reactions as involving an attack by an aromatic ring on a carbocation. The latter approach is the one used in the textbook, but the former approach is probably more common.
When more than one alkyl group is introduced into an aromatic ring during the course of a Friedel-Crafts alkylation reaction, polyalkylation is said to have occurred.
The four limitations on the use of Friedel-Crafts alkylations are as follows:
- vinyl and aryl halides cannot be used to form carbocations.
- the aromatic substrate must not contain a strongly deactivating group, or groups, such as NH 2 , NHR or NR 2 , which form complexes with the Lewis acid catalyst and in so doing become strongly deactivating.
- polyalkylation, which can be overcome by using a large excess of the aromatic substrate.
- carbocation rearrangements may occur in any reaction that involves a carbocation.
The reaction of an aromatic substrate with an acid chloride (or acid anhydride) in the presence of an aluminum chloride catalyst is used to introduce an acyl group (C=O) into the aromatic ring through an electrophilic aromatic substitution mechanism. Such reactions are Friedel-Crafts acylation reactions.


15.1: Basic Concepts
-
- Last updated
- Save as PDF
British philosopher Karl Popper said that theories can never be proven, only disproven. As an example, how can we prove that the sun will rise tomorrow? Popper said that just because the sun has risen every single day that we can remember does not necessarily mean that it will rise tomorrow, because inductively derived theories are only conjectures that may or may not be predictive of future phenomena. Instead, he suggested that we may assume a theory that the sun will rise every day without necessarily proving it, and if the sun does not rise on a certain day, the theory is falsified and rejected. Likewise, we can only reject hypotheses based on contrary evidence, but can never truly accept them because the presence of evidence does not mean that we will not observe contrary evidence later. Because we cannot truly accept a hypothesis of interest (alternative hypothesis), we formulate a null hypothesis as the opposite of the alternative hypothesis, and then use empirical evidence to reject the null hypothesis to demonstrate indirect, probabilistic support for our alternative hypothesis.
A second problem with testing hypothesised relationships in social science research is that the dependent variable may be influenced by an infinite number of extraneous variables and it is not plausible to measure and control for all of these extraneous effects. Hence, even if two variables may seem to be related in an observed sample, they may not be truly related in the population, and therefore inferential statistics are never certain or deterministic, but always probabilistic.
How do we know whether a relationship between two variables in an observed sample is significant, and not a matter of chance? Sir Ronald A. Fisher—one of the most prominent statisticians in history—established the basic guidelines for significance testing. He said that a statistical result may be considered significant if it can be shown that the probability of it being rejected due to chance is 5% or less. In inferential statistics, this probability is called the value , 5 per cent is called the significance level ( ), and the desired relationship between the -value and is denoted as: . The significance level is the maximum level of risk that we are willing to accept as the price of our inference from the sample to the population. If the -value is less than 0.05 or 5 per cent it means that we have a 5 per cent chance of being incorrect in rejecting the null hypothesis or having a Type I error. If 0.05" decoding="async" height="17" loading="lazy" src=" https://usq.pressbooks.pub/app/uploa...a5aee31_l3.svg " title="Rendered by QuickLaTeX.com" width="65">, we do not have enough evidence to reject the null hypothesis or accept the alternative hypothesis.
We must also understand three related statistical concepts: sampling distribution, standard error, and confidence interval. A sampling distribution is the theoretical distribution of an infinite number of samples from the population of interest in your study. However, because a sample is never identical to the population, every sample always has some inherent level of error, called the standard error . If this standard error is small, then statistical estimates derived from the sample—such as sample mean— are reasonably good estimates of the population. The precision of our sample estimates is defined in terms of a confidence interval (CI). A 95% CI is defined as a range of plus or minus two standard deviations of the mean estimate, as derived from different samples in a sampling distribution. Hence, when we say that our observed sample estimate has a CI of 95%, what we mean is that we are confident that 95% of the time, the population parameter is within two standard deviations of our observed sample estimate. Jointly, the -value and the CI give us a good idea of the probability of our result and how close it is from the corresponding population parameter.


23: Electromagnetic Induction, AC Circuits, and Electrical Technologies
Historically, it was very shortly after Oersted discovered currents cause magnetic fields that other scientists asked the following question: Can magnetic fields cause currents? The answer was soon found by experiment to be yes. In 1831, some 12 years after Oersted’s discovery, the English scientist Michael Faraday (1791–1862) and the American scientist Joseph Henry (1797–1878) independently demonstrated that magnetic fields can produce currents. The basic process of generating emfs (electromotive force) and, hence, currents with magnetic fields is known as induction ; this process is also called magnetic induction to distinguish it from charging by induction, which utilizes the Coulomb force.
-
- 23.0: Prelude to Electromagnetic Induction, AC Circuits and Electrical Technologies
- Today, currents induced by magnetic fields are essential to our technological society. The ubiquitous generator—found in automobiles, on bicycles, in nuclear power plants, and so on—uses magnetism to generate current. Other devices that use magnetism to induce currents include pickup coils in electric guitars, transformers of every size, certain microphones, airport security gates, and damping mechanisms on sensitive chemical balances.
-
- 23.5: Faraday’s Law of Induction- Lenz’s Law
- Faraday’s experiments showed that the emf induced by a change in magnetic flux depends on only a few factors. First, emf is directly proportional to the change in flux ΔΦ. Second, emf is greatest when the change in time Δt is smallest—that is, emf is inversely proportional to Δt . Finally, if a coil has N turns, an emf will be produced that is Ntimes greater than for a single coil, so that emf is directly proportional to N.
-
- 23.6: Motional Emf
- As we have seen, any change in magnetic flux induces an emf opposing that change—a process known as induction. Motion is one of the major causes of induction. For example, a magnet moved toward a coil induces an emf, and a coil moved toward a magnet produces a similar emf. In this section, we concentrate on motion in a magnetic field that is stationary relative to the Earth, producing what is loosely called motional emf.
