User:Chetvorno/work8

= For Electric power =

An alternate way to derive this formula is to note that voltage is defined as the amount of work that a unit charge (one coulomb) does when it moves between the two terminals, $$\scriptstyle V\,=\,dW/dQ$$, and the current is defined as the number of coulombs flowing per second, $$\scriptstyle I\,=\,dQ/dt$$, so


 * $$P =\quad {dW \over dt} \qquad\qquad (\text{rate of work done per unit time}) \;$$


 * $$ = \quad {dW \over dQ} \times {dQ \over dt} \quad\; (\text{work done per unit charge}\,\times\,\text{charge flowing per unit time})$$


 * $$ = \quad\; V \; \times \; I \qquad\;  (\text{voltage}\,\times\,\text{current}) $$

= For World Wireless System =

Overview
Serbian-American inventor Nikola Tesla performed the first experiments in wireless power transmission around the turn of the 20th century, and was its most tireless proponent. In dramatic lectures before crowds at scientific meetings in the 1890s he demonstrated lights and motors powered wirelessly by electricity transmitted for short distances by capacitive and inductive coupling.

Beginning as early as 1891, Tesla conceived a long range wireless electric power distribution system, in which electric power could be transmitted through the air from power plants directly to homes and factories, without wires, and used to run vehicles, even aircraft. In its most developed form, which he called the "World Wireless System", he envisioned a global network of power stations that would transmit both electric power and information to any point on Earth. He predicted telecommunication capabilities for this system which sound remarkably like modern technology; he said in 1909:
 * "It will soon be possible, for instance, for a business man in New York to dictate instructions and have them appear instantly in type in London or elsewhere. He will be able to call up from his desk and talk with any telephone subscriber in the world. It will only be necessary to carry an inexpensive instrument not bigger than a watch, which will enable its bearer to hear anywhere on sea or land for distances of thousands of miles. One may listen or transmit speech or song to the uttermost parts of the world."

In addition to these wireless communication capabilities, Tesla claimed these stations would be able to transmit electric power through the air to users for hundreds of miles around. They would make power lines obsolete. He even suggested that by transmitting hydropower economically from remote sources such as Victoria Falls in Africa to industrial users in developed countries, the world might be able to get all its energy from natural sources such as water power, making steam power plants unnecessary.

In 1901 with the backing of Wall Street banker J. P. Morgan, Tesla began construction of a radiotelegraphy transmitting station, now called the Wardenclyffe tower, in Shoreham, New York on Long Island, to achieve transatlantic radio communication. However he went beyond his contract with Morgan and over budget to make the device a prototype wireless power transmitter. It consisted of a powerhouse with a 187 foot tower topped by a dome-shaped metal capacitive "antenna". By 1904 Tesla had lost funding and the facility was never completed; it was torn down in 1916. In the 100 years since then, although his futuristic predictions of a global wireless telecommunication network have become reality through somewhat different technology, no one has been able to transmit electric power through the air as Tesla claimed to be able to do. For the rest of his life until his death in 1943, Tesla maintained that his system would have worked, and wireless power was just around the corner.

Background
In order to understand Tesla's ideas, it is necessary to see them in the context of the technological knowledge of the time.

At the time Tesla developed his wireless power ideas, 1891, electric power had been used in industry for 20 years and was fueling the Industrial Revolution, and the first electric power grids were being built to distribute power to people in cities. Circuit theory, the laws of electricity flowing through wires and other conductors, was becoming well understood. Direct current (DC) and low frequency alternating current (AC) were widely used.

However, the science of "wireless" electricity, electromagnetic waves, was in a much earlier state of development. In 1865 James Clerk Maxwell had proposed a complicated mathematical theory that light was "electromagnetic waves". In 1887, only 4 years previously, Heinrich Hertz had discovered a new form of electromagnetic waves, "Hertzian waves" (radio waves), which confirmed Maxwell's theory. At the time of Tesla's research, Maxwell's equations, which are now accepted as the foundation of electromagnetic theory, were only understood by a few physicists. Due to Maxwell, physicists thought of radio waves as analogous to light rays. Like light they obeyed the inverse-square law; their intensity decreased as the square of the distance from the transmitter. It was believed they only traveled in straight lines, and therefore could not be used for long distance transmission beyond line-of-sight, certainly not over the horizon.

The kind of "wireless power" which Tesla and other engineers were most familiar with is the kind that occurs in a transformer or electric motor - electromagnetic induction. An alternating current applied to a coil of wire will create a magnetic field, which can cross space and induce a current in a second coil. This was the basis of the alternating current electric power technology and the induction motor which Tesla invented, and a number of researchers had attempted to use it to create wireless communication and power transmission systems.

Tesla's grandiose ambitions to build a worldwide system were in line with the spirit of the Gilded Age, a period of big infrastructure projects. Tesla had examples of other huge networks before him; the first transcontinental railroads, telegraph networks, telephone networks, and submarine telephone cables, which were all constructed during his lifetime.

(Sources)
Tesla describing his wireless power transmitter in Electrical Experimenter magazine: "it is a resonant transformer which... is accurately proportioned to fit the globe and its electrical constants and properties, by virtue of which design it becomes highly efficient and effective at wireless transmission of energy. Distance is then absolutely eliminated, there being no diminution of the transmitted impulses. It is even possible to make the actions increase with the distance from the plant according to an exact mathematical law."  (Cheney "Man out of time", p. 175)

Tesla in "Experiments with Alternate Currents of Very High Frequency and Their Application to Methods of Artificial Illumination" "I adhere to the idea that there is a thing which we have been in the habit of calling electricity. The question is, What is that thing? Or, What, of all things the existence of which we know, have we the best reason to call electricity? We know that it acts like an incompressible fluid; that there must be a constant quantity of it in nature; that it can be neither produced nor destroyed; ...” The last words are the basis on which Tesla developed his hypothesis about possibility to transmit currents through the earth (Marinčić)

Thomas H. White (2012) Nikola Tesla - the guy who DIDN'T "invent radio" has a lot of great quotes by Tesla and others, plus a table comparing the Tesla and Hertzian view of wireless.

Development of Tesla's theories
Tesla was an expert in the new field of alternating current (AC) electric power engineering; he had studied at Austrian Polytechnic 1875-1878, had worked for Edison (1882-1885), had invented the AC induction motor (1888) and three-phase power transmission, and consulted for Westinghouse Corporation designing an early AC power system, the Niagara Falls hydroelectric plant (1888-1897). Wealthy from royalties on his alternating current inventions paid by Westinghouse, Tesla's research interest turned to high frequency current. In 1886 Heinrich Hertz had discovered radio waves, and Tesla set up a lab in New York City to investigate these high frequencies, initially duplicating Hertz's experiments.

