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Glans penis

In male human anatomy, the glans penis or penile glans, commonly referred to as the glans, (from Latin glans meaning "acorn") is the bulbous structure at the distal end of the human penis that is the human male's most sensitive erogenous zone and primary anatomical source of sexual pleasure. The glans penis is present in the male reproductive organs of humans and most other mammals where it may appear smooth, spiny, elongated or divided. It is externally lined with mucosal tissue, which creates a smooth texture and glossy appearance. In humans, the glans is located over the distal ends of the corpora cavernosa and is a continuation of the corpus spongiosum of the penis. At the summit appears the urinary meatus and at the base forms the corona glandis. An elastic band of tissue, known as the frenulum, runs on its ventral surface. In men who are not circumcised, it is completely or partially covered by a fold of skin called the foreskin. In adults, the foreskin can generally be retracted over and past the glans manually or sometimes automatically during an erection.

The glans penis develops as the terminal end of the genital tubercle during the embryonic development of the male fetus. The tubercle is present in the embryos of both sexes as an outgrowth in the caudal region that later develops into a primordial phallus. Exposure to male hormones (androgens) initiates the tubercle's development into a penis making the glans penis anatomically homologous to the clitoral glans in females.

The glans is more commonly known as the "head" or the "tip" of the penis, and colloquially referred to in British English and Irish English as the "bellend".

Structure
The glans penis is a body of spongy erectile tissue that is moulded on the rounded ends of the two corpora cavernosa penis, extending farther on their upper than on their lower surfaces. It is the expanded cap of the corpus spongiosum, a sponge-like region that surrounds the male urethra within the penis maintaining it as a viable channel for ejaculation. The glans is covered by a stratified squamous epithelium and a dense layer of connective tissue equivalent to the dermis of typical skin. The papillary layer of the dermis blends into the dense connective tissue forming the tunica albuginea of the corpus spongiosum behind the glans. The external lining with mucosal tissue is responsible for its typical smooth texture and appearance.

The increase of arterial flow during erection fills the erectile tissue with blood causing the glans to grow in size and sensitivity. While the penis is rigid when erect, the glans itself remains slightly softer. The soft cushiony texture of the glans absorbs impact during rigorous instances of copulation. The proportional size of the glans penis can vary among males. While the shape of the glans is typically acorn-like, in some men it might be wider in circumference than the shaft, giving the penis a mushroom-like appearance, while in others it might be narrower and more akin to a probe in shape. Some researchers have suggested that the glans has evolved to become acorn-, mushroom- or cone-shaped so that during copulation it acts to remove any semen still there from previous sex partners, but this is not supported when looking at primate relatives who have different mating behaviors.

At the summit of the glans is the slit-like vertical external urethral orifice, called the urinary meatus, through which urine, semen and pre-ejaculatory fluid exit the penis. The circumference of the base of the glans forms a rounded projecting border, the corona glandis, overhanging a deep retroglandular groove known as the coronal sulcus. Behind the corona is the neck of the penis, which separates the glans and the penile shaft. Ventrally, the two glans wings merge on the midline forming the septum glandis and a triangle or a V-shaped area under it. The frenulum is the highly vascularized elastic band of tissue located on the underside of the glans that connects the foreskin to the head of the penis. The frenulum is supple enough to allow the retraction of the foreskin over the glans and pull it back when the erection is gone. In flaccid state, it tightens to narrow the foreskin opening.

Innervation
The glans and the frenulum are innervated by the bilateral dorsal nerve of the penis and the perineal nerve, both divisions of the pudendal nerve. Branches of the dorsal nerve extend through the glans ventrolaterally displaying a three-dimensional innervation pattern. The main branches form smaller bundles of nerves that expand outwards into the tissue of the glans. The rich innervation of the glans penis reveals its function as a primary anatomical source of male sexual pleasure. Yang & Bradley argue; "the distinct pattern of innervation of the glans emphasizes its role as a sensory structure". While Yang & Bradley's (1998) report "showed no areas in the glans to be more densely innervated than others.", Halata & Munger (1986) report that the density of several nerve terminals is greatest in the corona glandis.

Halata & Spathe (1997) reported; "the glans penis contains a predominance of free nerve endings, numerous genital end bulbs and rarely Pacinian and Ruffinian corpuscles. Merkel nerve endings and Meissner's corpuscles (mechanoreceptors typically found in thick glabrous skin) are not present". The genital end bulbs, that are present throughout the glans, are most numerous in the corona and near the frenulum. Simple, Pacinian and Ruffinian corpuscles are identified predominantly in the corona glandis. The most numerous nerve terminals are free nerve endings present in almost every dermal papilla of the glans, as well as scattered throughout the deeper dermis.

Blood supply
The glans penis receives blood from the internal pudental artery through its branch, the dorsal artery of the penis, which also supplies the foreskin, and the penile shaft. Behind the corona, the terminal branches of the dorsal arteries anastomose with the axial arteries through perforating branches before they end in the glans. Branches of the dorsal artery curve around each side of the distal shaft to enter the glans and the frenulum ventrally. Venous drainage of the penis begins at the base of the glans. Small tributaries deriving from the corona form a venous plexus at the neck of the penis, known as the retro-coronal, or retro-balanic, plexus. Smaller paired venules run into the frenulum and the glans from its ventral surface. The deep dorsal vein, one of the two dorsal veins of the penis, serves as a common vessel receiving blood drained from the glans and the two corpora cavernosa through the circumflex veins that surround them.

Foreskin
The glans is completely or partially covered by a double-layered fold of skin, known as the foreskin. In adults, glans exposure can be easily achieved by manual retraction of the foreskin or sometimes automatically during erection. The degree of automatic foreskin retraction varies considerably depending on the foreskin length. The foreskin can be characterized as long when the preputial orifice extends beyond the glans during erection or medium when the orifice is located around the meatus. The primary purpose of the foreskin is considered to be the covering of the glans and the urinary meatus, while also maintaining the mucosa in a moist environment. Foreskin rectractability gradually increases with age. In infancy the foreskin is fused to the glans, it remains non retractable in early childhood and it continues to be tight during preadolescence. The skin begins to loosen up significantly during puberty allowing the glans to be completely exposed when needed. By the age of eighteen most boys will have a fully retractable foreskin.

In some cases, for cultural, medical, or prophylactic reasons some men undergo circumcision or were circumcised as infants, a procedure where the foreskin is partially or completely removed from the penis. The glans of circumcised men remains fully exposed and dry. Several studies have suggested the glans is generally equally sensitive in both circumcised and uncircumcised penises,  though physical changes do occur, including keratinization of the glans.

Development
The glans develops as the terminal end of a phallic structure, called the genital tubercle, which forms in the embryo regardless of sex during the early weeks of pregnancy. Initially undifferentiated, the tubercle develops into a penis during the development of the reproductive system depending on the exposure to male hormones, such as androgens. In mammals, sexual differentiation is determined by the sperm that carries either an X or a Y (male) chromosome. The Y chromosome contains a sex-determining gene (SRY) that encodes a transcription factor for the protein TDF (testis determining factor) and triggers the creation of testosterone for the embryo's development into a male.

Although the sex of the infant is determined from the moment of conception, the complete external differentiation of the organs begins about eight or nine weeks after conception. Some sources state that the process will be completed by the twelfth week, while others state that it is clearly evident by the thirteenth week and that the sex organs are fully developed by the sixteenth week.

Both the penis and clitoris develop from the same tissues that become the glans and shaft of the penis and this shared embryonic origin makes these two organs homologous (different versions of the same structure). In the female fetus the absence of testosterone will stop the growth of the phallus causing the tubercle to shrink and form the clitoris. In the male fetus the presence of a Y chromosome leads to the development of the testes, which secrete a large amount of hormones called androgens. These hormones will cause the masculinization of the phenotypically indifferent organs. When exposed to testosterone, the genital tubercle elongates to form the penis. By fusion of the urogenital folds – elongated spindle-shaped structures that contribute to the formation of the urethral groove on the belly aspect of the genital tubercle – the urogenital sinus closes completely to form the spongy urethra and the labioscrotal swellings unite to form the scrotum. The secretion of testosterone during this phase plays a decisive role in the final shaping of the penis. After birth, testosterone levels drop significantly until puberty.

Clinical significance
The epithelium of the glans penis consists of mucosal tissue. Birley et al. report that excessive washing with soap may dry the mucous membrane which covers the glans penis and cause non-specific dermatitis. The condition is described as an inflammation of the skin, often caused by an irritating substance or a contact allergy. Sensitivity to chemicals in certain products can cause an allergic reaction, including irritation, itching and rash.