-
- 23.7: Eddy Currents and Magnetic Damping
- Motional emf is induced when a conductor moves in a magnetic field or when a magnetic field moves relative to a conductor. If motional emf can cause a current loop in the conductor, we refer to that current as an eddy current. Eddy currents can produce significant drag, called magnetic damping, on the motion involved.
-
- 23.10: Transformers
- Transformers do what their name implies—they transform voltages from one value to another. Transformers are also used at several points in the power distribution systema. . Power is sent long distances at high voltages, because less current is required for a given amount of power, and this means less line loss, as was discussed previously. But high voltages pose greater hazards, so that transformers are employed to produce lower voltage at the user’s location.
-
- 23.11: Electrical Safety - Systems and Devices
- Electricity has two hazards. A thermal hazard occurs when there is electrical overheating. A shock hazard occurs when electric current passes through a person. Both hazards have already been discussed. Here we will concentrate on systems and devices that prevent electrical hazards.
Thumbnail: Small cheap inductor. (CC-SA-BY 3.0; FDominec).


1.2: Project Organization
The LibreTexts consists of 12 field-specific libraries, a K-12 library (available in English) and a Spanish (Español) library. Table \(\PageIndex{1}\) lists the libraries and the URLs for reaching them directly:
| Library | URL |
|---|---|
| Biology | https://bio.libretexts.org/ |
| Business | https://biz.libretexts.org/ |
| Chemistry | https://chem.libretexts.org/ |
| Engineering | https://eng.libretexts.org/ |
| Español | https://espanol.libretexts.org/ |
| Geosciences | https://geo.libretexts.org/ |
| Global | https://query.libretexts.org/ |
| Humanities | https://human.libretexts.org/ |
| K12 Edducation | https://K12.libretexts.org/ |
| Mathematics | https://math.libretexts.org/ |
| Medicine (Human Health) | https://med.libretexts.org/ |
| Physics | https://phys.libretexts.org/ |
| Social Sciences | https://socialsci.libretexts.org/ |
| Statistics | https://stats.libretexts.org/ |
| Ukrayinska | https://ukrayinska.libretexts.org/ |
| Workforce | https://workforce.libretexts.org/ |
There are several other ways of reaching the library of your choice which we will describe on the next page
With the exception of the K-12 and Español, Global and Ukrayinska libraries, the content is organized in three primary sections: the Campus Bookshelves , the Bookshelves and Learning Objects . The Español, Global and Ukrayinska libraries contain machine translations of books found in the Bookshelves areas.
Campus Bookshelves
The content in the Campus Bookshelves is formulated as campus-dependent and faculty-dependent course shells with customized textbook remixes. These textbooks are customized by faculty, often in collaboration with developers on the LibreTexts team and may include content from any other sections of the LibreTexts libraries (both within and outside of the respective library where it resides).
For the student, the Course shell is the “decoder” to hundreds of thousands of pages of content in the libraries. While the texts in the Bookshelves are curated by the LibreTexts team and the greater LibreTexts community, the content in the Course Shells are intended to be curated by the faculty and campuses that "own" them. The full freedom to customize Course Shells enables faculty to construct Textbooks and Shells as they see fit. For example, the Chem 2BH course Shell at UCD includes much more than a simple textbook; it also includes an Agenda, homework assignments, worksheets, and the laboratory manual.
Bookshelves
This is where the LibreTexts books are stored. Instructors can either use these texts as constructed or they can use the OER Remixer to modify these materials and create a remix text that is customized for their course.
Some of the texts in the Bookshelves are titled "Book:" and some "Map:". The "books" are original OER textbooks that have been created by LibreTexts, or integrated into the LibreTexts (e.g., texts from OpenStax or the Open Textbook Library referatory). The content in these Books are largely original, although often heavily edited by editors to conform to project standards on presentation and error correction. Alternatively, "Maps" are remixes that are constructed and organized around existing commercial textbooks, but using existing OER content with light or heavy filling in of gaps in the library. The principal goal of the TextMaps is to provide an easy-to-adopt alternative to faculty’s existing textbook choices. Faculty that are too invested in commercial texts to switch to a new organization can adopt a Map.
Learning Objects
These secondary sections contain all materials other than books and homework. The content in this section varies from library to library and may include:
- Worksheets : The LibreTexts worksheets are documents with questions or exercises for students to complete and record answers and are intended to help a student become proficient in a particular skill that was taught to them in class .
- Exemplars and Case Studies : Exemplars are subject-specific examples that illustrate specific concepts. Students can choose from a wide variety of “exemplars”—subject-specific examples that illustrate each concept. Exemplars are arranged in “tracks” that run through the entire course. One track contains most of the topics and concepts included in a typical general chemistry course; topics are arranged by headings similar to textbook chapter titles and subheadings similar to textbook chapter sections.
- Visualization : These are interactive web-based tutorials and simulations that take advantage of the computational resources of an online resource. This involves other active feedback simulations, including virtual laboratories, tutorials, and real-time concept tests.
- References : Tables of constants typically found in textbook appendices or reference books are found in this section.
- Laboratories Experiments and Demonstrations : These are traditional experimental write-ups and demonstrations to increase student attention and engagement in class.
Homework Exercises Sections
These secondary sections are where we organize homework exercises and solutions. A "Faculty Only" section may exist that is accessible to faculty only for use in homework and exams. The homework infrastructure has been migrated to a new server ( https://adapt.libretexts.org ). A studio for creating H5P questions is also available ( https://studio.libretexts.org ).


14.4: John Brown and the Election of 1860
Events in the late 1850s did nothing to quell the country’s sectional unrest, and compromise on the issue of slavery appeared impossible. Lincoln’s 1858 speeches during his debates with Douglas made the Republican Party’s position well known; Republicans opposed the extension of slavery and believed a Slave Power conspiracy sought to nationalize the institution. They quickly gained political momentum and took control of the House of Representatives in 1858. Southern leaders were divided on how to respond to Republican success. Southern extremists, known as “Fire-Eaters,” openly called for secession. Others, like Mississippi senator Jefferson Davis, put forward a more moderate approach by demanding constitutional protection of slavery.