Tesla was secretive about the details of his wireless research; he accused several people, particularly Marconi, of stealing his ideas. He worked alone, with a few assistants whom he swore to secrecy, and did not publish details of his research in scientific journals. However he discussed the general outlines of his scheme in public lectures, articles in popular magazines, and in many interviews in the press over the years, as well as his patents, and these descriptions are generally consistent.

First wireless power experiments
Unlike most wireless inventors whose efforts focused on radio communication (called wireless telegraphy), from the beginning Tesla was interested in a more radical idea: wireless power transmission. His work on electric power grids at Westinghouse gave him experience in how expensive it was to build power lines to transport energy from where it was produced to the cities where it was used. If electric power could be transmitted through the air or ground from the power plant directly to the consumer without wires, it would revolutionize electrical technology.

In 1891 attempting to develop a wireless lighting system he invented the Tesla coil circuit, consisting of a spark-excited resonant transformer, with its secondary winding connected to the ground at one end, with the other end connected to an elevated capacitive electrode. He  He observed that the oscillating Tesla coil, working against the elevated capacitive terminal, drove pulses of current into the ground, and that sparks could be drawn from grounded metal objects like water pipes at some distance from the coil.

Ground currents vs radio waves
During the same period as Tesla was researching wireless power, 1894-1901, radio pioneer Guglielmo Marconi developed the first practical radio communication systems. The first primitive radio transmitters, such as those built by Heinrich Hertz and Oliver Lodge around 1887, used short ungrounded dipole antennas and generated high frequencies in the VHF and UHF bands, which had very short range of less than a mile. In 1895 Marconi had discovered that the range of his transmitters could be increased by grounding one terminal of the transmitter and receiver and connecting the other to a large elevated vertical wire antenna. This had the effect of reducing the frequency of the waves. In a series of experiments from 1895 to 1901 Marconi by reducing the frequency of his transmitters increased the range of transmission, culminating in his historic transatlantic radio transmission on 12 December 1901 from Poldhu, Cornwall to St. John's, Newfoundland, a distance of 2200 miles. At this time scientists believed that radio waves traveled in straight lines, so it was not understood how Marconi's waves had managed to travel around the curve of the Earth between Britain and Canada.

Although he performed some of the early experiments with radio, Tesla rejected the "Hertzian wave" (radio wave) theory of radio transmission, and instead attributed long distance radio communication to oscillating currents conducted through the Earth. He clung to the common view that, like light rays, radio waves only traveled in straight lines, so they could not reach points on Earth beyond the horizon. Also since their intensity decreased as the inverse square of distance,(Carlson) they were a "weak" effect, which would never be useful for wireless power transmission. Instead he focused on the oscillating ground currents which a grounded generator produced. In Tesla's view, during the late 1890s Marconi's wireless transmitters were becoming similar to Tesla's grounded Tesla coil oscillator, which also used low frequencies and an elevated capacitive electrode. In spite of Tesla's 1891 patent, Marconi used the Tesla coil circuit in his long distance transmitters, and claimed rights to it in his 26 April 1900 "four circuit" patent   This led Tesla to accuse Marconi of infringing his patents. Tesla believed that the long distance communication of "Hertzian" transmitters was not due to radio waves at all, but to ground currents. "The plain fact is that the Hertz waves emitted by the aerial are just as much of a loss of power as the... heat due to frictional waste [resistance] in the wire." In Tesla's opinion, the first transmitters, which were ungrounded, emitted their energy solely as radio waves, which was why they had limited range. As longer vertical grounded antennas and lower frequency tuned circuits were used, later wireless transmitters radiated most of their energy as "ground currents" and little as radio waves; "...the Hertzian effect has been gradually reduced through the lowering of frequency so as to be negligible when the usual frequencies are employed"   According to Tesla, these "ground currents" and not the radio waves accounted for over-the-horizon communication, such as Marconi's 1901 transatlantic transmission.

Obvious objections to this theory was how Earth currents could account for radio transmission to aircraft, and the observation that wireless signals were affected by atmospheric conditions; for example, they could be received at much longer distances at night than during the day. To explain these facts he proposed a theory of charged particles evaporating from the surface of the Earth during the day, reducing the ground currents, which was never scientifically accepted. 

In the 1920s it was discovered that radio waves can travel beyond the horizon, around the curve of the Earth, by two methods: either as ground waves which due to refraction follow the curving terrain, or as skywaves which reflect from layers of charged particles in the ionosphere. Marconi's transatlantic transmission was carried by radio waves, not earth currents; the longer radio waves which Marconi's later transmitters radiated were simply more effective at these modes of transmission.

Winans, Wireless Power, NY Tribune Sunday Mag, 3/3/1912 Interview with Tesla with a lot of quotes of advantages and uses of wireless power

Narodny, Marconi's plans for the world, Technical World Mag, Oct 1912 Marconi also claimed that wireless power transmission was possible by radio waves. His system sounds like Teslas, with oscillating "magnifying transmitter" of 15-20 million volts.

Marconi lights a lamp 6 miles away, Electrical Experimenter, April 1914 Claim not believed by many.

"Shaking the Earth"


Tesla came to believe that the entire Earth could act as an electrical resonator, and that its potential could oscillate at one or more natural resonant frequencies. If driven by pulses of current at one of these resonant frequencies from a grounded electrical oscillator such as his Tesla coil, he thought, standing waves of voltage would be generated in the Earth. A receiver consisting of a resonant circuit connected between the Earth and an elevated capacitive electrode, if tuned to resonance with the Earth oscillations, might receive power at any point over the surface of the Earth.

Tesla reached this idea in two stages. In his March 1893 lecture On light and other high frequency phenomena, Tesla demonstrated how a single wire terminated by a capacitive plate can conduct alternating current power, powering a light bulb in series with it without a return path for the current, which is required in lower frequency circuits. The capacitive plate functions as a charge reservoir, allowing alternating current to pass along the wire. Tesla claimed the ground can serve the same function, conducting power between a driving grounded oscillator working against a capacitive plate, and a receiver consisting of a grounded tuned circuit and a second capacitive plate. He claimed this technology could be exploited now to power a city by ground current, possibly using its water mains as conductors. He speculated that the Earth itself might also serve as the resonator, but that it would take further research to determine its capacitance and resonant frequency.