Inflammation of the glans penis is known as balanitis. It is a treatable condition that occurs in about 3–11% of males (up to 35% of diabetic males). Edwards reported that it is generally more common in males who have poor hygiene habits or have not been circumcised. It has many causes, including irritation or infection with a wide variety of pathogens. Symptoms of balanitis may appear suddenly or develop gradually. They might include pain, irritation, redness or red patches on the glans penis. Careful identification of the cause with the aid of patient history, physical examination, swabs and cultures, and biopsy are essential in order to determine the proper treatment.

The meatus (opening) of the urethra located at the tip of the glans might become subject to meatal stenosis, a condition mostly seen as a late complication of circumcision. It occurs in about 2–20% of circumcised boys and it is rarely seen in uncircumcised men. It is characterized by a narrowing of the meatus, which might cause sudden or often urges to urinate and burning during the process.

For some individuals who experience difficulty in achieving full glanular engorgement of glans penis, they may be diagnosed with soft glans syndrome (glans insufficiency syndrome). It is often undiagnosed in the general population due to the lack of a standardized nomenclature.

Carnivores
Male felids are able to urinate backwards by curving the tip of the glans penis backward. In cats, the glans penis is covered with spines. Penile spines also occur on the glans of male and female spotted hyenas. In male dogs the glans penis is smooth and consists of two parts called the bulbus glandis and pars longa glandis. The glans of a fossa's penis extends about halfway down the shaft and is spiny except at the tip. In comparison, the glans of felids is short and spiny, while that of viverrids is smooth and long.

Rodents
The glans penis of the marsh rice rat is long and robust, averaging 7.3 mm (0.29 in) long and 4.6 mm (0.18 in) broad. Winkelmann's mouse can most readily be distinguished from its close relatives by its partially corrugated glans penis. In Thomasomys ucucha, the glans penis is rounded, short, and small and is superficially divided into left and right halves by a trough at the top and a ridge at the bottom. Most of the glans is covered with spines, except for an area near the tip. The glans penis of a male cape ground squirrel is large with a prominent baculum.

Perissodactyls
When erect, the glans of a horse's penis increases by 3 to 4 times. The urethra opens within the urethral fossa, a small pouch at the distal end of the glans. Unlike the human glans, the glans of a horse's penis extends backwards on its shaft.

Marsupials, monotremes and bats
The shape of the glans varies among different marsupial species. In most marsupials, the glans is divided, but male macropods have an undivided glans penis.

The glans penis is also divided into two parts in platypuses and echidnas.

Males of Racey's pipistrelle bat have a narrow, egg-shaped glans penis.

title: Glans penis


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Palette knife

A palette knife is a blunt tool used for mixing or applying paint, with a flexible steel blade. It is primarily used for applying paint to the canvas, mixing paint colors, adding texture to the painted surface, paste, etc., or for marbling, decorative endpapers, etc. The "palette" in the name is a reference to an artist's palette which is used for mixing oil paint and acrylic paints.

Art knives come primarily in two types:
 * palette knife resembling a putty knife with a rounded tip, suited for mixing paints on the palette;
 * painting knife with a pointed tip, lowered or "cranked" like a trowel, suited to painting on canvas.

While palette knives are manufactured without sharpened cutting edges, with prolonged use they may become "sharpened" by the action of abrasive pigments such as earth colors.

Palette knives are also used in cooking, where their flexibility allows them to easily slide underneath pastries or other items. See frosting spatula.

license: Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/

title: Palette knife


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Libanius

Libanius (Λιβάνιος; c. 314–392 or 393) was a teacher of rhetoric of the Sophist school in the Eastern Roman Empire. His prolific writings make him one of the best documented teachers of higher education in the ancient world and a critical source of history of the Greek East during the 4th century AD. During the rise of Christian hegemony in the later Roman Empire, he remained unconverted and in religious matters was a pagan Hellene.

Life
Libanius was born in Antioch, Coele-Syria located near the modern-day city of Antakya, Turkey. He was born into a deeply cultured and once-influential family that had experienced substantial recent decline. In 303 AD, eleven years before his birth, his family had participated in resisting an insurrection by a local army garrison. In the end, Roman Imperial authorities were equally concerned by local aristocrats arming themselves as they were by the rebellious troops. Libanius' family fell out of favor and his grandfather was executed. Libanius' father died when he was eleven, leaving his upbringing to his mother and maternal uncles, who were in the process of rebuilding his family's reputation.

At fourteen years old he began his study of rhetoric, for which he withdrew from public life and devoted himself to philosophy. Unfamiliar with Latin literature, he deplored its influence.

He studied in Athens under Diophantus the Arab and began his career in Constantinople as a private tutor. He was exiled to Nicomedia in 346 (or earlier) for around five years but returned to Constantinople and taught there until 354. Before his exile, Libanius was a friend of the emperor Julian, with whom some correspondence survives, and in whose memory he wrote a series of orations; they were composed between 362 and 365. In 354 he accepted the chair of rhetoric in Antioch, his birthplace, where he stayed until his death. His pupils included both pagans and Christians.

Libanius used his arts of rhetoric to advance various private and political causes. He attacked the increasing imperial pressures on the traditional city-oriented culture that had been supported and dominated by the local upper classes. He is known to have protested against the persecution of pagans in the late Roman Empire. In 386, he appealed without success to emperor Theodosius to prevent the destruction of a temple in Edessa, and pleaded for toleration and the preservation of the temples against the predation of Christian monks, who he claimed:

"[...]hasten to attack the temples with sticks and stones and bars of iron, and in some cases, disdaining these, with hands and feet. Then utter desolation follows, with the stripping of roofs, demolition of walls, the tearing down of statues and the overthrow of altars, and the priests must either keep quiet or die. After demolishing one, they scurry to another, and to a third, and trophy is piled on trophy, in contravention of the law. Such outrages occur even in the cities, but they are most common in the countryside. Many are the foes who perpetrate the separate attacks, but after their countless crimes this scattered rabble congregates and they are in disgrace unless they have committed the foulest outrage...Temples, Sire, are the soul of the countryside: they mark the beginning of its settlement, and have been passed down through many generations to the men of today. In them the farming communities rest their hopes for husbands, wives, children, for their oxen and the soil they sow and plant. An estate that has suffered so has lost the inspiration of the peasantry together with their hopes, for they believe that their labour will be in vain once they are robbed of the gods who direct their labours to their due end. And if the land no longer enjoys the same care, neither can the yield match what it was before, and, if this be the case, the peasant is the poorer, and the revenue jeopardized."

- Libanius, Pro Templis

The surviving works of Libanius, which include over 1,600 letters, 64 speeches and 96 progymnasmata (rhetorical exercises), are valuable as a historical source for the changing world of the later 4th century. His oration "A Reply To Aristides On Behalf Of The Dancers" is one of the most important records of Roman concert dance, particularly that immensely popular form known as pantomime. His first Oration I is an autobiographical narrative, first written in 374 and revised throughout his life, a scholar's account that ends as an old exile's private journal. Progymnasma 8 (see below for explanation of a "progymnasma") is an imaginary summation of the prosecution's case against a physician charged with poisoning some of his patients.

Although Libanius was not a Christian his students included such notable Christians as John Chrysostom and Theodore of Mopsuestia. Despite his friendship with the pagan restorationist Emperor Julian he was made an honorary praetorian prefect by the Christian Emperor Theodosius I.

Works

 * 64 orations in the three fields of oratory: judicial, deliberative and epideictic, both orations as if delivered in public and orations meant to be privately read (aloud) in the study. The two volumes of selections in the Loeb Classical Library devote one volume to Libanius' orations that bear on the emperor Julian, the other on Theodosius; the most famous is his "Lamentation" about the desecration of the temples (Περὶ τῶν Ἱερῶν);
 * 51, a traditional public-speaking format of Rhetoric in Antiquity, taking set topics with historical and mythological themes (translations into English by e.g. D.A. Russell, "Libanius: Imaginary Speeches"; M. Johansson, "Libanius' Declamations 9 and 10";
 * 96 or compositional exercises for students of rhetoric, used in his courses of instruction and widely admired as models of good style;
 * 57 or introductions to Demosthenes' orations (written c. 352), in which he sets them in historical context for the novice reader, without polemics;
 * 1545 letters have been preserved, more letters than those of Cicero. Some 400 additional letters in Latin were later accepted, purporting to be translations, but a dispassionate examination of the texts themselves shows them to be misattributed or forgeries, by the Italian humanist Francesco Zambeccari in the 15th century. Among his correspondents there was Censorius Datianus.