JOHN BROWN
In October 1859, the radical abolitionist John Brown and eighteen armed men, both blacks and whites, attacked the federal arsenal in Harpers Ferry, Virginia. They hoped to capture the weapons there and distribute them among slaves to begin a massive uprising that would bring an end to slavery. Brown had already demonstrated during the 1856 Pottawatomie attack in Kansas that he had no patience for the nonviolent approach preached by pacifist abolitionists like William Lloyd Garrison. Born in Connecticut in 1800, Brown (Figure 14.4.1) spent much of his life in the North, moving from Ohio to Pennsylvania and then upstate New York as his various business ventures failed. To him, slavery appeared an unacceptable evil that must be purged from the land, and like his Puritan forebears, he believed in using the sword to defeat the ungodly.
Brown had gone to Kansas in the 1850s in an effort to stop slavery, and there, he had perpetrated the killings at Pottawatomie. He told other abolitionists of his plan to take Harpers Ferry Armory and initiate a massive slave uprising. Some abolitionists provided financial support, while others, including Frederick Douglass, found the plot suicidal and refused to join. On October 16, 1859, Brown’s force easily took control of the federal armory, which was unguarded (Figure 14.4.2). However, his vision of a mass uprising failed completely. Very few slaves lived in the area to rally to Brown’s side, and the group found themselves holed up in the armory’s engine house with townspeople taking shots at them. Federal troops, commanded by Colonel Robert E. Lee, soon captured Brown and his followers. On December 2, Brown was hanged by the state of Virginia for treason.
Visit the Avalon Project on Yale Law School’s website to read the impassioned speech that Henry David Thoreau delivered on October 30, 1859, arguing against the execution of John Brown. How does Thoreau characterize Brown? What does he ask of his fellow citizens?
John Brown’s raid on Harpers Ferry generated intense reactions in both the South and the North. Southerners grew especially apprehensive of the possibility of other violent plots. They viewed Brown as a terrorist bent on destroying their civilization, and support for secession grew. Their anxiety led several southern states to pass laws designed to prevent slave rebellions. It seemed that the worst fears of the South had come true: A hostile majority would stop at nothing to destroy slavery. Was it possible, one resident of Maryland asked, to “live under a government, a majority of whose subjects or citizens regard John Brown as a martyr and Christian hero?” Many antislavery northerners did in fact consider Brown a martyr to the cause, and those who viewed slavery as a sin saw easy comparisons between him and Jesus Christ.
THE ELECTION OF 1860
The election of 1860 triggered the collapse of American democracy when the elevation of Abraham Lincoln to the presidency inspired secessionists in the South to withdraw their states from the Union.
Lincoln’s election owed much to the disarray in the Democratic Party. The Dred Scott decision and the Freeport Doctrine had opened up huge sectional divisions among Democrats. Though Brown did not intend it, his raid had furthered the split between northern and southern Democrats. Fire-Eaters vowed to prevent a northern Democrat, especially Illinois’s Stephen Douglas, from becoming their presidential candidate. These proslavery zealots insisted on a southern Democrat.
The Democratic nominating convention met in April 1860 in Charleston, South Carolina. However, it broke up after northern Democrats, who made up a majority of delegates, rejected Jefferson Davis’s efforts to protect slavery in the territories. These northern Democratic delegates knew that supporting Davis on this issue would be very unpopular among the people in their states. A second conference, held in Baltimore, further illustrated the divide within the Democratic Party. Northern Democrats nominated Stephen Douglas, while southern Democrats, who met separately, put forward Vice President John Breckinridge from Kentucky. The Democratic Party had fractured into two competing sectional factions.
By offering two candidates for president, the Democrats gave the Republicans an enormous advantage. Also hoping to prevent a Republican victory, pro-Unionists from the border states organized the Constitutional Union Party and put up a fourth candidate, John Bell, for president, who pledged to end slavery agitation and preserve the Union but never fully explained how he’d accomplish this objective. In a pro-Lincoln political cartoon of the time (Figure 14.4.3), the presidential election is presented as a baseball game. Lincoln stands on home plate. A skunk raises its tail at the other candidates. Holding his nose, southern Democrat John Breckinridge holds a bat labeled “Slavery Extension” and declares “I guess I’d better leave for Kentucky, for I smell something strong around here, and begin to think, that we are completely skunk’d.”
The Republicans nominated Lincoln, and in the November election, he garnered a mere 40 percent of the popular vote, though he won every northern state except New Jersey. (Lincoln’s name was blocked from even appearing on many southern states’ ballots by southern Democrats.) More importantly, Lincoln did gain a majority in the Electoral College (Figure 14.4.4). The Fire-Eaters, however, refused to accept the results. With South Carolina leading the way, Fire-Eaters in southern states began to withdraw formally from the United States in 1860. South Carolinian Mary Boykin Chesnut wrote in her diary about the reaction to the Lincoln’s election.  Her statement revealed many southerners’ fear that with Lincoln as President, the South could expect more mayhem like the John Brown raid.
Section Summary
A new level of animosity and distrust emerged in 1859 in the aftermath of John Brown’s raid. The South exploded in rage at the northern celebration of Brown as a heroic freedom fighter. Fire-Eaters called openly for disunion. Poisoned relations split the Democrats into northern and southern factions, a boon to the Republican candidate Lincoln. His election triggered the downfall of the American experiment with democracy as southern states began to leave the Union.
Review Questions
Why did John Brown attack the armory at Harpers Ferry?
to seize weapons to distribute to slaves for a massive uprising
to hold as a military base against proslavery forces
in revenge after the sacking of Lawrence
to prevent southern states from seceding
A
Which of the following did not contribute to Lincoln’s victory in the election of 1860?
the split between northern and southern democrats
the defeat of the Whig party
Lincoln’s improved national standing after his senatorial debates with Stephen Douglas
the Constitutional Union party’s further splintering the vote
B
What were southerners’ and northerners’ views of John Brown?
Antislavery northerners tended to view Brown as a martyr for the antislavery cause; some saw in him a Christ-like figure who died for his beliefs. Southerners, for their part, considered Brown a terrorist. They felt threatened by northerners’ deification of Brown and worried about the potential for other, similar armed insurrections.