 * "...(using) the Earth itself as the medium for conducting the currents, thus dispensing with wires and all other artificial conductors ... a machine which, to explain its operation in plain language, resembled a pump in its action, drawing electricity from the Earth and driving it back into the same at an enormous rate, thus creating ripples or disturbances which, spreading through the Earth as through a wire, could be detected at great distances by carefully attuned receiving circuits. In this manner I was able to transmit to a distance, not only feeble effects for the purposes of signaling, but considerable amounts of energy, and later discoveries I made convinced me that I shall ultimately succeed in conveying power without wires, for industrial purposes, with high economy, and to any distance, however great." Nikola Tesla, "Talking with planets", Collier's Weekly, 9 February 1901

Colorado Springs experiments
To research power transmission, in 1899 Tesla moved to Colorado Springs, Colorado and built a high voltage laboratory. At this laboratory he constructed one of the largest Tesla coils ever made, which he called a "magnifying transmitter" because it was intended to transmit power.

Evidence from lightning
On July 3, 1899 in Colorado Springs Tesla made an observation that he always afterwards claimed confirmed his theory. While using his coherer radio receiver to record pulses of radio static from lightning strikes due to a thunderstorm moving for hundreds of miles across the prairie, he observed that the intensity of the pulses varied periodically as the storm moved away from him, repeatedly getting stronger and weaker. He interpreted these as nodes and antinodes of electrical standing waves (stationary waves) excited in the Earth by the discharges. To Tesla this was proof that the Earth itself could act and in fact did act as an electrical resonator.


 * It was on the third of July--the date I shall never forget--when I obtained the first decisive experimental evidence of a truth of overwhelming importance for the advancement of humanity. A dense mass of strongly charged clouds gathered in the west and towards the evening a violent storm broke loose which, after spending its fury in the mountains, was driven away with great velocity over the plains. [...] The recording apparatus being properly adjusted, its indications became fainter and fainter with the increasing distance of the storm until they ceased altogether. I was watching in eager expectation. Surely enough, in a little while the indications again began, grew stronger and stronger and, after passing thru a maximum, gradually decreased and ceased once more. Many times, in regularly recurring intervals, the same actions were repeated until the storm, which, as evident from simple computations, was moving with nearly constant speed, had retreated to a distance of about three hundred kilometers...  No doubt whatever remained: I was observing stationary waves.

Tesla compared the Earth to the metal balls he used as capacitive resonators on his spark oscillators. He thought that, like a metal ball, the Earth had negligible resistance to these "terrestrial" waves; in modern terms it was a high Q resonator. When excited by a grounded Tesla transformer, they would pass through the entire Earth and reflect from the antipode point on the globe opposite the transmitter, bouncing back and forth to create "terrestrial standing waves". There would be little power loss
 * A popular error which I often have the opportunity to correct is the belief that the energy of such a plant would dissipate itself in all directions. This is not so. Electricity is displaced by the transmitter in all directions through the earth and air, but energy is expended only at the place where it is collected and used to perform some work.  A plant of 10,000 horsepower might be running full blast at Niagara, and one flying machine of 50 horsepower might be in another place.  Only 50 horsepower would be furnished by the plant.  Although the electrical oscillations would manifest themselves all over the earth at the surface as well as high in the air, virtually no power would be consumed.  My experiments have shown that the entire electrical movement which keeps the whole globe a-tremble can be maintained with but a few horsepower.   Arthur Reeve "Tesla and his Wireless Age", Popular Electricity, June 1911, p. 100

Tesla believed he had figured out the law of propagation of these waves: "the projections of the wavelengths (measured along the surface) on the Earth's diameter or axis of symmetry are all equal". In other words the wave front traveled along the diameter of the Earth at the speed of light, but the apparent speed of the waves along the Earth's surface would be greater than the speed of light, and would vary. Near the transmitter and the antipode point opposite it the velocity would approach infinity, while at the halfway point they would decrease to the speed of light.

Atmospheric conduction
"''The earth is 4,000 miles radius. Around this conducting earth is an atmosphere.  The earth is a conductor; the atmosphere above is a conductor, only there is a little stratum between the conducting atmosphere and the conducting earth which is insulating. . . . Now, you realize right away that if you set up differences of potential at one point, say, you will create in the media corresponding fluctuations of potential.  But, since the distance from the earth's surface to the conducting atmosphere is minute, as compared with the distance of the receiver at 4,000 miles, say, you can readily see that the energy cannot travel along this curve and get there, but will be immediately transformed into conduction currents, and these currents will travel like currents over a wire with a return.  The energy will be recovered in the circuit, not by a beam that passes along this curve and is reflected and absorbed,. . . but it will travel by conduction and will be recovered in this way.''" Tesla, quoted in Leland Anderson, Nikola Tesla On His Work With Alternating Currents and Their Application to Wireless Telegraphy, Telephony, and Transmission of Power

By using a transmitter producing hundreds of millions of volts, the current could be transformed down, reducing resistance losses in the long ionized path through the atmosphere: ". . . by such means as have been described practically any potential that is desired may be obtained, the currents through the air strata may be rendered very small, whereby the loss in the transmission may be reduced." SYSTEM OF TRANSMISSION OF ELECTRICAL ENERGY, Sept. 2, 1897, U.S. Patent No. 645,576, Mar. 20, 1900

Modern scientific views
The ocnsensus of modern electrical engineers is that Tesla's wireless transmission system as he designed it would not have worked as a long distance transmitter of either information or electric power. It is now understood that radio waves do not necessarily travel in straight lines as Tesla thought. Wavelengths used by Tesla and Marconi can travel beyond the horizon by two methods: ground waves which travel just above the Earth's surface (not in the Earth) following the contour of the terrain, and skywaves, which reflect from layers of charged particles in the ionosphere and can return to Earth beyond the horizon. These, not the ground currents proposed by Tesla, are responsible for long distance radio communication. As a radiotelegraph transmitter, Tesla's Wardenclyffe station operated in the right frequency range (~150 kHz) and would have had sufficient power to communicate across the Atlantic. However to transmit radio waves of these frequencies efficiently requires large wire antennas, like Marconi used. The opinion of modern wireless sources is that Wardenclyffe tower's 61 foot diameter capacitive terminal was too small and would not have radiated enough radio power to be received at transatlantic distances.

Tesla's wireless power system is even more improbable. Tesla's Earth transmission ideas have never been confirmed or demonstrated. Although Tesla claimed his wireless power ideas were proven, he had a history of making claims that he had not confirmed by experiment, and there seems to be no evidence that he ever transmitted significant power beyond the short-range demonstrations mentioned above. The few reports of long-distance power transmission by Tesla are not from reliable sources. For example, a widely repeated myth is that in 1899 he wirelessly lit 200 light bulbs at a distance of 26 miles. There is no independent confirmation of this supposed demonstration; Tesla did not mention it, and it does not appear in his laboratory notes. It originated in 1944, 40 years after the fact, from Tesla's first biographer, John J. O'Neill, who said he pieced it together from "fragmentary material... in a number of publications".