English editions

 * Scott Bradbury, Selected Letters of Libanius. Liverpool, University Press, 2004. ISBN 0-85323-509-0
 * Raffaella Cribiore, The School of Libanius in Late Antique Antioch. Princeton: Princeton University Press, 2007. (Includes translation of c. 200 letters dealing with the school and its students. Reviewed in Bryn Mawr Classical Reviews.)
 * Margaret E. Molloy: Libanius and the Dancers, Olms-Weidmann, Hildesheim 1996 ISBN 3-487-10220-X
 * A.F. Norman, Libanius: Selected Works, 2 volumes. Cambridge, Massachusetts: Loeb Classical Library, 1969–1977.
 * A.F. Norman, Libanius: Autobiography and Selected Letters, 2 volumes. Cambridge, Massachusetts: Loeb Classical Library, 1993. Reviewed in Bryn Mawr Classical Reviews.)
 * Lieve Van Hoof, Libanius: a critical introduction (Cambridge University Press, 2014)

title: Libanius


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College Bowl

College Bowl (which has carried a naming rights sponsor, initially General Electric and later Capital One) is a radio, television, and student quiz show. College Bowl first aired on the NBC Radio Network in 1953 as College Quiz Bowl. It then moved to American television broadcast networks, airing from 1959 to 1963 on CBS and from 1963 to 1970 on NBC. In 1977, the president of College Bowl, Richard Reid, developed it into a non-televised national championship competition on campuses across America through an affiliation with the Association of College Unions International (ACUI), which lasted for 31 years. In 1989, College Bowl introduced a (sponsored) version of College Bowl for historically black colleges and universities (HBCUs) called Honda Campus All-Star Challenge (HCASC) which is ongoing. In 2007, College Bowl produced a new version and format of the game as an international championship in Africa, called Africa Challenge (Celtel Africa Challenge, Zain Africa Challenge). The College Bowl Campus Program and National Championship ran until 2008.

In November 2020, NBC announced a revival of the show, developed from the format of Honda Campus All-Star Challenge and Africa Challenge, with Peyton Manning as host and a ten-episode run ordered. The revival, Capital One College Bowl, aired from June 22, 2021 to October 28, 2022.

History
College Bowl originated as a USO activity created by Canadian Don Reid for soldiers serving in World War II. Reid and John Moses then developed the game into a radio show. Grant Tinker, later President of NBC and MTM Enterprises, got his start as an assistant on the show.

Richard Reid has led the College Bowl since 1975. He has created, produced, and supervised all versions of College Bowl innovated since then (except for a 1984 NBC special).

Two four-member teams representing various colleges and universities competed; one member of each team was its captain. The game began with a "toss-up" question for ten points. The first player to buzz in got the right to answer, but if the contestant was wrong, the other team could try to answer (if a player buzzed in before the host finished reading the question and was wrong, the team was penalized five points). Answering a "toss-up" correctly earned the team the right to answer a multi-part "bonus" question worth up to thirty points; the team members could collaborate, but only the captain was allowed to answer. The game continued in this manner and was played in halves. During halftime, the players were allowed to show a short promotional film of their school or they might talk about career plans or the like.

The first College Quiz Bowl match was played on NBC radio on October 10, 1953, when Northwestern University defeated Columbia University, 135–60. Twenty-six episodes ran in that first season, with winning teams receiving $500 grants for their school. Good Housekeeping magazine became the sponsor for the 1954–55 season, and a short third season in the autumn of 1955 finished the run. The most dominant team was the University of Minnesota, which had teams appear in 23 of the 68 broadcast matches. The 1953–55 series had a powerful appeal because it used remote broadcasts; each team was located at their college where they were cheered on by their wildly enthusiastic classmates. The effect was akin to listening to a football game.

Television
Though a pilot was shot in the spring of 1955, the game did not move to television until 1959. As G.E. College Bowl with General Electric as the primary sponsor, the show ran on CBS from 1959 to 1963, and moved back to NBC from 1963 to 1970. Allen Ludden was the original host, but left to do Password full-time in 1962. Robert Earle was the moderator for the rest of the run. The norm developed in the Ludden-Earle era of undefeated teams retiring after winning five games. Each winning team earned $1,500 in scholarship grants from General Electric with runner-up teams receiving $500. A team's fifth victory awarded $3,000 from General Electric plus $1,500 from Gimbels department stores for a grand total of $10,500. On April 16, 1967, Seventeen magazine matched GE's payouts so that each victory won $3,000 and runners-up earned $1,000. The payouts from Gimbel's department stores remained the same so that five-time champions retired with a grand total of $19,500.

Colgate University was the first team to win five consecutive contests and become "retired undefeated champions," defeating New York University in Colgate's first appearance in April 1960 when NYU was going for its fifth win. Rutgers was the second college to win five contests and be retired. Colgate later defeated Rutgers in a special one-time playoff contest to become the only six-time winner in a "five-win-limit" competition. An upset occurred in 1961, when the small liberal arts colleges of Hobart and William Smith in Geneva, New York, defeated Baylor University to become the third college to retire undefeated. Pomona College began its five-game G.E. College Bowl winning streak on October 15, 1961, by first defeating Texas Christian University followed by the University of Washington, Hood College, Amherst College, and Washington and Lee University. In another surprise, Lafayette College retired undefeated in fall 1962 after beating the University of California, Berkeley for its fifth victory, a David and Goliath event. Ohio Wesleyan University retired undefeated easily beating Bard, Marymount, UCLA, Michigan Tech, and Alfred. Another upset occurred in 1966 when the all-female Agnes Scott College from Georgia defeated an all-male team from Princeton University.

The show licensed and spun off three other academic competitions in the U.S.:
 * Alumni Fun, which appeared on ABC and CBS TV networks in the 1960s and featured former college students
 * Bible Bowl, which has evolved into at least three separate national competitions and used the Bible as a source
 * High School Bowl, which was broadcast in some local TV markets and featured high school students

Honda Campus All-Star Challenge
In 1989, College Bowl introduced its academic team championship for historically black colleges and universities (HBCUs) called Honda Campus All-Star Challenge (HCASC) sponsored by American Honda Motor Company. From 1990 to 1995, Honda Campus All-Star Challenge was broadcast on BET, featuring the top 16 HBCUs, survivors of regional tournaments, competing in a single-elimination tournament. The game was played under the same rules as College Bowl. Starting in 1996 and until the present, HCASC has been played as a live-event national championship. Originally, sixty-four HBCUs traveled to and competed at the national championship. Now, forty-eight schools travel and compete. Due to the pandemic, in 2020 the national championship was suspended and the 2021 version was a virtual event.

In 2011, HCASC adopted the Africa Challenge format of the game created by Richard Reid: the highlights of the format were three rounds of Face-Off (Toss-up) and Bonus questions played in categories followed by a catch-up round called the Ultimate Challenge.

University Challenge
A British version of the televised College Bowl competition was launched as University Challenge in 1962. The program, presented by Bamber Gascoigne, produced by Granada Television and broadcast across the ITV network, was very popular and ran until it was taken off the air in 1987. In 1994, the show was resurrected by the BBC with Jeremy Paxman (who was also hosting Newsnight at that time) as the new quizmaster. In 2022, it was announced that Amol Rajan would be taking over as host, after Paxman announced that he was stepping down, owing to Parkinson's disease. Since 2011, a Christmas-time edition has also existed, titled Christmas University Challenge, featuring university alumni.

University Challenge New Zealand
A New Zealand version of University Challenge ran from 1976 to 1989, hosted by Peter Sinclair. It had a three-year revival beginning in 2014, with Tom Conroy as host.

University Challenge Australia
University Challenge in Australia ran on ABC from 1987 until 1989, hosted by Magnus Clarke.

Africa Challenge
Launched in 2007, Africa Challenge was an international championship version of College Bowl featuring schools from across the continent that finished at the top of nationwide, non-televised championship tournaments. The format for Africa Challenge was created by Richard Reid. It featured three players playing three rounds of Face-Off and Bonus questions, and it culminated in a catch-up round called the Ultimate Challenge.

The program was sponsored by the mobile phone company Celtel, its headquarters in The Netherlands. In the first year, schools from Kenya, Tanzania, and Uganda competed. In the second year schools from Malawi and Zambia were added. In the third year, schools from Ghana, Nigeria, and Sierra Leone were added. After the second year, Celtel was sold to the mobile phone company Zain, headquartered in Bahrain. The name of the game changed to Zain Africa Challenge. Season five, which was set to be telecast in 2011, failed to make it past pre-production after Zain sold its African network operations to Bharti Airtel.

Challenging Times
An Irish version of the competition called Challenging Times ran between 1991 and 2002. It was sponsored by The Irish Times newspaper and presented by Kevin Myers, then a columnist with that newspaper. Throughout the show, University College Cork had the most wins, with three, while National University of Ireland, Galway qualified for the most finals, winning twice and placing second twice.