Critical Thinking Questions
Why would Americans view the Compromise of 1850 as a final solution to the sectional controversy that began with the Wilmot Proviso in 1846?
If you were a proslavery advocate, how would you feel about the platform of the newly formed Republican Party?
Based on the text of the Lincoln-Douglas debates, what was the position of the Republican Party in 1858? Was the Republican Party an abolitionist party? Why or why not?
John Brown is often described as a terrorist. Do you agree with this description? Why or why not? What attributes might make him fit this profile?
Was it possible to save American democracy in 1860? What steps might have been taken to maintain unity? Why do you think these steps were not taken?
Glossary
- Harpers Ferry
- the site of a federal arsenal in Virginia, where radical abolitionist John Brown staged an ill-fated effort to end slavery by instigating a mass uprising among slaves
- Fire-Eaters
- radical southern secessionists


2.1.8: Case Study- Betty
-
- Last updated
- Save as PDF
- Anonymous
- LibreTexts
- Master Textbook & Whatcom Community College
- Whatcom Community College via WCC Library Press
WCC HLTH 285
Case Study Bio
Betty is a 60 year old Caucasian woman referred for substance use disorder assessment by her primary care doctor because he is concerned about seemingly increased mania and diabetes which is not well managed when she becomes manic. In periods of mania the doctor reported Betty binge drinks, does not eat and does not check her blood glucose levels. She was recently hospitalized with a blood sugar of 782 and was almost unconscious. When she started to feel better and stabilize, she attempted to leave the hospital AMA (against medical advice) so the unit physician contacted the county designated mental health team to evaluate Betty for detainment, enacting the Involuntary Treatment Act. Evaluators did commit Betty for 72 hours but upon release from the hospital she was placed on a Least Restrictive Mental Health Court Order which expires in 90 days.
Betty is an only child born to married parents who have both recently passed. Betty was recently separated from her husband of 34 years, but not legally; they are living apart “only to give me time to get squared away again”. When asked why she has come for an assessment, Betty explained, “My recent hospitalization due to the diabetes and the doctor is worried about my drinking, apparently. When asked if she thinks she may have a problem with drinking or other substances, she replied, “I don’t think it is my drinking but really it is the Bipolar that is the problem. I just haven’t found the right doctor to treat me for that yet.” She went on to expand quite extensively about her medical history and many of her accomplishments in her life which she described as having come to “a screaming halt” when she gave birth to a stillborn daughter 15 years ago. She had tried to get pregnant for years and finally did only to lose the baby. She stated she was unable to have any more children due to her age. At this point during assessment Betty said she couldn’t continue with the assessment and asked if she could leave. This writer reminded her she is required by her LR court order to complete the assessment. She was offered a short break prior to continuing the assessment. Betty appeared very agitated during the assessment and often talked over this writer and interrupted. She presented somewhat manic although she stated, “I am better than I was before going into the hospital.”
Betty reported she was hospitalized once as a teen and once as a young adult. She admitted to attempting suicide on only two occasions. When asked how she tried to suicide, she reported she overdosed on her mother’s pain pills when she was a teen but as an adult she tried cutting her wrists. She reported she still sometimes thinks about suicide, especially around April 14th, because that is when her baby girl was born. Betty became very emotional when talking about her stillborn daughter. According to Betty, her daughter was a full term, healthy baby but in the last 30 minutes of delivery, the baby stressed and the cord was wrapped around her neck and she was not yet turned face down. She stated when the doctor tried to turn the baby manually, she further stressed and that is when the heartbeat stopped. She gave birth to her daughter but she could not be revived.
Betty graduated from Friday Harbor High School and went on to a four year college, Washington State University. She was studying nursing but was a year shy of her bachelor’s degree when she had to quit. When asked why, she reported she just had some personal problems and she did something “stupid and ended up in the hospital”. When asked what, Betty said she had a minor car accident. This writer then called Betty’s attention to the discharge paperwork from the hospital and mentioned it has most of her history in the document. This writer asked again, if it may have been when she tried to hurt herself. Betty then became quite defensive and agitated and yelled, “that’s none of your business and it has nothing to do with why I am here. “ This writer reassured Betty the information is very confidential and it is important to have all the information about her history in order to be of the most help possible. Betty calmed down and then said she was hospitalized because she slit her forearms open and was found unconscious. Betty stated the man she loved and wanted to marry cheated on her and she caught him.
Betty reported that she enjoys dancing and listening to music, sometimes swims at the local gym, and used to love hiking and bicycling, although she has not done these activities in a long time.


Sections
-
- Introduction
- Æschylus, the first of the great Grecian writers of tragedy, was born at Eleusis, in 525 B.C. He was the son of Euphorion, who was probably a wealthy owner of rich vineyards. The poet's early employment was to watch the grapes and protect them from the ravages of men and other animals, and it is said that this occupation led to the development of his dramatic genius. It is more easy to believe that it was responsible for the development of certain other less admirable qualities of the poet.
-
- Prometheus Chained
- Prometheus having, by his attention to the wants of men, provoked the anger of Jove, is bound down in a cleft of a rock in a distant desert of Scythia. Here he not only relates the wanderings, but foretells the future lot of Io, and likewise alludes to the fall of Jove's dynasty. Disdaining to explain his meaning to Mercury, he is swept into the abyss amid terrific hurricane and earthquake.
-
- The Seven Against Thebes
- The siege of the city of Thebes, and the description of the seven champions of the Theban and Argive armies, The deaths of the brothers Polynices and Eteocles, the mournings over them, by their sisters Antigone and Ismene, and the public refusal of burial to the ashes of Polynices, against which Antigone boldly protests, conclude the play.
Thumbnail: Scene from The Seven against Thebes by Aeschylus: Capaneus scales the city walls to overthrow King Creon who looks down from the battlements. Campanian red-figure neck-amphora, ca. 340 BC. Image cropped. (CC BY_SA 3.0 Unported; Wolfgang Sauber via Wikipedia )


7.12: A Martian Controversy
Before spacecraft and more-advanced equipment, we depended on the eye and photography to decipher and understand what we saw in the Universe. Mars was – and it still is – interesting because it is so close to Earth and somewhat “Earth-like. ” Some astronomers focused on the planets and specifically Mars. Percival Lowell was one such astronomer.