Modern researchers such as Robert Golka  have built large tesla coils similar to Tesla's Colorado Springs equipment, but long-distance power transmission has not been demonstrated,   and the scientific consensus is his World Wireless system would not have worked. Contemporary scientists point out that while Tesla's coils (with appropriate antennas) can function as radio transmitters, transmitting energy in the form of radio waves, the frequency he used, around 150 kHz, is far too low for practical long-range power transmission. At these wavelengths the radio waves spread out in all directions and cannot be focused on a distant receiver. Physics professor Dennis Papadopoulos, interviewed for the 2001 PBS television biography on Tesla, said
 * "His major defect was that he was dreaming but he was doing very few calculations on paper. Because on paper he could have realized that because the dimensions of the wave guide, are so enormous, you can transmit power, but not very much power. You can transfer power to hear the radio, or for television, or for a telephone. But once you want to start turning on lights in which you really need high currents, the power gets diluted because the space is very large. It's a standard defect of dreamers, geniuses, not like Einstein, the other type of geniuses, the inventors, who visualize things, but have difficulty putting numbers [on paper]. And actually, I think that was his downfall."

Tesla's world power transmission scheme remains today what it was in Tesla's time: a bold, fascinating dream.

Legacy
Tesla abandoned research on wireless power transmission after the failure of Wardenclyffe, but until his death in 1938 continued to claim his ideas were correct and wireless power was just around the corner.

Tesla was rediscovered in the 1970s and is a popular cult figure today. Due to his mystique, many fan websites and New Age books promote pseudoscientific conspiracy theories that Tesla did achieve long distance wireless power transmission but that the technology was either misunderstood and ignored, or suppressed. One common myth is that corporate interests deliberately suppressed the technology because there would be no way to charge for electricity transmitted through the air. According to this myth, Tesla's backer, the banker J. P. Morgan, is supposed to have said "Where would we put the electric meter?" There is no evidence for any of this. Morgan's contract with Tesla was for a transatlantic radio station. After Marconi beat Tesla to transatlantic radio communication, and Tesla violated his contract by going over budget, Morgan declined to continue funding. There is no evidence he cared about Tesla's unproven wireless power ideas. [https://books.google.com/books?id=ZNqo1zaZRTYC&pg=PA147&dq=tesla+wireless+power#v=onepage&q=tesla%20wireless%20power&f=false  Kurt Van Voorhies 1991 World Wireless Power Prospects, Proc. of the IECEC]

History
Although crude electromechanical amplifier devices were developed in the late 1800s, practical amplifiers were made possible by the invention of the first amplifying vacuum tube, the Audion (triode) in 1906 by Lee De Forest. The achievement of amplification was a key accomplishment in electrical technology; it created the new field of electronics and made possible radio and television broadcasting, long distance telephone service, public address systems, talking motion pictures, the recording industry, radar, and eventually computers and the internet. Vacuum tubes dominated electronics until they were replaced by the transistor in the 1960s and 70s. The transistor is the most widely used amplifying component today, although vacuum tubes are still used in higher power applications.

The first term used for this new device was "electron relay", because the only previous device which had an analogous signal-strengthening action was a relay, which was used as a repeater in long telegraph lines to rejuvenate the signal. The terms amplification and amplifier, (from the Latin amplificare, 'to enlarge or expand' ) began to be used for this new effect when triode tubes came into wide use after 1915.

Pre-vacuum tube amplifiers
The development of electronic voice (audio) communication technologies; telephone and intercom systems around 1880 and amplitude modulated (AM) radio around 1900, created a need to increase the amplitude of an audio signal, to make the sound louder. Telephone companies played a large role in developing the first amplifiers. The unamplified telephone circuits used up to that time, consisting of a microphone, earphone and battery connected by long wires, were adequate for calls between neighboring cities, but the length of a telephone line was limited to several hundred miles because of power lost in the resistance of the long wires. Telephone companies' goal was to develop a "repeater", a device analogous to the relay used in telegraph circuits that could be inserted in a telephone line when the signal got weak, to amplify the signal to its original strength. Corporate laboratories belonging to the telephone companies attacked the problem systematically around 1900.

The first crude amplifiers were electromechanical devices based on the carbon microphone, which had been used in telephone systems since the 1870s. It consisted of a "cell" with electrodes on either face, containing loose carbon granules. When sound waves vibrated a diaphragm attached to one electrode it caused the resistance of the carbon granules to vary. A constant voltage from a battery was applied to the microphone, and the varying resistance caused the current through the microphone to vary. The audio signal produced was proportional to the DC current through the microphone.

Around the late 1800s primitive electromechanical amplifiers were built by coupling an earphone driver to a carbon microphone. The incoming audio signal to be amplified was applied to an electromagnet that vibrated a steel diaphragm attached to the carbon microphone, and a battery passed enough current through the microphone that the output audio signal was larger than the input. Since the carbon microphone didn't generate its own current but modulated the current from an external source (the battery), it could produce more audio power than the sound waves striking it, and thus amplify.

These devices were very unsatisfactory amplifiers. The mass of the acoustic driver system gave them a sharp resonance peak that emphasized some frequencies, distorting voices. They were insensitive to weak signals, and could not be used in telephone lines with loading coils. They also had DC level offset problems because the carbon's resistance varied with temperature. As the carbon got warmer due to power dissipation by the current, its resistance decreased, increasing the DC current through it; this thermal runaway process could cause the receiver to "saturate". In addition the carbon microphone produced electronic noise, called the "carbon hiss", which sounded like a roaring in the background.

One of the most widely used of these devices was the Shreeve repeater, developed in 1903 by Herbert Shreeve at Western Electric, the manufacturing arm of AT&T, and used in the first long distance telephone lines until about 1920. To reduce resonance peaks Shreeve omitted the diaphragm and used a light piston to apply pressure to the carbon granules. A bimetallic strip compensated for temperature-caused offsets by applying a pressure to the piston that decreased with temperature. However only one repeater could be used in a line before the sound quality became unacceptable, so they were unsuitable for long transcontinental lines.

Another early attempt was the Arnold mercury arc tube, designed by Harold D. Arnold at Western Electric. In this device an arc between a pool of mercury in the bottom was split between two electrodes at top. The incoming signal to be amplified was applied to a pair of deflection coils on either side of the tube, creating a varying magnetic field across the arc. As in a cathode ray tube (CRT), this deflected the beam from one electrode to the other, creating a varying current in the electrode. The tube functioned adequately as an amplifier, but it was so finicky to adjust that it was installed in only one telephone line.