Later history
The game returned to radio from 1979 to 1982, hosted by Art Fleming, with the 1978 and 1979 national tournament semi-finals and finals appearing on syndicated television. The two champions from those years earned $5,000 for their school and competed against teams from the UK for a $7,500 grant in the "College Bowl World Championship," which was also televised. In 1978, Stanford defeated Yale with a score of 260–200 at the US Championships, earning the right to represent the US against a UK all-star team. The UK all-star team, composed of two students from Oxford, one student from Nottingham, and one from Durham, defeated the US team with a score of 385–55. The game played in 1978 between the US and the UK followed College Bowl rules. In 1979, Davidson College emerged as the winner of the US Championships and competed against Sidney Sussex College, Cambridge University under University Challenge rules. Once again, the UK team secured a win. There have been two television appearances since then; the 1984 tournament semi-finals and finals aired on NBC, hosted by Pat Sajak, and the entire 1987 tournament on Disney Channel, hosted by Dick Cavett. The University of Minnesota won both iterations.

In 1970, modern quiz bowl invitational tournaments began with the Southeastern Invitational Tournament, and the circuit expanded through the 1970s, 1980s, and 1990s. These tournaments increasingly made various modifications to the College Bowl format and came to be known as quiz bowl. Earlier invitational tournaments, such as the Syra-quiz at Syracuse University, had occurred in the 1950s and 1960s.

In 1976, the program became affiliated with the Association of College Unions International (ACUI), which continued to promote the competition as a non-broadcast event after the demise of the radio and television experiments. That affiliation ended in 2008, and the College Bowl campus program is no longer active. The College Bowl Company continues to create, produce and license versions of College Bowl in the United States and elsewhere, including Africa Challenge (2007–10), which featured schools from Ghana, Kenya, Malawi, Nigeria, Sierra Leone, Tanzania, Uganda and Zambia; University Challenge in New Zealand and India; University Challenge in the United Kingdom, which is seen every week in primetime on BBC 2; and the Honda Campus All-Star Challenge at historically black colleges and universities, sponsored by American Honda, which has awarded over $10,000,000 in institutional grants since its debut in 1989.

In the 1990s with the rise of the Academic Competition Federation and National Academic Quiz Tournaments, both with their national championships, several schools (such as the University of Maryland, the University of Chicago, both former national champions, and recent runner up Georgia Tech) "de-affiliated" from College Bowl. Factors that contributed to this process included, among other issues, eligibility rules for College Bowl (which limited the number of graduate students who could compete and required a minimum course load), higher participation costs for College Bowl relative to these other formats, and concerns regarding the quality and difficulty of the questions used in College Bowl competitions.

2021 revival
On November 24, 2020, it was announced that a 10-episode revival of the series had been ordered at NBC, with Peyton Manning as host (and producer) and his brother Cooper serving as sidekick. The revival, titled Capital One College Bowl, premiered on June 22, 2021. Twelve teams compete for $1 million in scholarship funds; each team fields four players, three of whom are starters. The fourth player is a backup player who replaces a team player if it is necessary to be replaced.

On April 28, 2022, NBC renewed the series for a second season, with Harry Friedman being named executive producer. The second season premiered on September 9, 2022.

Gameplay
The game is played using a modified version of the current Honda Campus All-Star Challenge rules, between two teams of three players, with no penalty for wrong answers. In each of the first two rounds, the teams are shown four categories. This round was known as the "Face-Off" in season one but was renamed the "Kickoff" in season two, with only one round being played. Each category contains one "Face-Off" question and two bonus "Follow-Up" questions, all worth 10 points each. The Face-Off questions are asked on the buzzers to all players, with no conferring allowed. The first player to buzz in and answer correctly wins control of the Follow-Ups, for which conferring is allowed. An incorrect response at any time gives the opposing team a chance to steal the points with a correct answer. In season two, a category that was used is replaced with a new one. One category is secretly designated as "Extra Credit," awarding 20 points per question if chosen. This was removed in the second season. Each of these rounds ends after three categories have been played.

A random draw decides which team will choose first in Round 1, and the trailing team at the end of this round chooses first in Round 2. During both rounds, the team that correctly answers the Face-Off question in a category earns the right to select the next one, regardless of the outcome of the Follow-Ups.

The second "Kickoff" round was replaced in season two by a new round dubbed "One-on-One," in which three categories, each with three questions, are shown and both teams determine who will participate. Correct answers are worth 20 points, while an incorrect response gives the opponent control.

The third and final round is the "Two-Minute Drill," in which each team has two minutes to answer as many questions as possible. In season one, teams chose their categories ("majors") from a group of six before the game began, and the trailing team went first. Teammates may confer on the questions, but only the captain may buzz in and answer. Each correct response scores 25 points, and a bonus is awarded after every fifth such answer, starting with 50 for the fifth and increasing by 25 for each additional set of five. The bonus was set to a flat 100 points in the second season.

Each qualifying contest consists of two complete games, with two new teams per game. The two highest-scoring teams from each contest advance to an eight-team elimination bracket, regardless of whether they won their respective games. The two highest scorers from the remaining six teams also advance as wild cards.

All matches beyond the qualifiers in the first season had three Face-Off rounds, with five categories available; each round ended when four of them have been played. The higher-seeded team in each match starts the first round. In season one, the Face-Offs were followed by a "Dropout Round," in which the host asks a question with multiple correct answers (e.g. naming the 30 teams in the NBA) and calls on one member at a time from alternating teams to respond. A miss or repetition of any previously given answer eliminates the contestant from the round. When called on, a contestant may challenge any one opposing team member to respond instead; if the opponent gives a correct answer, the challenger is eliminated. The first team to be eliminated must choose one of its members to sit out the Two-Minute Drill, while the opposing team plays the round with all three.

The "Dropout Round" was replaced in season two starting with the quarterfinals with a new round dubbed "The Handoff," in which three categories each feature a question with three correct answers. The trailing team goes first. Each correct response is worth 20 points, while an incorrect response moves on to the next teammate until all three chances have finished. Just like "The Knockout," a used category gets replaced with a new one. In subsequent turns, the contestant in the position corresponding to the question number goes first. The round ends after six questions.

In season two, a new round was added in between the quarterfinals and semifinals, called "The Blitz," consisting of the Kickoff and Two-Minute Drill. 3 matches are played, with the winner of each advancing to the semifinals, along with the runner-up team that scored the most points.

In the second season's semifinals, a new round dubbed "Pass-Play" was introduced. In this round, two categories are shown, and the controlling team chooses which one to play, with the remaining category being given to the opposing team. The trailing team goes first. Correct answers are worth 20 points, while an incorrect answer awards the opponents those points. The round consists of eight questions, with the latter half including two "Extra Credit" categories that not only double the points but also feature two answers.

In season one, members of teams eliminated in the qualifiers, quarterfinals, or semifinals each receive $5,000, $10,000, or $15,000, respectively. In season two, members of teams eliminated in the qualifiers, quarterfinals, "The Blitz," or semifinals each receive $5,000, $10,000, $15,000, or $20,000 respectively. Members of the tournament champion and runner-up teams each receive $125,000 and $25,000, respectively. Alternates receive the same monetary award as their teammates. In addition, four contestants are chosen to receive $5,000 Merit Awards based on their character and leadership ability.

Season 1
The revival's first season took place over four rounds, with a qualifier round followed by a traditional playoff bracket including quarterfinals, semifinals, and a final.

After the end of the qualifier rounds, eight of the original 12 teams advanced to the quarterfinals. Listed by team name (seed), these were: Ole Miss (1), Tennessee (2), Michigan (3), USC (4), UCLA (5), Alabama (6), Columbia (7), and Auburn (8). Morehouse, UVA, Minnesota, and XULA were eliminated.

The quarterfinal results were as follows:
 * Auburn (8) def. Ole Miss (1) 665–535
 * Columbia (7) def. Tennessee (2) 925–510
 * Alabama (6) def. Michigan (3) 860–540
 * USC (4) def. UCLA (5) 760–500

For the semifinals, the remaining four teams were re-seeded based on their combined scores from the qualifier and quarterfinal rounds. The resulting order was: Columbia (1), USC (2), Alabama (3), and Auburn (4). USC was the only team to avoid being defeated by a lower-seeded opponent in the quarterfinals.

The semifinal results were as follows:
 * Columbia (1) def. Auburn (4) 735–695
 * USC (2) def. Alabama (3) 950–600

For the final, the team with the higher cumulative score in all previous matches (USC) won the right to choose the first category in Round 1. Columbia won the championship with a score of 790–775; team members Tamarah Wallace, Shomik Ghose, and Jake Fisher received $125,000 scholarships to put towards their college education, as did alternate Addis Boyd. USC's Ann Nguyen, Karan Menon, and Brendan Glascock, along with alternate Astrid, received $25,000 scholarships as the runner-up team.

Season 2
The revival's second season took place over five rounds, with a qualifier round followed by a traditional playoff bracket including quarterfinals, "The Blitz," semifinal, and a final.

After the end of the qualifier rounds, twelve of the original 16 teams advanced to the quarterfinals. Listed by team name (seed), these were: Notre Dame (1), Oklahoma (2), Texas (3), Georgia (4), Washington (5), BYU (6), Penn State (7), Spelman (8), Syracuse (9), UC Santa Barbara (10), Columbia (11), and Duke (12). Ohio State, Florida, Albany State, and Morehouse were eliminated.