Lowell Observatory in Flagstaff, Arizona
In the late 1800s, Percival Lowell was the first to seek site with good air clarity and superior seeing (steady air or lack of turbulence). After testing many different site tests, Lowell gave the lumber town of Flagstaff, AZ (population of 1,000) high marks. He selected a hill 300 feet above town and built an observatory, which is still in use today.
Lowell initiated Mars Mania; he sensationalized what he perceived as Martian canals . The canals were first described by the Italian astronomer Giovanni Schiaparelli during the close Martian approach in 1877. Schiaparelli gave them the Italian name ‘canali. ’ The translation was taken as canals; canals refer to something made by someone, something, but not Earth-natural…. Lowell pressed the idea that these canals were made by intelligent beings – Martians. There were books and articles that promoted the Martian canals as well as Martian seasonal vegetation growth. This, and surrounding controversy, stimulated interest in Red Planet. And in 1898, science fiction author H. G. Wells wrote “The War of the Worlds,” furthering the controversy.
Other observers also saw the Martian Canals. Lowell astronomer Earl Slipher’s book, The Photographic Story of Mars , shows adjacent direct photographs and drawings. In 1962, Slipher asked readers to judge:
History … shows that every skilled observer … has had no great difficulty of seeing and convincing himself of the reality of the canals. Photographs have recorded traces of so many of the canals and oases … that they should remove all doubt of the reality — E. C. Slipher (1962).
So what happened? Were the canals real? When spacecraft first started exploring Mars in 1965, Mariner 4, no canals were seen. It is now believed that the eye fooled the astronomer, he connected features on Mars that were there – but the canals were not. It was like connecting the dots, an optical illusion.


3.6: Resonance Structures
-
- Last updated
- Save as PDF
Learning Objectives
- To understand the concept of resonance.
Resonance structures are a set of two or more Lewis Structures that collectively describe the electronic bonding of a single polyatomic species including fractional bonds and fractional charges. Resonance structures are capable of describing delocalized electrons that cannot be expressed by a single Lewis formula with an integral number of covalent bonds.
Sometimes one Lewis Structure is not Enough
Sometimes, even when formal charges are considered, the bonding in some molecules or ions cannot be described by a single Lewis structure. Resonance is a way of describing delocalized electrons within certain molecules or polyatomic ions where the bonding cannot be expressed by a single Lewis formula. A molecule or ion with such delocalized electrons is represented by several contributing structures (also called resonance structures or canonical forms). Such is the case for ozone (\(\ce{O3}\)), an allotrope of oxygen with a V-shaped structure and an O–O–O angle of 117.5°.
Ozone (\(O_3\))
1. We know that ozone has a V-shaped structure, so one O atom is central:
2. Each O atom has 6 valence electrons, for a total of 18 valence electrons.
3. Assigning one bonding pair of electrons to each oxygen–oxygen bond gives
with 14 electrons left over.
4. If we place three lone pairs of electrons on each terminal oxygen, we obtain
and have 2 electrons left over.
5. At this point, both terminal oxygen atoms have octets of electrons. We therefore place the last 2 electrons on the central atom:
6. The central oxygen has only 6 electrons. We must convert one lone pair on a terminal oxygen atom to a bonding pair of electrons—but which one? Depending on which one we choose, we obtain either
Which is correct? In fact, neither is correct. Both predict one O–O single bond and one O=O double bond. As you will learn, if the bonds were of different types (one single and one double, for example), they would have different lengths. It turns out, however, that both O–O bond distances are identical, 127.2 pm, which is shorter than a typical O–O single bond (148 pm) and longer than the O=O double bond in O 2 (120.7 pm).
Equivalent Lewis dot structures, such as those of ozone, are called resonance structures . The position of the atoms is the same in the various resonance structures of a compound, but the position of the electrons is different. Double-headed arrows link the different resonance structures of a compound:
The double-headed arrow indicates that the actual electronic structure is an average of those shown, not that the molecule oscillates between the two structures.
When it is possible to write more than one equivalent resonance structure for a molecule or ion, the actual structure is the average of the resonance structures.
The Carbonate (\(CO_3^{2−} \)) Ion
Like ozone, the electronic structure of the carbonate ion cannot be described by a single Lewis electron structure. Unlike O 3 , though, the actual structure of CO 3 2 − is an average of three resonance structures.
1. Because carbon is the least electronegative element, we place it in the central position:
2. Carbon has 4 valence electrons, each oxygen has 6 valence electrons, and there are 2 more for the −2 charge. This gives 4 + (3 × 6) + 2 = 24 valence electrons.
3. Six electrons are used to form three bonding pairs between the oxygen atoms and the carbon:
4. We divide the remaining 18 electrons equally among the three oxygen atoms by placing three lone pairs on each and indicating the −2 charge:
5. No electrons are left for the central atom.
6. At this point, the carbon atom has only 6 valence electrons, so we must take one lone pair from an oxygen and use it to form a carbon–oxygen double bond. In this case, however, there are three possible choices:
As with ozone, none of these structures describes the bonding exactly. Each predicts one carbon–oxygen double bond and two carbon–oxygen single bonds, but experimentally all C–O bond lengths are identical. We can write resonance structures (in this case, three of them) for the carbonate ion:
The actual structure is an average of these three resonance structures.
The Nitrate (\(NO_3^-\)) ion
1. Count up the valence electrons: (1*5) + (3*6) + 1(ion) = 24 electrons
2. Draw the bond connectivities:
3. Add octet electrons to the atoms bonded to the center atom:
4. Place any leftover electrons (24-24 = 0 ) on the center atom:
5. Does the central atom have an octet?
- NO , it has 6 electrons
- Add a multiple bond (first try a double bond) to see if the central atom can achieve an octet:
6. Does the central atom have an octet?
- YES
- Are there possible resonance structures? YES
Note: We would expect that the bond lengths in the \(\ce{NO_3^{-}}\) ion to be somewhat shorter than a single bond.