Vacuum tubes
The first widely used amplifying device was the triode vacuum tube invented in 1906 by Lee De Forest. In 1905-6 De Forest, trying to create a more sensitive detector for use in early radio receivers, devised vacuum tubes he called Audions by adding a third electrode to the Fleming valve detector, the first vacuum tube, invented by John Ambrose Fleming in 1904. He tried placing the third electrode in many positions in the tube. He found that placing a zigzag wire, which he called the grid, between the heated filament and the plate electrode in the tube increased the sensitivity greatly. This device, which he patented _____, was the first triode vacuum tube.

In operation, a battery connected between the filament and plate caused electrons emitted by the hot filament to be attracted to the plate, creating a current through the tube. The signal to be amplified was applied to the grid. The grid, located between the filament and plate, acted as a "gate" to control the current of electrons. When the voltage on the grid was positive, it allowed more electrons to flow through to the plate. When the grid was negative, it repelled the electrons, so fewer got through to the plate. Since a small voltage on the grid could control a large current from filament to plate, the tube could amplify.

The Audion was little used until its amplifying ability was recognized around 1912. Fritz Lowenstein probably built the first Audion amplifier in 1911 by replacing De Forest's grid capacitor with a bias battery, making a class A amplifier. Exactly how the Audion amplifier worked was not understood until Edwin Armstrong clarified it in a 1914 paper. The first Audions were primitive amplifiers due to residual air left in the tube by De Forest, who believed gas ionization was essential to it's operation. This caused nonlinear amplification and erratic operation. De Forest sold the rights to the Audion in 1913 to the Western Electric Co., whose head, Harold Arnold, thought that it could be developed into a practical telephone line repeater. Arnold, as well as Irving Langmuir at General Electric laboratories, realized that the residual air in the tube which caused the problems was not necessary, and that the Audion could operate on electron conduction alone. Both laboratories developed methods of better evacuation, and by 1914 produced the first "hard vacuum" triode tubes. Triode repeaters from Western Electric made long distance telephone lines possible, allowing AT&T to open the first transcontinental telephone line from New York to Los Angeles 2 years later, on January 25, 1915.

Beginning of electronics
The discovery of the triode's amplifying ability revolutionized electrical technology, creating the new field of electronics, the technology of active (amplifying) electrical devices. The triode was immediately applied to many areas of communication. Vacuum tube "continuous wave" radio transmitters replaced the cumbersome inefficient "damped wave" spark gap transmitters, allowing the transmission of sound by amplitude modulation (AM). Amplifying radio receivers, which had the power to drive loudspeakers, replaced weak crystal radios, which had to be listened to with earphones, allowing families to listen together. This resulted in the beginning of radio broadcasting around 1920, the first mass communication medium. The triode served as the technological base from which later vacuum tubes developed, such as the tetrode (Walter Schottky, 1916) and pentode (Bernardus Tellegen, 1926). Other inventions made possible by vacuum tube amplification were television, public address systems, electric phonographs, home audio systems, talking motion pictures, radar, and the first computers.

The mathematical theory of amplifiers was developed in the 1930s largely at Bell Telephone Laboratories which evolved from Arnold's lab at AT&T. The first amplifiers had large amounts of distortion until Bell Labs engineer Harold Stephen Black applied negative feedback to amplifiers in 1927, which allowed control of amplifiers' gain, distortion, and other characteristics. Analytical techniques for feedback amplifiers developed at Bell Labs include the Nyquist stability criterion (1932) by Harry Nyquist, Bode plots by Hendrik Wade Bode, and the root locus method (1948) by Walter R. Evans.

Amplidyne
The Amplidyne was an electromechanical amplifier invented by Ernst Alexanderson in World War 2; a motor-generator which acted as a power amplifier. It was used as a high power amplifier in servo applications where vacuum tubes were inadequate; in elevators, steel mills, and naval gun mounts during World War 2. In the Amplidyne, power from an electric motor turned a generator, generating electric current. The signal to be amplified was applied to the generator's field winding, thus the voltage generated by the armature winding was proportional to the input current. Due to the iron core its frequency response and linearity was of course too poor for audio applications. The Amplidyne is now obsolete, replaced by power semiconductor devices.

Transistors
Vacuum tubes were bulky, fragile, expensive, and due to their filament they had a limited life, consumed a lot of energy and produced a great deal of waste heat, and required a heavy transformer power supply for the plate voltage. They were largely replaced as amplifiers in the 1960s and 1970s by the transistor, another product of telephone company research, invented in 1947 by  John Bardeen, Walter Brattain, and William Shockley at Bell Labs. The transistor created another revolution in electronic technology, making possible the first truly portable electronic devices: transistor radios, walkie-talkies, boom boxes, CD players, cell phones. Today vacuum tubes are only used in a few high-power applications for which semiconductor devices are unsuitable, such as radio transmitters and industrial heating equipment. The development of integrated circuits (ICs) in the 1970s allowed an entire amplifier to be placed on a semiconductor chip, and saw the evolution of versatile general purpose negative feedback IC amplifiers called op-amps.

Theory
(under construction)

The first mathematical analyses of the Tesla coil circuit were done by Anton Oberbeck (1895) and Paul Drude (1904). Drude showed that maximum voltage was produced with a coupling coefficient $$k$$ = 0.6 and a unity tuning ratio $$f_{sec}/f_{pri}\,$$ (resonance between primary and secondary). Phung et al (1991) generalized this result, showing that maximum voltage occurred at a number of different coupling coefficients, the difference being that the smaller the coupling coefficient the longer the time interval for "ring up" (the more cycles of the radio frequency) before the maximum is reached. Finkelstein et al (1966) used a different metric to optimize the circuit, determining the conditions for a complete transfer of the energy in the primary to the secondary.

Lumped-element analysis
Traditional analyses of the Tesla coil use the ordinary lumped element circuit model, in which the components are modeled by discrete circuit elements. In the simplest circuit model the Tesla circuit has four energy storage elements: $$L1, C1, L2, C2\;$$ with $$C2\;$$ representing the capacitance of the secondary winding and the capacitive top-load (the supply transformer has very large impedance at radio frequency and so is not included). So an exact analysis requires a fourth order differential equation, (or using Laplace transforms a fourth order algebraic equation) which is extremely laborious to solve in closed form. Therefore most paper analyses use the simplifying assumption of no resistance ($$R1\;=\;R2\;=\;0$$) which results in a 2nd order equation. This is justified since practical coils are designed with low resistance and have high Q_factor, and at high Qs the results desired, the peak output voltage and frequency, depend little on the resistance. Computer Tesla coil simulation programs use numerical methods to solve the full fourth order equation, although the AC resistances of the primary and secondary coils at radio frequency, $$R1$$ and $$R2$$, are difficult to estimate without expensive lab equipment.