The quarterfinal results were as follows:
 * BYU (6) def. Notre Dame (1) 650–605
 * Georgia (4) def. Texas (3) 465–455
 * Penn State (7) def. Oklahoma (2) 690–650
 * Syracuse (9) def. Duke (12) 660–640
 * UC Santa Barbara (10) def. Spelman (8) 530–445
 * Columbia (11) def. Washington (5) 740–710

For "The Blitz," the remaining six teams were re-seeded based on their combined scores from the qualifier and quarterfinal rounds. The resulting order was: Penn State (1), Columbia (2), BYU (3), Georgia (4), Syracuse (5), and UC Santa Barbara (6). Penn State advanced to the semifinal, with the most points out of the runner-up teams.

The "Blitz" results were as follows:
 * BYU (3) def. Penn State (1) 725–590
 * Georgia (4) def. Syracuse (5) 565–540
 * Columbia (2) def. UC Santa Barbara (6) 755–225

For the semifinal, the remaining four teams were re-seeded based on their combined scores from the qualifier, quarterfinal, and "Blitz" rounds. The resulting order was: Columbia (1), Penn State (2), BYU (3), and Georgia (4).

The semifinal results were as follows:
 * Columbia (1) def. BYU (3) 925–590
 * Georgia (4) def. Penn State (2) 805–610

For the final, Georgia won the championship with a score of 910–855; team members Layla Parsa, Aidan Leahy, and Elijah Odunade received $125,000 scholarships to put towards their college education. Columbia's Akshay Manglik, Albert Zhang, and Forrest Weintraub received $25,000 scholarships as the runner-up team.

In popular culture

 * In 2009, brief scenes from the early 1960s episodes of College Bowl with Allen Ludden appeared in the film Gifted Hands.
 * A brief scene of GE College Bowl with Allen Ludden appeared in the 1982 film Diner.

Criticism
In the 1987 regional tournament, College Bowl was accused of recycling questions from previous tournaments, thereby possibly compromising the integrity of results. Questions for tournaments need to be new for all teams involved, or certain teams could have a competitive advantage from having heard some questions previously. The 1987 National Tournament on the Disney Channel saw additional controversy, as several protested matches proved to strain the television format. Especially in the early 1990s, The College Bowl Company attempted to collect licensing fees based on copyright and trade dress claims from invitational tournaments that employed formats that it claimed were similar to College Bowl and threatened not to allow schools that failed to pay these fees to compete in College Bowl events. As it was, the company's intellectual property claims were never tested in court. These events and the growing Internet community of quiz bowl players led to a great increase in teams, tournaments, and formats.

Top four finishers of CBI National Championship Tournament (1978–2008)

 * Source: "NCT Results: Season = 1977-1978 to 2007-2008"

No tournament was held in 1983 or 1985, though regional tournaments were held each year.

†Tied for third (lost in semifinals, no playoff for third place).

††In 1994, Brigham Young University finished second in the round-robin, qualifying for the final series. However, as the final best-two-out-of-three series was held on Sunday, the team declined to participate, and the University of Virginia took their place instead. Brigham Young was awarded third place.

title: College Bowl


text:
Cox's theorem

Cox's theorem, named after the physicist Richard Threlkeld Cox, is a derivation of the laws of probability theory from a certain set of postulates. This derivation justifies the so-called "logical" interpretation of probability, as the laws of probability derived by Cox's theorem are applicable to any proposition. Logical (also known as objective Bayesian) probability is a type of Bayesian probability. Other forms of Bayesianism, such as the subjective interpretation, are given other justifications.

Cox's assumptions
Cox wanted his system to satisfy the following conditions:

 * 1) Divisibility and comparability – The plausibility of a proposition is a real number and is dependent on information we have related to the proposition.
 * 2) Common sense – Plausibilities should vary sensibly with the assessment of plausibilities in the model.
 * 3) Consistency – If the plausibility of a proposition can be derived in many ways, all the results must be equal.

The postulates as stated here are taken from Arnborg and Sjödin. "Common sense" includes consistency with Aristotelian logic in the sense that logically equivalent propositions shall have the same plausibility.

The postulates as originally stated by Cox were not mathematically rigorous (although more so than the informal description above), as noted by Halpern. However it appears to be possible to augment them with various mathematical assumptions made either implicitly or explicitly by Cox to produce a valid proof.

Cox's notation:
 * The plausibility of a proposition $$A$$ given some related information $$X$$ is denoted by $$A\mid X$$.

Cox's postulates and functional equations are:
 * The plausibility of the conjunction $$AB$$ of two propositions $$A$$, $$B$$, given some related information $$X$$, is determined by the plausibility of $$A$$ given $$X$$ and that of $$B$$ given $$AX$$.
 * In form of a functional equation
 * $$AB\mid X=g(A\mid X,B\mid AX)$$
 * Because of the associative nature of the conjunction in propositional logic, the consistency with logic gives a functional equation saying that the function $$g$$ is an associative binary operation.

 * Additionally, Cox postulates the function $$g$$ to be monotonic.
 * All strictly increasing associative binary operations on the real numbers are isomorphic to multiplication of numbers in a subinterval of $[0, +∞]$, which means that there is a monotonic function $$w$$ mapping plausibilities to $[0, +∞]$ such that
 * $$w(AB\mid X)=w(A\mid X)w(B\mid AX)$$

 * In case $$A$$ given $$X$$ is certain, we have $$AB\mid X=B\mid X$$ and $$B\mid AX=B\mid X$$ due to the requirement of consistency. The general equation then leads to
 * $$w(B\mid X)=w(A\mid X)w(B\mid X)$$
 * This shall hold for any proposition $$B$$, which leads to
 * $$w(A\mid X)=1$$

 * In case $$A$$ given $$X$$ is impossible, we have $$AB\mid X=A\mid X$$ and $$A\mid BX=A\mid X$$ due to the requirement of consistency. The general equation (with the A and B factors switched) then leads to
 * $$w(A\mid X)=w(B\mid X)w(A\mid X)$$
 * This shall hold for any proposition $$B$$, which, without loss of generality, leads to a solution
 * $$w(A\mid X)=0$$
 * Due to the requirement of monotonicity, this means that $$w$$ maps plausibilities to interval $[0, 1]$.

 * The plausibility of a proposition determines the plausibility of the proposition's negation.
 * This postulates the existence of a function $$f$$ such that
 * $$w(\text{not } A\mid X)=f(w(A\mid X))$$
 * Because "a double negative is an affirmative", consistency with logic gives a functional equation
 * $$f(f(x))=x,$$
 * saying that the function $$f$$ is an involution, i.e., it is its own inverse.

 * Furthermore, Cox postulates the function $$f$$ to be monotonic.

 * The above functional equations and consistency with logic imply that
 * $$w(AB\mid X)=w(A\mid X)f(w(\text{not }B\mid AX))=w(A\mid X)f\left( {w(A\text{ not }B\mid X) \over w(A\mid X)} \right)$$
 * Since $$AB$$ is logically equivalent to $$BA$$, we also get
 * $$w(A\mid X)f\left( {w(A\text{ not }B\mid X) \over w(A\mid X)} \right)=w(B\mid X)f\left( {w(B\text{ not }A\mid X) \over w(B\mid X)} \right)$$

 * If, in particular, $$B=\text{ not }(AD)$$, then also $$A\text{ not } B = \text{not }B$$ and $$B\text{ not }A=\text{ not }A$$ and we get
 * $$w(A\text{ not }B\mid X)=w(\text{not }B\mid X)=f(w(B\mid X))$$
 * and
 * $$w(B\text{ not }A\mid X)=w(\text{not }A\mid X)=f(w(A\mid X))$$
 * Abbreviating $$w(A\mid X)=x$$ and $$w(B\mid X)=y$$ we get the functional equation
 * $$x\,f\left({f(y) \over x}\right)=y\,f\left({f(x) \over y}\right)$$

Implications of Cox's postulates
The laws of probability derivable from these postulates are the following. Let $$A\mid B$$ be the plausibility of the proposition $$A$$ given $$B$$ satisfying Cox's postulates. Then there is a function $$w$$ mapping plausibilities to interval [0,1] and a positive number $$m$$ such that

 * 1) Certainty is represented by $$w(A\mid B)=1.$$
 * 2) $$w^m(A|B)+w^m(\text{not }A\mid B)=1.$$
 * 3) $$w(AB\mid C)=w(A\mid C)w(B\mid AC)=w(B\mid C)w(A\mid BC).$$

It is important to note that the postulates imply only these general properties. We may recover the usual laws of probability by setting a new function, conventionally denoted $$P$$ or $$\Pr$$, equal to $$w^m$$. Then we obtain the laws of probability in a more familiar form:

 * 1) Certain truth is represented by $$\Pr(A\mid B)=1$$, and certain falsehood by $$\Pr(A\mid B)=0.$$
 * 2) $$\Pr(A\mid B)+\Pr(\text{not }A\mid B)=1.$$
 * 3) $$\Pr(AB\mid C)=\Pr(A\mid C)\Pr(B\mid AC)=\Pr(B\mid C)\Pr(A\mid BC).$$

Rule 2 is a rule for negation, and rule 3 is a rule for conjunction. Given that any proposition containing conjunction, disjunction, and negation can be equivalently rephrased using conjunction and negation alone (the conjunctive normal form), we can now handle any compound proposition.