Example \(\PageIndex{1}\): Benzene
Benzene is a common organic solvent that was previously used in gasoline; it is no longer used for this purpose, however, because it is now known to be a carcinogen. The benzene molecule (\(\ce{C6H6}\)) consists of a regular hexagon of carbon atoms, each of which is also bonded to a hydrogen atom. Use resonance structures to describe the bonding in benzene.
Given: molecular formula and molecular geometry
Asked for: resonance structures
Strategy:
- Draw a structure for benzene illustrating the bonded atoms. Then calculate the number of valence electrons used in this drawing.
- Subtract this number from the total number of valence electrons in benzene and then locate the remaining electrons such that each atom in the structure reaches an octet.
- Draw the resonance structures for benzene.
Solution:
A Each hydrogen atom contributes 1 valence electron, and each carbon atom contributes 4 valence electrons, for a total of (6 × 1) + (6 × 4) = 30 valence electrons. If we place a single bonding electron pair between each pair of carbon atoms and between each carbon and a hydrogen atom, we obtain the following:
Each carbon atom in this structure has only 6 electrons and has a formal charge of +1, but we have used only 24 of the 30 valence electrons.
B If the 6 remaining electrons are uniformly distributed pairwise on alternate carbon atoms, we obtain the following:
Three carbon atoms now have an octet configuration and a formal charge of −1, while three carbon atoms have only 6 electrons and a formal charge of +1. We can convert each lone pair to a bonding electron pair, which gives each atom an octet of electrons and a formal charge of 0, by making three C=C double bonds.
C There are, however, two ways to do this:
Each structure has alternating double and single bonds, but experimentation shows that each carbon–carbon bond in benzene is identical, with bond lengths (139.9 pm) intermediate between those typically found for a C–C single bond (154 pm) and a C=C double bond (134 pm). We can describe the bonding in benzene using the two resonance structures, but the actual electronic structure is an average of the two. The existence of multiple resonance structures for aromatic hydrocarbons like benzene is often indicated by drawing either a circle or dashed lines inside the hexagon:
Exercise \(\PageIndex{1}\): Nitrite Ion
The sodium salt of nitrite is used to relieve muscle spasms. Draw two resonance structures for the nitrite ion (NO 2 − ).
- Answer
-
Resonance structures are particularly common in oxoanions of the p -block elements, such as sulfate and phosphate, and in aromatic hydrocarbons, such as benzene and naphthalene.
Warning
If several reasonable resonance forms for a molecule exists, the "actual electronic structure" of the molecule will probably be intermediate between all the forms that you can draw. The classic example is benzene in Example \(\PageIndex{1}\). One would expect the double bonds to be shorter than the single bonds, but if one overlays the two structures, you see that one structure has a single bond where the other structure has a double bond. The best measurements that we can make of benzene do not show two bond lengths - instead, they show that the bond length is intermediate between the two resonance structures.
Resonance structures is a mechanism that allows us to use all of the possible resonance structures to try to predict what the actual form of the molecule would be. Single bonds, double bonds, triple bonds, +1 charges, -1 charges, these are our limitations in explaining the structures, and the true forms can be in between - a carbon-carbon bond could be mostly single bond with a little bit of double bond character and a partial negative charge, for example.
Summary
Some molecules have two or more chemically equivalent Lewis electron structures, called resonance structures. Resonance is a mental exercise and method within the Valence Bond Theory of bonding that describes the delocalization of electrons within molecules. These structures are written with a double-headed arrow between them, indicating that none of the Lewis structures accurately describes the bonding but that the actual structure is an average of the individual resonance structures. Resonance structures are used when one Lewis structure for a single molecule cannot fully describe the bonding that takes place between neighboring atoms relative to the empirical data for the actual bond lengths between those atoms. The net sum of valid resonance structures is defined as a resonance hybrid, which represents the overall delocalization of electrons within the molecule. A molecule that has several resonance structures is more stable than one with fewer. Some resonance structures are more favorable than others.


6: Vacuum Solutions
In this chapter we investigate general relativity in regions of space that have no matter to act as sources of the gravitational field. We will not , however, limit ourselves to calculating spacetimes in cases in which the entire universe has no matter. For example, we will be able to calculate general-relativistic effects in the region surrounding the earth, including a full calculation of the geodetic effect, which was estimated in Section 5.5 only to within an order of magnitude. We can have sources, but we just won’t describe the metric in the regions where the sources exist, e.g., inside the earth. The advantage of accepting this limitation is that in regions of empty space, we don’t have to worry about the details of the stress-energy tensor or how it relates to curvature. As should be plausible based on the physical motivation given in Section 5.1 , the field equations in a vacuum are simply \(R_{ab} = 0\).
-
- 6.1: Event Horizons
- One seemingly trivial way to generate solutions to the field equations in vacuum is simply to start with a flat Lorentzian spacetime and do a change of coordinates. This might seem pointless, since it would simply give a new description (and probably a less convenient and descriptive one) of the same old, boring, flat spacetime. It turns out, however, that some very interesting things can happen when we do this.
-
- 6.2: The Schwarzschild Metric (Part 1)
- We now set ourselves the goal of finding the metric describing the static spacetime outside a spherically symmetric, nonrotating, body of mass m. This problem was first solved by Karl Schwarzschild in 1915. One byproduct of finding this metric will be the ability to calculate the geodetic effect exactly, but it will have more far reaching consequences, including the existence of black holes.
-
- 6.4: Black Holes (Part 1)
- A provocative feature of the Schwarzschild metric is that it has elements that blow up at r=0 and at r=2m. If this is a description of the sun, for example, then these singularities are of no physical significance, since we only solved the Einstein field equation for the vacuum region outside the sun, whereas r=2m would lie about 3 km from the sun’s center. Furthermore, it is possible that one or both of these singularities is nothing more than a spot where our coordinate system misbehaves.