Streamer length
In Tesla coils used for entertainment the air discharges from the HV terminal are the desired output, and the design goal is usually to produce the longest streamer arcs. The length of air gap that a DC spark can jump, given by Paschen's law, is proportional to the voltage, so traditionally Tesla coils have been designed to produce the highest voltage. However, experience has shown that, due to the pulsed output waveform, the longest arcs from Tesla coils are not necessarily achieved by simply maximizing the output voltage at the expense of current. The long streamers produced by Tesla coils are thought to build up over several output pulses, with each pulse of voltage extending the ionized channel produced by previous pulses. The amount of ionization is dependent on the current as well as the voltage. Therefore it has been observed that the length of streamers is more closely related to the total power than the voltage.

The air discharges from the top terminal can be regarded as the "load" on the circuit. Until the air breakdown voltage is reached the resistance from the output terminal to ground is very high. Once arcs break out from the terminal they reduce the resistance to a lower value, limiting the voltage an open-air Tesla coil can produce. Higher voltages can be produced by enclosing the coil in a pressurized tank of insulating gas such as sulfur hexafluoride or liquid such as transformer oil.

The secondary as a resonant transmission line
The secondary winding of the Tesla coil is long enough that it is a significant fraction of a wavelength which is about 300 - 3000 meters at the radio frequencies at which the coil operates. Therefore it does not behave as a lumped element inductor, as transformer windings at lower frequencies do, but must be viewed using the distributed element model. Tesla was the first to point out that the secondary acts as an open-ended quarter-wave transmission line; it resonates at the frequency at which the length of the secondary winding (when uncoiled) is one quarter of a wavelength. Waves of RF current pass up the coil and are reflected from the top terminal, and the reflected waves interfere with the direct waves. Thus the current and voltage along the coil is not uniform as assumed by the lumped element model, but forms a standing wave. At the fundamental resonant frequency the current has the profile of one quarter of a sine wave; it has a maximum (a "loop" or antinode) at the bottom, decreasing to zero (a node) at the top terminal (this applies to an unloaded coil; with a capacitive top load the current doesn't go to zero at the top). Similarly the voltage has a node at the bottom and an antinode at the top of the coil. Although the speed of the current and voltage waves along the wire is close to the speed of light, because the wire is wound into a coil (helix) the speed of the waves when measured in an axial direction, along the coil axis, is much slower, so the coil is called a "slow-wave" resonator.

Thus the "lumped-element" equations above normally used to design Tesla coils, which treat the secondary as a simple inductor in series with a capacitor, are inaccurate. How much error is introduced depends on the geometry of the secondary coil and top load. A short, wide coil with a large capacitive load may be accurately modeled by lumped elements, but as the coil is made longer and thinner and the top torus is made smaller the results depart further from the lumped model. It has long been recognized that in Tesla coils, as in RF circuits generally, the lumped element equations are just a starting point, and the design must be refined by trial and error to achieve the best performance. In recent years computer design programs for Tesla coils have begun to incorporate distributed models which treat the secondary as a transmission line and calculate output voltage from the standing wave ratio, with the goal of more accurate designs.

Overtone modes
One of the consequences of the secondary acting as a transmission line is that it has more than one resonant frequency. Tesla coils normally operate at their lowest resonant frequency or fundamental mode, at which the secondary is a quarter wavelength long, as described above. However the secondary can also resonate at a series of discrete frequencies above the fundamental frequency, called overtones in which there are more standing waves along the coil. Bipolar coils have antinodes at both ends, so they have only even numbered overtones. Unipolar coils have an antinode at the top and a node at the bottom grounded end, so they have only odd overtones. For example, the next higher mode of a unipolar (grounded) secondary is the third overtone, in which the standing wave along the coil is 3/4 wavelengths long. In addition to its maximum (antinode) at the top terminal of the coil, the voltage has a minimum (node) 2/3 of the way up the coil, and a second maximum (antinode) 1/3 of the way up the coil. The presence of voltage maxima on the coil can cause arcs to break out from the windings which can damage the thin wire, so overtone modes are undesirable.

Another circumstance in which the voltage maximum (antinode) occurs on the winding and not at the top of the coil is when the coil is out of resonance, so the operating frequency is higher than the secondary's resonant frequency. This also can cause the commonly-seen problem of discharges and arcs breaking out from the winding. It is important when adjusting the resonant frequency of the primary that it be tuned to the fundamental frequency of the secondary and not to an overtone.

Right-hand rule
This is a relation between three directions in space. The directions are usually represented by vectors, depicted as arrows. The right hand is held as shown (right) with thumb, index, and middle fingers perpendicular. If $$(\mathbf{a},\;\mathbf{b},\;\mathbf{c})$$ are three noncoplanar vectors
 * The index finger is pointed in the direction of $$ \mathbf{a}$$
 * The middle finger is pointed in the direction of $$\mathbf{b}$$
 * If the thumb points in the direction of $$\mathbf{c}$$ then the three vectors obey the right-hand rule; the ordered triple $$(\mathbf{a},\;\mathbf{b},\;\mathbf{c})$$ is called a right-hand system.  If the thumb points opposite to the direction of  $$\mathbf{c}$$ then the three vectors are called a left-hand system.

The order of the three vectors matters. If the order of the first two vectors $$\mathbf{a}$$ and $$\mathbf{b}$$ is switched, the thumb will point in the opposite direction, so if $$(\mathbf{a},\;\mathbf{b},\;\mathbf{c})$$ is a right-hand system, then $$(\mathbf{b},\;\mathbf{a},\;\mathbf{c})$$ is a left-hand system;  $$(\mathbf{b},\;\mathbf{a},\;\mathbf{-c})$$ is a right-hand system.

The right-hand rule appears in problems where the third vector $$\mathbf{c}$$ can have two possible directions; the right-hand rule specifies which direction is correct. The first two vectors define a plane. The right-hand rule tells which side of the plane the third vector is on.

Right-hand grip rule
This is a relation between a direction of circulation around a loop, and the direction of a vector perpendicular to the loop. One example would be a rotation axis and the direction of rotation.
 * The right hand is curled around the loop with the fingers pointing in the direction of circulation, the direction that the arrows point.
 * If the thumb points in the direction of the vector, it obeys the right-hand rule. The loop and vector constitute a right hand system.  If the vector passes through the loop in the other direction, opposite to the direction the thumb points, it is a left-hand system.