The laws thus derived yield finite additivity of probability, but not countable additivity. The measure-theoretic formulation of Kolmogorov assumes that a probability measure is countably additive. This slightly stronger condition is necessary for certain results. An elementary example (in which this assumption merely simplifies the calculation rather than being necessary for it) is that the probability of seeing heads for the first time after an even number of flips in a sequence of coin flips is $$\tfrac13$$.

Interpretation and further discussion
Cox's theorem has come to be used as one of the justifications for the use of Bayesian probability theory. For example, in Jaynes it is discussed in detail in chapters 1 and 2 and is a cornerstone for the rest of the book. Probability is interpreted as a formal system of logic, the natural extension of Aristotelian logic (in which every statement is either true or false) into the realm of reasoning in the presence of uncertainty.

It has been debated to what degree the theorem excludes alternative models for reasoning about uncertainty. For example, if certain "unintuitive" mathematical assumptions were dropped then alternatives could be devised, e.g., an example provided by Halpern. However Arnborg and Sjödin  suggest additional "common sense" postulates, which would allow the assumptions to be relaxed in some cases while still ruling out the Halpern example. Other approaches were devised by Hardy or Dupré and Tipler.

The original formulation of Cox's theorem is in, which is extended with additional results and more discussion in. Jaynes cites Abel for the first known use of the associativity functional equation. János Aczél provides a long proof of the "associativity equation" (pages 256-267). Jaynes reproduces the shorter proof by Cox in which differentiability is assumed. A guide to Cox's theorem by Van Horn aims at comprehensively introducing the reader to all these references.

Baoding Liu, the founder of uncertainty theory, criticizes Cox's theorem for presuming that the truth value of conjunction $$P \land Q$$ is a twice differentiable function $$f$$ of truth values of the two propositions $$P$$ and $$Q$$, i.e., $$T(P \land Q) = f(T(P), T(Q))$$, which excludes uncertainty theory's "uncertain measure" from its start, because the function $$f(x, y) = x \land y$$, used in uncertainty theory, is not differentiable with respect to $$x$$ and $$y$$. According to Liu, "there does not exist any evidence that the truth value of conjunction is completely determined by the truth values of individual propositions, let alone a twice differentiable function."

title: Cox's theorem


text:
Ohm's law

Ohm's law states that the electric current through a conductor between two points is directly proportional to the voltage across the two points. Introducing the constant of proportionality, the resistance, one arrives at the three mathematical equations used to describe this relationship:

$$V = IR \quad \text{or}\quad I = \frac{V}{R} \quad \text{or}\quad R = \frac{V}{I}$$

where $I$ is the current through the conductor, V is the voltage measured across the conductor and R is the resistance of the conductor. More specifically, Ohm's law states that the R in this relation is constant, independent of the current. If the resistance is not constant, the previous equation cannot be called Ohm's law, but it can still be used as a definition of static/DC resistance. Ohm's law is an empirical relation which accurately describes the conductivity of the vast majority of electrically conductive materials over many orders of magnitude of current. However some materials do not obey Ohm's law; these are called non-ohmic.

The law was named after the German physicist Georg Ohm, who, in a treatise published in 1827, described measurements of applied voltage and current through simple electrical circuits containing various lengths of wire. Ohm explained his experimental results by a slightly more complex equation than the modern form above (see  below).

In physics, the term Ohm's law is also used to refer to various generalizations of the law; for example the vector form of the law used in electromagnetics and material science:

$$\mathbf{J} = \sigma \mathbf{E},$$

where J is the current density at a given location in a resistive material, E is the electric field at that location, and σ (sigma) is a material-dependent parameter called the conductivity, defined as the inverse of resistivity ρ (rho). This reformulation of Ohm's law is due to Gustav Kirchhoff.

History
In January 1781, before Georg Ohm's work, Henry Cavendish experimented with Leyden jars and glass tubes of varying diameter and length filled with salt solution. He measured the current by noting how strong a shock he felt as he completed the circuit with his body. Cavendish wrote that the "velocity" (current) varied directly as the "degree of electrification" (voltage). He did not communicate his results to other scientists at the time, and his results were unknown until James Clerk Maxwell published them in 1879.

Francis Ronalds delineated "intensity" (voltage) and "quantity" (current) for the dry pile—a high voltage source—in 1814 using a gold-leaf electrometer. He found for a dry pile that the relationship between the two parameters was not proportional under certain meteorological conditions.

Ohm did his work on resistance in the years 1825 and 1826, and published his results in 1827 as the book Die galvanische Kette, mathematisch bearbeitet ("The galvanic circuit investigated mathematically"). He drew considerable inspiration from Joseph Fourier's work on heat conduction in the theoretical explanation of his work. For experiments, he initially used voltaic piles, but later used a thermocouple as this provided a more stable voltage source in terms of internal resistance and constant voltage. He used a galvanometer to measure current, and knew that the voltage between the thermocouple terminals was proportional to the junction temperature. He then added test wires of varying length, diameter, and material to complete the circuit. He found that his data could be modeled through the equation $$x = \frac{a}{b + \ell},$$ where x was the reading from the galvanometer, ℓ was the length of the test conductor, a depended on the thermocouple junction temperature, and b was a constant of the entire setup. From this, Ohm determined his law of proportionality and published his results. In modern notation we would write, $$ I = \frac {\mathcal E}{r+R},$$ where $$\mathcal E$$ is the open-circuit emf of the thermocouple, $$r$$ is the internal resistance of the thermocouple and $$R$$ is the resistance of the test wire. In terms of the length of the wire this becomes, $$ I = \frac {\mathcal E}{r+\mathcal R \ell},$$ where $$\mathcal R$$ is the resistance of the test wire per unit length. Thus, Ohm's coefficients are, $$ a = \frac {\mathcal E}{\mathcal R}, \quad b = \frac {\mathcal r}{\mathcal R} .$$ Ohm's law was probably the most important of the early quantitative descriptions of the physics of electricity. We consider it almost obvious today. When Ohm first published his work, this was not the case; critics reacted to his treatment of the subject with hostility. They called his work a "web of naked fancies" and the Minister of Education proclaimed that "a professor who preached such heresies was unworthy to teach science." The prevailing scientific philosophy in Germany at the time asserted that experiments need not be performed to develop an understanding of nature because nature is so well ordered, and that scientific truths may be deduced through reasoning alone. Also, Ohm's brother Martin, a mathematician, was battling the German educational system. These factors hindered the acceptance of Ohm's work, and his work did not become widely accepted until the 1840s. However, Ohm received recognition for his contributions to science well before he died.

In the 1850s, Ohm's law was widely known and considered proved. Alternatives such as "Barlow's law", were discredited, in terms of real applications to telegraph system design, as discussed by Samuel F. B. Morse in 1855.

The electron was discovered in 1897 by J. J. Thomson, and it was quickly realized that it was the particle (charge carrier) that carried electric currents in electric circuits. In 1900, the first (classical) model of electrical conduction, the Drude model, was proposed by Paul Drude, which finally gave a scientific explanation for Ohm's law. In this model, a solid conductor consists of a stationary lattice of atoms (ions), with conduction electrons moving randomly in it. A voltage across a conductor causes an electric field, which accelerates the electrons in the direction of the electric field, causing a drift of electrons which is the electric current. However the electrons collide with atoms which causes them to scatter and randomizes their motion, thus converting kinetic energy to heat (thermal energy). Using statistical distributions, it can be shown that the average drift velocity of the electrons, and thus the current, is proportional to the electric field, and thus the voltage, over a wide range of voltages.

The development of quantum mechanics in the 1920s modified this picture somewhat, but in modern theories the average drift velocity of electrons can still be shown to be proportional to the electric field, thus deriving Ohm's law. In 1927 Arnold Sommerfeld applied the quantum Fermi-Dirac distribution of electron energies to the Drude model, resulting in the free electron model. A year later, Felix Bloch showed that electrons move in waves (Bloch electrons) through a solid crystal lattice, so scattering off the lattice atoms as postulated in the Drude model is not a major process; the electrons scatter off impurity atoms and defects in the material. The final successor, the modern quantum band theory of solids, showed that the electrons in a solid cannot take on any energy as assumed in the Drude model but are restricted to energy bands, with gaps between them of energies that electrons are forbidden to have. The size of the band gap is a characteristic of a particular substance which has a great deal to do with its electrical resistivity, explaining why some substances are electrical conductors, some semiconductors, and some insulators.