-
- 6.6: Degenerate Solutions
- So it appears that the signature of spacetime is something that is not knowable a priori, and must be determined by experiment. When a thing is supposed to be experimentally observable, general relativity tells us that it had better be coordinate-independent. Is this so? A proposition from linear algebra called Sylvester’s law of inertia encourages us to believe that it is.
Thumbnail: This artist's concept illustrates a supermassive black hole with millions to billions times the mass of our sun. Supermassive black holes are enormously dense objects buried at the hearts of galaxies. (Public Domain; NASA/JPL-Caltech).


2.11: Subject-focused indexes
When you are doing in-depth research and need to find a comprehensive set of scholarly articles in a particular subject area, the best finding tool will almost always be an index focused on that subject. Here are a few examples of subject-focused indexes to give you a better idea of what they can cover:
- CAB Abstracts is one of the largest agricultural databases covering international research and development literature in the fields of agronomy, animal health, forestry, human health, nutrition, and management & conservation of natural resources.
- Design & Applied Arts Index (DAAI) is an international index that provides resources on topics related to design and applied arts such as advertising, product design, and architecture. It contains citations and abstracts of articles, news items, and reviews created from 1973 to the present.
- Compendex – Engineering Village is a highly comprehensive engineering database with over 14 million records from journals, conference proceedings and technical papers in all engineering disciplines. Coverage dates from 1884 to the present.
- Food Science and Technology Abstracts is produced by the International Food Information Service (IFIS), and is the world’s leading database of information on food science, food technology and nutrition.
- Sociological Abstracts provides full-text access to resources on topics related to sociology such as family and marriage, social change, and culture. This index covers thousands of journals dating back to 1952 and includes articles, book chapters, and conference papers.
- ABI/INFORM Global searches worldwide business periodicals for in-depth coverage of business and economic conditions, management techniques, theory, and practice of business, advertising, marketing, economics, human resources, finance, taxation, computers, and more.
- ERIC (Education Resource Information Center, EBSCOhost Access) is a comprehensive index of education literature and curriculum materials, containing more than 1.3 million records and links to more than 323,000 full-text documents dating back to 1966.
- Communication & Mass Media Complete is a robust communication studies database. It provides full-text, indexing and abstracts for many top communication journals covering all related disciplines, including media studies, linguistics, rhetoric and discourse.
Subject-focused indexes like these offer discipline-specific tools to help you search, sort, and focus your results. For example, history databases let you search by the date an event occurred, whereas chemistry databases let you search by chemical compound. You will want to familiarize yourself with the indexes that relate to your classes so you can efficiently find the best sources for your projects.
Interdisciplinary topics
Depending on your topic, you may need to use more than one subject-focused index. Let’s say you want to learn more about marketing new products to college students and influencing their buying behavior. In this example, you’d probably want to search a business-focused index (such as ABI/INFORM) and maybe also an index focused on sociology or psychology (such as Sociological Abstracts or PsycInfo). Each index will retrieve articles with different viewpoints, keywords, or subject emphases. Talk with your course instructor or a subject librarian if you need help identifying the right indexes for your topic.
Conclusion
To find out which tools are the best for your needs, you will probably need to explore a few of them and compare your search results in each. In scholarly research, one-stop-shopping doesn’t happen very often. Instead, you need to use the right tools for the right job and be willing to experiment. In the next chapter, we will discuss how to use these finding tools effectively by implementing effective search strategies.


22.3: Radiation from a Current Moment
-
- Last updated
- Save as PDF
In this section, we begin to address the following problem: Given a distribution of impressed current density \({\bf J}({\bf r})\), what is the resulting electric field intensity \({\bf E}({\bf r})\)? One route to an answer is via Maxwell’s equations. Viewing Maxwell’s equations as a system of differential equations, a rigorous mathematical solution is possible given the appropriate boundary conditions. The rigorous solution following that approach is relatively complicated, and is presented beginning in Section 9.2 of this book.
If we instead limit scope to a sufficiently simple current distribution, a simple informal derivation is possible. This section presents such a derivation. The advantage of tackling a simple special case first is that it will allow us to quickly assess the nature of the solution, which will turn out to be useful once we do eventually address the more general problem. Furthermore, the results presented in this section will turn out to be sufficient to tackle many commonly-encountered applications.
Figure \(\PageIndex{1}\): A \(+\hat{\bf{z}}\)-directed current moment located at the origin. ( CC BY-SA 4.0 ; C. Wang)The simple current distribution considered in this section is known as a current moment . An example of a current moment is shown in Figure \(\PageIndex{1}\) and in this case is defined as follows:
\[\Delta{\bf J}({\bf r}) = \hat{\bf z}~I~\Delta l~\delta({\bf r}) \label{m0194_eCM} \]
where \(I~\Delta l\) is the scalar component of the current moment, having units of current times length (SI base units of A\(\cdot\)m); and \(\delta({\bf r})\) is the volumetric sampling function 1 defined as follows:
\begin{align} ~ &\delta({\bf r}) \triangleq 0~~~ \mbox{for}~~~ {\bf r}\neq 0;~~~ \mbox{and} \label{m0194_eDelta1} \\ ~ &\int_{\mathcal{V}}\delta({\bf r})~dv \triangleq 1 \label{m0194_eDelta2}\end{align}
where \(\mathcal{V}\) is any volume which includes the origin (\({\bf r}=0\)). It is evident from Equation \ref{m0194_eDelta2} that \(\delta({\bf r})\) has SI base units of m\(^{-3}\). Subsequently, \(\Delta{\bf J}({\bf r})\) has SI base units of A/m\(^2\), confirming that it is a volume current density. However, it is the simplest possible form of volume current density, since – as indicated by Equation \ref{m0194_eDelta1} – it exists only at the origin and nowhere else.
Although some current distributions approximate the current moment, current distributions encountered in common engineering practice generally do not exist in precisely this form. Nevertheless, the current moment turns out to be generally useful as a “building block” from which practical distributions of current can be constructed, via the principle of superposition. Radiation from current distributions constructed in this manner is calculated simply by summing the radiation from each of the constituent current moments.