Relation between the two rules
The right-hand rule and right-hand grip rule are different; they apply to different situations. The right-hand rule gives the relation between the direction of a vector and two other vectors; the right-hand grip rule gives the relation between the direction of a vector and the direction of circulation around a loop. However they are closely related. Since they are so similar and are both defined by the right hand, many authors combine the two rules, applying one hand gesture to cover both situations, so only one rule need be memorized. The "gripping" hand gesture in the right-hand grip rule can also be used to define the right hand relation between three vectors, thus covering both rules. Given a triple of vectors $$(\mathbf{a},\;\mathbf{b},\;\mathbf{c})$$
 * The right hand is oriented so when the hand is open the fingers point to $$\mathbf{a}$$ but when the fingers are curled toward the palm they rotate toward $$\mathbf{b}$$.
 * If the thumb points in the direction of $$\mathbf{c}$$, then $$(\mathbf{a},\;\mathbf{b},\;\mathbf{c})$$ obeys the right-hand rule.

Cross product
A major reason for the appearance of the right-hand rule in physics and mathematics is that it defines the direction of the vector cross product. The cross product $$\mathbf{a} \times \mathbf{b}$$ of two vectors $$\mathbf{a}$$ and $$\mathbf{b}$$ is defined to be a vector that is perpendicular to the plane passing through both vectors and has a magnitude equal to $$|\mathbf{a}||\mathbf{b}|\sin \theta$$, where $$\theta$$ is the angle between the vectors. But there are two vectors, pointing in opposite directions, which are perpendicular to the plane; which is it? This is decided by the right-hand rule Thus the vector triple $$(\mathbf{a},\;\mathbf{b},\; \mathbf{a}\times\mathbf{b})$$ obeys the right-hand rule.
 * The index finger of the right hand is pointed in the direction of $$\mathbf{a}$$.
 * The middle finger is pointed in the direction of $$\mathbf{b}$$.
 * Then the thumb points to the side of the plane that the vector $$\mathbf{a} \times \mathbf{b}$$ is on.

Due to the right-hand rule, the order of the two arguments in the cross product matters. If $$\mathbf{a}$$ and $$\mathbf{b}$$ are switched, the thumb will point in the opposite direction, so the cross product is anticommutative
 * $$\mathbf{a} \times \mathbf{b} = -(\mathbf{b} \times \mathbf{a})\;$$

Relation between the two rules
The right-hand rule and the right-hand grip rule are interrelated; together they constitute a single sign convention appearing throughout physics and mathematics. The choice of right hand over the left hand was an arbitrary one made by physicists in the late 19th century. If one of these two rules was changed to the "left-hand" version, the physics and math formulas using them would no longer work, but if both of them were changed there would be no consequence; physics and math would still work, but the definition of quantities which involve the cross product (called pseudovectors), for example the magnetic field $$\mathbf{B}$$ and angular momentum $$\mathbf{L}$$, would have a minus sign.

The right-hand grip rule's appearance in electromagnetics formulas such as Ampere's and Faraday's laws, as well as fluid mechanics, comes from the Stokes theorem, in which the direction of integration around a loop is related by the right-hand grip rule to the direction of a surface integral over the surface bounding the loop. This in turn is related to the right-hand rule through the cross product.

The use of the right-hand grip rule in mechanics to define the direction of quantities such as the angular velocity $$\mathbf{\Omega}$$ and angular momentum  $$\mathbf{L}$$ vectors is related to the right-hand rule through the cross product. If the $$\mathbf{\Gamma}$$ and $$\mathbf{n}$$ are an oriented circle and an axis that are related by the right-hand grip rule, and $$\mathbf{r}$$ is a radius vector from the axis to the circle and $$\mathbf{s}$$ points in the direction of rotation, then $$(\mathbf{r},\;\mathbf{s},\;\mathbf{n})$$ obey the right-hand rule.

History
The historical origin of the right-hand rule lies in the predominance of righthandedness, that is the preference for using the right hand, over lefthandedness, in the human population. Screw fasteners were developed in the 15th century, which could be made in either handedness. Right-hand screws, that is screws with threads which obey the right-hand grip rule, became the default handedness for screw fasteners. The reason for this is thought to be that for a right-handed person, tightening a right-hand screw is easier than tightening a left-hand screw, because it uses the stronger supinator muscle of the arm rather than the weaker pronator muscle. The right-hand grip rule was invented to distinguish right- from left-hand screws.

Immanuel Kant was one of the first to suggest that the distinction between right and left might be a characteristic of space itself. In 1768 he wrote a paper in which he argued that our ability to distinguish between objects which are mirror images of each other, such as right- and left-handed objects, was an argument for absolute space

John Ambrose Fleming invented the right hand rule

Pseudovectors and parity
The choice of the right hand over the left hand was an arbitrary choice or sign convention of physicists. If the right-hand rule (including right-hand grip rule) were switched for the left-hand rule in all of physics and mathematics, there would be no consequence; all of the formulas that use the rules would still be valid, except that pseudovectors like magnetic field $$\mathbf{B}$$ and angular momentum $$\mathbf{L}$$ would have the opposite sign.

Applications
These are some of the more important of the many applications of these rules:

Screw handedness
Screws come in two different types; the screw threads around the shaft can twist in two possible directions. This is called a screw's handedness. The handedness of a screw is determined by the right-hand grip rule
 * The right hand is wrapped around the screw shaft so the fingers point in the direction it is turning.
 * If the thumb points in the direction the screw shaft is moving, it is a right-handed screw. If the screw shaft moves in the opposite direction to the thumb, it is a left-handed screw.

Ampere's circuital law
Ampere's circuital law says that an electric current $$\mathbf{I}$$ in a wire generates a magnetic field $$\mathbf{B}$$ circling the wire. The magnetic field lines could circle the wire in two possible directions. The direction of the magnetic field is given by the right hand grip rule:
 * The right hand is curled around the wire so that the thumb points in the direction of the electric current $$\mathbf{I}$$ (conventional current, flow of positive charge)
 * Then the fingers will point in the direction that the magnetic field $$\mathbf{B}$$ circles the wire.

One of the sources is the Biot-Savart law, an equation which gives the magnetic field due to the current $$\mathbf{dI}$$ through a small section of the circuit:
 * $$B = {\mu_0 \over 4\pi} \int {\mathbf{dI} \times \mathbf{r} \over r^3} \,$$

Here $$\mathbf{r}$$ is the radial vector from $$\mathbf{dI}$$ to the point where $$\mathbf{B}$$ is measured. Due to the cross product, the vectors $$(\mathbf{I},\;\mathbf{r},\;\mathbf{B})$$ obey the right-hand rule.