While the old term for electrical conductance, the mho (the inverse of the resistance unit ohm), is still used, a new name, the siemens, was adopted in 1971, honoring Ernst Werner von Siemens. The siemens is preferred in formal papers.

In the 1920s, it was discovered that the current through a practical resistor actually has statistical fluctuations, which depend on temperature, even when voltage and resistance are exactly constant; this fluctuation, now known as Johnson–Nyquist noise, is due to the discrete nature of charge. This thermal effect implies that measurements of current and voltage that are taken over sufficiently short periods of time will yield ratios of V/I that fluctuate from the value of R implied by the time average or ensemble average of the measured current; Ohm's law remains correct for the average current, in the case of ordinary resistive materials.

Ohm's work long preceded Maxwell's equations and any understanding of frequency-dependent effects in AC circuits. Modern developments in electromagnetic theory and circuit theory do not contradict Ohm's law when they are evaluated within the appropriate limits.

Scope
Ohm's law is an empirical law, a generalization from many experiments that have shown that current is approximately proportional to electric field for most materials. It is less fundamental than Maxwell's equations and is not always obeyed. Any given material will break down under a strong-enough electric field, and some materials of interest in electrical engineering are "non-ohmic" under weak fields.

Ohm's law has been observed on a wide range of length scales. In the early 20th century, it was thought that Ohm's law would fail at the atomic scale, but experiments have not borne out this expectation. As of 2012, researchers have demonstrated that Ohm's law works for silicon wires as small as four atoms wide and one atom high.

Microscopic origins

The dependence of the current density on the applied electric field is essentially quantum mechanical in nature; (see Classical and quantum conductivity.) A qualitative description leading to Ohm's law can be based upon classical mechanics using the Drude model developed by Paul Drude in 1900.

The Drude model treats electrons (or other charge carriers) like pinballs bouncing among the ions that make up the structure of the material. Electrons will be accelerated in the opposite direction to the electric field by the average electric field at their location. With each collision, though, the electron is deflected in a random direction with a velocity that is much larger than the velocity gained by the electric field. The net result is that electrons take a zigzag path due to the collisions, but generally drift in a direction opposing the electric field.

The drift velocity then determines the electric current density and its relationship to E and is independent of the collisions. Drude calculated the average drift velocity from p = −eEτ where p is the average momentum, −e is the charge of the electron and τ is the average time between the collisions. Since both the momentum and the current density are proportional to the drift velocity, the current density becomes proportional to the applied electric field; this leads to Ohm's law.

Hydraulic analogy
A hydraulic analogy is sometimes used to describe Ohm's law. Water pressure, measured by pascals (or PSI), is the analog of voltage because establishing a water pressure difference between two points along a (horizontal) pipe causes water to flow. The water volume flow rate, as in liters per second, is the analog of current, as in coulombs per second. Finally, flow restrictors—such as apertures placed in pipes between points where the water pressure is measured—are the analog of resistors. We say that the rate of water flow through an aperture restrictor is proportional to the difference in water pressure across the restrictor. Similarly, the rate of flow of electrical charge, that is, the electric current, through an electrical resistor is proportional to the difference in voltage measured across the resistor. More generally, the hydraulic head may be taken as the analog of voltage, and Ohm's law is then analogous to Darcy's law which relates hydraulic head to the volume flow rate via the hydraulic conductivity.

Flow and pressure variables can be calculated in fluid flow network with the use of the hydraulic ohm analogy. The method can be applied to both steady and transient flow situations. In the linear laminar flow region, Poiseuille's law describes the hydraulic resistance of a pipe, but in the turbulent flow region the pressure–flow relations become nonlinear.

The hydraulic analogy to Ohm's law has been used, for example, to approximate blood flow through the circulatory system.

Circuit analysis

In circuit analysis, three equivalent expressions of Ohm's law are used interchangeably:

$$I = \frac{V}{R} \quad \text{or}\quad V = IR \quad \text{or} \quad R = \frac{V}{I}. $$

Each equation is quoted by some sources as the defining relationship of Ohm's law, or all three are quoted, or derived from a proportional form, or even just the two that do not correspond to Ohm's original statement may sometimes be given.

The interchangeability of the equation may be represented by a triangle, where V (voltage) is placed on the top section, the I (current) is placed to the left section, and the R (resistance) is placed to the right. The divider between the top and bottom sections indicates division (hence the division bar).

Resistive circuits
Resistors are circuit elements that impede the passage of electric charge in agreement with Ohm's law, and are designed to have a specific resistance value R. In schematic diagrams, a resistor is shown as a long rectangle or zig-zag symbol. An element (resistor or conductor) that behaves according to Ohm's law over some operating range is referred to as an ohmic device (or an ohmic resistor) because Ohm's law and a single value for the resistance suffice to describe the behavior of the device over that range.

Ohm's law holds for circuits containing only resistive elements (no capacitances or inductances) for all forms of driving voltage or current, regardless of whether the driving voltage or current is constant (DC) or time-varying such as AC. At any instant of time Ohm's law is valid for such circuits.

Resistors which are in series or in parallel may be grouped together into a single "equivalent resistance" in order to apply Ohm's law in analyzing the circuit.

Reactive circuits with time-varying signals
When reactive elements such as capacitors, inductors, or transmission lines are involved in a circuit to which AC or time-varying voltage or current is applied, the relationship between voltage and current becomes the solution to a differential equation, so Ohm's law (as defined above) does not directly apply since that form contains only resistances having value R, not complex impedances which may contain capacitance (C) or inductance (L).

Equations for time-invariant AC circuits take the same form as Ohm's law. However, the variables are generalized to complex numbers and the current and voltage waveforms are complex exponentials.

In this approach, a voltage or current waveform takes the form Ae$st$, where t is time, s is a complex parameter, and A is a complex scalar. In any linear time-invariant system, all of the currents and voltages can be expressed with the same s parameter as the input to the system, allowing the time-varying complex exponential term to be canceled out and the system described algebraically in terms of the complex scalars in the current and voltage waveforms.

The complex generalization of resistance is impedance, usually denoted Z; it can be shown that for an inductor, $$Z = sL$$ and for a capacitor, $$Z = \frac{1}{sC}.$$

We can now write, $$V = Z\,I$$ where V and I are the complex scalars in the voltage and current respectively and Z is the complex impedance.

This form of Ohm's law, with Z taking the place of R, generalizes the simpler form. When Z is complex, only the real part is responsible for dissipating heat.

In a general AC circuit, Z varies strongly with the frequency parameter s, and so also will the relationship between voltage and current.

For the common case of a steady sinusoid, the s parameter is taken to be $$j\omega$$, corresponding to a complex sinusoid $$Ae^{\mbox{ } j \omega t}$$. The real parts of such complex current and voltage waveforms describe the actual sinusoidal currents and voltages in a circuit, which can be in different phases due to the different complex scalars.

Linear approximations
Ohm's law is one of the basic equations used in the analysis of electrical circuits. It applies to both metal conductors and circuit components (resistors) specifically made for this behaviour. Both are ubiquitous in electrical engineering. Materials and components that obey Ohm's law are described as "ohmic" which means they produce the same value for resistance (R = V/I) regardless of the value of V or I which is applied and whether the applied voltage or current is DC (direct current) of either positive or negative polarity or AC (alternating current).

In a true ohmic device, the same value of resistance will be calculated from R = V/I regardless of the value of the applied voltage V. That is, the ratio of V/I is constant, and when current is plotted as a function of voltage the curve is linear (a straight line). If voltage is forced to some value V, then that voltage V divided by measured current I will equal R. Or if the current is forced to some value I, then the measured voltage V divided by that current I is also R. Since the plot of I versus V is a straight line, then it is also true that for any set of two different voltages V1 and V2 applied across a given device of resistance R, producing currents I1 = V1/R and I2 = V2/R, that the ratio (V1 − V2)/(I1 − I2) is also a constant equal to R. The operator "delta" (Δ) is used to represent a difference in a quantity, so we can write ΔV = V1 − V2 and ΔI = I1 − I2. Summarizing, for any truly ohmic device having resistance R, V/I = ΔV/ΔI = R for any applied voltage or current or for the difference between any set of applied voltages or currents.