Now let us consider the electric field intensity \(\Delta{\bf E}({\bf r})\) that is created by this current distribution. First, if the current is steady (i.e., “DC”), this problem falls within the domain of magnetostatics; i.e., the outcome is completely described by the magnetic field, and there can be no radiation. Therefore, let us limit our attention to the “AC” case, for which radiation is possible. It will be convenient to employ phasor representation. In phasor representation, the current density is
\[\Delta\widetilde{\bf J}({\bf r}) = \hat{\bf z}~\widetilde{I}~\Delta l~\delta({\bf r}) \label{m0194_eCM2} \]
where \(\widetilde{I}~\Delta l\) is simply the scalar current moment expressed as a phasor.
Now we are ready to address the question “What is \(\Delta\widetilde{\bf E}({\bf r})\) due to \(\Delta\widetilde{\bf J}({\bf r})\)?” Without doing any math, we know quite a bit about \(\Delta\widetilde{\bf E}({\bf r})\). For example:
- Since electric fields are proportional to the currents that give rise to them, we expect \(\Delta\widetilde{\bf E}({\bf r})\) to be proportional to \(\left|\widetilde{I}~\Delta l\right|\).
- If we are sufficiently far from the origin, we expect \(\Delta\widetilde{\bf E}({\bf r})\) to be approximately proportional to \(1/r\) where \(r\triangleq\left|{\bf r}\right|\) is the distance from the source current. This is because point sources give rise to spherical waves, and the power density in a spherical wave would be proportional to \(1/r^2\). Since time-average power density is proportional to \(\left|\Delta\widetilde{\bf E}({\bf r})\right|^2\), \(\Delta\widetilde{\bf E}({\bf r})\) must be proportional to \(1/r\).
- If we are sufficiently far from the origin, and the loss due to the medium is negligible, then we expect the phase of \(\Delta\widetilde{\bf E}({\bf r})\) to change approximately at rate \(\beta\) where \(\beta\) is the phase propagation constant \(2\pi/\lambda\). Since we expect spherical phasefronts, \(\Delta\widetilde{\bf E}({\bf r})\) should therefore contain the factor \(e^{-j\beta r}\).
- Ampere’s law indicates that a \(\hat{\bf z}\)-directed current at the origin should give rise to a \(\hat{\bf \phi}\)-directed magnetic field in the \(z=0\) plane. 2 At the same time, Poynting’s theorem requires the cross product of the electric and magnetic fields to point in the direction of power flow. In the present problem, this direction is away from the source; i.e., \(+\hat{\bf r}\). Therefore, \(\Delta\widetilde{\bf E}(z=0)\) points in the \(-\hat{\bf z}\) direction. The same principle applies outside of the \(z=0\) plane, so in general we expect \(\Delta\widetilde{\bf E}({\bf r})\) to point in the \(\hat{\bf \theta}\) direction.
- We expect \(\Delta\widetilde{\bf E}({\bf r})=0\) along the \(z\) axis. Subsequently \(\left|\Delta\widetilde{\bf E}(\hat{\bf r})\right|\) must increase from zero at \(\theta=0\) and return to zero at \(\theta=\pi\). The symmetry of the problem suggests \(\left|\Delta\widetilde{\bf E}(\hat{\bf r})\right|\) is maximum at \(\theta=\pi/2\). This magnitude must vary in the simplest possible way, leading us to conclude that \(\Delta\widetilde{\bf E}(\hat{\bf r})\) is proportional to \(\sin\theta\). Furthermore, the radial symmetry of the problem means that \(\Delta\widetilde{\bf E}(\hat{\bf r})\) should not depend at all on \(\phi\).
Putting these ideas together, we conclude that the radiated electric field has the following form:
\[\Delta\widetilde{\bf E}({\bf r}) \approx \hat{\bf \theta} C \left(\widetilde{I}~\Delta l\right) \left(\sin\theta\right) \frac{e^{-j\beta r}}{r} \nonumber \]
where \(C\) is a constant which accounts for all of the constants of proportionality identified in the preceding analysis. Since the units of \(\Delta\widetilde{\bf E}({\bf r})\) are V/m, the units of \(C\) must be \(\Omega\)/m. We have not yet accounted for the wave impedance of the medium \(\eta\), which has units of \(\Omega\), so it would be a good bet based on the units that \(C\) is proportional to \(\eta\). However, here the informal analysis reaches a dead end, so we shall simply state the result from the rigorous solution: \(C=j\eta\beta/4\pi\). The units are correct, and we finally obtain:
\[\Delta\widetilde{\bf E}({\bf r}) \approx \hat{\bf \theta} \frac{j\eta\beta}{4\pi} \left(\widetilde{I}~\Delta l\right) \left(\sin\theta\right) \frac{e^{-j\beta r}}{r} \nonumber \]
Additional evidence that this solution is correct comes from the fact that it satisfies the wave equation \(\nabla^2 \Delta\widetilde{\bf E}({\bf r}) + \beta^2 \Delta\widetilde{\bf E}({\bf r}) = 0\). 3
Note that the expression we have obtained for the radiated electric field is approximate (hence the “\(\approx\)”). This is due in part to our presumption of a simple spherical wave, which may only be valid at distances far from the source. But how far? An educated guess would be distances much greater than a wavelength (i.e., \(r\gg\lambda\)). This will do for now; in another section, we shall show rigorously that this guess is essentially correct.
We conclude this section by noting that the current distribution analyzed in this section is sometimes referred to as a Hertzian dipole . A Hertzian dipole is typically defined as a straight infinitesimally-thin filament of current with length which is very small relative to a wavelength, but not precisely zero. This interpretation does not change the solution obtained in this section, thus we may view the current moment and the Hertzian dipole as effectively the same in practical engineering applications.
Additional Reading:
- “Dirac delta function” on Wikipedia.
- “Dipole antenna” (section entitled “Hertzian Dipole”) on Wikipedia.
- Also a form of the Dirac delta function ; see “Additional Reading” at the end of this section.↩
- This is sometimes described as the “right hand rule” of Ampere’s law.↩
- Confirming this is straightforward (simply substitute and evaluate) and is left as an exercise for the student.↩