There are, however, components of electrical circuits which do not obey Ohm's law; that is, their relationship between current and voltage (their I–V curve) is nonlinear (or non-ohmic). An example is the p–n junction diode (curve at right). As seen in the figure, the current does not increase linearly with applied voltage for a diode. One can determine a value of current (I) for a given value of applied voltage (V) from the curve, but not from Ohm's law, since the value of "resistance" is not constant as a function of applied voltage. Further, the current only increases significantly if the applied voltage is positive, not negative. The ratio V/I for some point along the nonlinear curve is sometimes called the static, or chordal, or DC, resistance, but as seen in the figure the value of total $V$ over total $I$ varies depending on the particular point along the nonlinear curve which is chosen. This means the "DC resistance" V/I at some point on the curve is not the same as what would be determined by applying an AC signal having peak amplitude $ΔV$ volts or $ΔI$ amps centered at that same point along the curve and measuring $ΔV/ΔI$. However, in some diode applications, the AC signal applied to the device is small and it is possible to analyze the circuit in terms of the dynamic, small-signal, or incremental resistance, defined as the one over the slope of the V–I curve at the average value (DC operating point) of the voltage (that is, one over the derivative of current with respect to voltage). For sufficiently small signals, the dynamic resistance allows the Ohm's law small signal resistance to be calculated as approximately one over the slope of a line drawn tangentially to the V–I curve at the DC operating point.

Temperature effects
Ohm's law has sometimes been stated as, "for a conductor in a given state, the electromotive force is proportional to the current produced. "That is, that the resistance, the ratio of the applied electromotive force (or voltage) to the current, "does not vary with the current strength."The qualifier "in a given state" is usually interpreted as meaning "at a constant temperature," since the resistivity of materials is usually temperature dependent. Because the conduction of current is related to Joule heating of the conducting body, according to Joule's first law, the temperature of a conducting body may change when it carries a current. The dependence of resistance on temperature therefore makes resistance depend upon the current in a typical experimental setup, making the law in this form difficult to directly verify. Maxwell and others worked out several methods to test the law experimentally in 1876, controlling for heating effects. Usually, the measurements of a sample resistance are carried out at low currents to prevent Joule heating. However, even a small current causes heating(cooling) at the first(second) sample contact due to the Peltier effect. The temperatures at the sample contacts become different, their difference is linear in current. The voltage drop across the circuit includes additionally the Seebeck thermoelectromotive force which again is again linear in current. As a result, there exists a thermal correction to the sample resistance even at negligibly small current. The magnitude of the correction could be comparable with the sample resistance.

Relation to heat conductions
Ohm's principle predicts the flow of electrical charge (i.e. current) in electrical conductors when subjected to the influence of voltage differences; Jean-Baptiste-Joseph Fourier's principle predicts the flow of heat in heat conductors when subjected to the influence of temperature differences.

The same equation describes both phenomena, the equation's variables taking on different meanings in the two cases. Specifically, solving a heat conduction (Fourier) problem with temperature (the driving "force") and flux of heat (the rate of flow of the driven "quantity", i.e. heat energy) variables also solves an analogous electrical conduction (Ohm) problem having electric potential (the driving "force") and electric current (the rate of flow of the driven "quantity", i.e. charge) variables.

The basis of Fourier's work was his clear conception and definition of thermal conductivity. He assumed that, all else being the same, the flux of heat is strictly proportional to the gradient of temperature. Although undoubtedly true for small temperature gradients, strictly proportional behavior will be lost when real materials (e.g. ones having a thermal conductivity that is a function of temperature) are subjected to large temperature gradients.

A similar assumption is made in the statement of Ohm's law: other things being alike, the strength of the current at each point is proportional to the gradient of electric potential. The accuracy of the assumption that flow is proportional to the gradient is more readily tested, using modern measurement methods, for the electrical case than for the heat case.

Other versions
Ohm's law, in the form above, is an extremely useful equation in the field of electrical/electronic engineering because it describes how voltage, current and resistance are interrelated on a "macroscopic" level, that is, commonly, as circuit elements in an electrical circuit. Physicists who study the electrical properties of matter at the microscopic level use a closely related and more general vector equation, sometimes also referred to as Ohm's law, having variables that are closely related to the V, I, and R scalar variables of Ohm's law, but which are each functions of position within the conductor. Physicists often use this continuum form of Ohm's Law:

$$ \mathbf{E} = \rho \mathbf{J} $$

where $E$ is the electric field vector with units of volts per meter (analogous to $V$ of Ohm's law which has units of volts), $J$ is the current density vector with units of amperes per unit area (analogous to $I$ of Ohm's law which has units of amperes), and ρ "rho" is the resistivity with units of ohm·meters (analogous to $R$ of Ohm's law which has units of ohms). The above equation is also written as $J = σE$ where $σ$ "sigma" is the conductivity which is the reciprocal of $ρ$.

The voltage between two points is defined as: $${\Delta V} = -\int {\mathbf E \cdot d \boldsymbol \ell} $$ with $$d \boldsymbol \ell$$ the element of path along the integration of electric field vector E. If the applied E field is uniform and oriented along the length of the conductor as shown in the figure, then defining the voltage V in the usual convention of being opposite in direction to the field (see figure), and with the understanding that the voltage V is measured differentially across the length of the conductor allowing us to drop the Δ symbol, the above vector equation reduces to the scalar equation:

$$V = {E}{\ell} \ \  \text{or} \ \ E = \frac{V}{\ell}. $$

Since the $E$ field is uniform in the direction of wire length, for a conductor having uniformly consistent resistivity ρ, the current density $J$ will also be uniform in any cross-sectional area and oriented in the direction of wire length, so we may write: $$ J = \frac{I}{a}.$$

Substituting the above 2 results (for E and J respectively) into the continuum form shown at the beginning of this section: $$\frac{V}{\ell} = \frac{I}{a}\rho \qquad \text{or} \qquad V = I \rho \frac{\ell}{a}.$$

The electrical resistance of a uniform conductor is given in terms of resistivity by: $${R} = \rho \frac{\ell}{a} $$ where ℓ is the length of the conductor in SI units of meters, $a$ is the cross-sectional area (for a round wire $a = πr^{2}$ if $r$ is radius) in units of meters squared, and ρ is the resistivity in units of ohm·meters.

After substitution of R from the above equation into the equation preceding it, the continuum form of Ohm's law for a uniform field (and uniform current density) oriented along the length of the conductor reduces to the more familiar form: $$V= I R. $$

A perfect crystal lattice, with low enough thermal motion and no deviations from periodic structure, would have no resistivity, but a real metal has crystallographic defects, impurities, multiple isotopes, and thermal motion of the atoms. Electrons scatter from all of these, resulting in resistance to their flow.

The more complex generalized forms of Ohm's law are important to condensed matter physics, which studies the properties of matter and, in particular, its electronic structure. In broad terms, they fall under the topic of constitutive equations and the theory of transport coefficients.

Magnetic effects
If an external B-field is present and the conductor is not at rest but moving at velocity $v$, then an extra term must be added to account for the current induced by the Lorentz force on the charge carriers. $$\mathbf{J} = \sigma (\mathbf{E} + \mathbf{v}\times\mathbf{B})$$

In the rest frame of the moving conductor this term drops out because $v = 0$. There is no contradiction because the electric field in the rest frame differs from the E-field in the lab frame: $E′ = E + v × B$. Electric and magnetic fields are relative, see Lorentz transformation.

If the current $J$ is alternating because the applied voltage or E-field varies in time, then reactance must be added to resistance to account for self-inductance, see electrical impedance. The reactance may be strong if the frequency is high or the conductor is coiled.

Conductive fluids
In a conductive fluid, such as a plasma, there is a similar effect. Consider a fluid moving with the velocity $$\mathbf{v}$$ in a magnetic field $$\mathbf{B}$$. The relative motion induces an electric field $$\mathbf{E}$$ which exerts electric force on the charged particles giving rise to an electric current $$\mathbf{J}$$. The equation of motion for the electron gas, with a number density $$n_e$$, is written as $$ m_e n_e {d\mathbf{v}_e\over dt} = -n_e e \mathbf{E} + n_e m_e \nu (\mathbf{v}_i - \mathbf{v}_e) - e n_e \mathbf{v}_e\times \mathbf{B}, $$

where $$e$$, $$m_e$$ and $$\mathbf{v}_e$$ are the charge, mass and velocity of the electrons, respectively. Also, $$\nu$$ is the frequency of collisions of the electrons with ions which have a velocity field $$\mathbf{v}_i$$. Since, the electron has a very small mass compared with that of ions, we can ignore the left hand side of the above equation to write $$ \sigma(\mathbf{E} + \mathbf{v} \times \mathbf{B}) = \mathbf{J}, $$

where we have used the definition of the current density, and also put $$\sigma = {n_e e^2\over \nu m_e}$$ which is the electrical conductivity. This equation can also be equivalently written as $$ \mathbf{E}+\mathbf{v}\times \mathbf{B}=\rho\mathbf{J}, $$ where $$\rho = \sigma^{-1}$$ is the electrical resistivity. It is also common to write $$\eta$$ instead of $$\rho$$ which can be confusing since it is the same notation used for the magnetic diffusivity defined as $$\eta = 1 / \mu_0\sigma$$.

title: Ohm's law