Faraday's law of induction (Fleming's right hand rule)
When a wire attached to a circuit is moved through a magnetic field $$\mathbf{B}$$ in a direction $$\mathbf{v}$$, the field induces a current $$\mathbf{I}$$ in the wire due to Faraday's law of induction. The current could be in two possible directions through the wire. The direction of the current is given by the right-hand rule. This application is called Fleming's right hand rule because it was invented by John Ambrose Fleming
 * The index finger of the right hand is pointed in the direction of motion $$\mathbf{v}$$ of the wire
 * The middle finger is pointed in the direction of the magnetic field $$\mathbf{B}$$
 * Then the thumb will point in the direction of the current $$\mathbf{I}$$ through the wire.

The reason for the rule is that the mobile charge carriers (electrons) in the wire move with the wire in the direction $$\mathbf{v}$$ and so the magnetic field exerts a sideways force on them by the Lorentz force
 * $$\mathbf{F} = q\mathbf{v} \times \mathbf{B}\,$$

The component of force $$\mathbf{F}$$ along the wire causes the charges to move along the wire, inducing the current $$\mathbf{I}$$. Since the equation above has the cross product in it, the three vectors $$(\mathbf{v},\;\mathbf{B},\;\mathbf{I})$$ are related by the right-hand rule.

Lorentz force law
The Lorentz force law says that a magnetic field $$\mathbf{B}$$ will exert a force on a charged particle moving through it with a velocity $$\mathbf{v}$$, as long as the field is not parallel to $$\mathbf{v}$$. The force is perpendicular to both the particle's path and the magnetic field, but this leaves two possible directions. The direction of the force is given by the right hand rule. For positively charged particles For a negatively charged particle, the direction is opposite. The reason for the rule is that the Lorentz force is defined by the cross product:
 * The index finger of the right hand is pointed in the direction of the particle velocity $$\mathbf{v}$$
 * The middle finger is pointed in the direction of the magnetic field $$\mathbf{B}$$
 * Then the thumb will point in the direction of the force on the particle $$\mathbf{F}$$.
 * $$\mathbf{F} = q\mathbf{v} \times \mathbf{B}\,$$

thus the three vectors $$(\mathbf{v},\;\mathbf{B},\;\mathbf{F})$$ form a right hand system

Coriolis force
The Coriolis force $$\boldsymbol{F_C}$$ is a fictitious force that appears to act on moving objects in a rotating coordinate system, due to being in a noninertial reference frame. The Coriolis force depends on the velocity of the object $$\mathbf{v}$$ in the rotating coordinate system, and the angular velocity vector $$\boldsymbol{\Omega}$$ which is defined to be a vector parallel to the rotation axis equal in magnitude to the rotation rate, with its direction defined by the right-hand grip rule. The Coriolis force vector is
 * $$\boldsymbol{F_C} = 2m\boldsymbol{v \times \Omega}$$

Since it is defined by the cross product Thus on a surface rotating counterclockwise, for example, $$\boldsymbol{\Omega}$$ is directed up, and the Coriolis force is always to the right as one faces in the direction of motion $$\mathbf{v}$$.
 * When the index finger of the right hand points in the $$\mathbf{v}$$ direction
 * And the middle finger points in the $$\boldsymbol{\Omega}$$ direction
 * Then the thumb will point in the direction of the Coriolis force $$\boldsymbol{F_C}$$.

A specific example is application to Earth's atmosphere to explain why cyclones rotate counter-clockwise in the Northern Hemisphere. The Earth rotates from West to East. Curling the right hand around the Earth's axis so the fingers point in the direction of the Earth's rotation, the thumb points north. Therefore applying the right-hand grip rule the direction of the Earth's angular velocity vector $$\boldsymbol{\Omega}$$  points out of the the North Pole. In the Northern Hemisphere a parcel of air moving north has a velocity $$\mathbf{v}$$ toward the axis. Applying the right-hand rule, the direction of the Coriolis force $$\mathbf{F_C}$$ is toward the east; to the right when facing in the direction of motion. Similarly, applying the right hand rule to a parcel of air moving south, $$\mathbf{F_C}$$ is directed to the west, also to the right. Thus the Coriolis force on a moving parcel of air in the Northern Hemisphere is always toward the right side, when facing in the direction of motion. In a cyclone (low pressure system) the Coriolis force is what maintains the pressure difference between the center (eye) of the storm and the outside, so the force $$\boldsymbol{F_C}$$ must be directed everywhere outward from the center of the cyclone. Since $$\boldsymbol{F_C}$$ is directed to the right, the center of the cyclone must always be located to the left when facing in the direction wind is blowing. This means the wind circulates in a counterclockwise direction.

Right-handed and left-handed coordinate systems
A three-dimensional coordinate system can be either right-handed or left-handed, depending on whether the coordinate axes $$\scriptstyle (\mathbf{x},\;\mathbf{y},\;\mathbf{z})$$ obey the right-hand rule
 * The index finger of the right hand is pointed along the positive $$\scriptstyle \boldsymbol{x}$$ axis.
 * The middle finger is pointed along the positive $$\scriptstyle \boldsymbol{y}$$ axis.
 * If the thumb points in the direction of the positive $$\scriptstyle \boldsymbol{z}$$ axis, then it is a right-handed coordinate system. If it points in the opposite $$\scriptstyle \boldsymbol{-z}$$ direction, it is a left-handed coordinate system.

Permutations: Alternate finger assignments
A source of confusion is that sometimes different authors may apply the right-hand rule to the same physical problem with the vectors assigned to the fingers in a different sequence. Because of symmetry, in applying the right-hand rule the vectors $$\scriptstyle (\mathbf{a},\;\mathbf{b},\; \mathbf{c})$$  can be assigned to the fingers (index finger, middle finger, thumb) in three different ways, and the result will still obey the right hand rule as long as the order is kept the same. In other words the three cyclic permutations of the vectors $$\scriptstyle (\mathbf{a},\;\mathbf{b},\; \mathbf{c})$$, $$\scriptstyle (\mathbf{b},\;\mathbf{c},\; \mathbf{a})$$, and  $$\scriptstyle (\mathbf{c},\;\mathbf{a},\; \mathbf{b})$$ have the same handedness; they are all right-hand systems, if one of them is. Thus, the right-hand rule for these vectors could be expressed in these different ways; they are all equivalent However, just switching two of the vectors will not give a right-hand system, so this will be false
 * When the index finger points in the direction $$\scriptstyle \mathbf{a}$$ and the middle finger points in the direction $$\scriptstyle \mathbf{b}$$, then the thumb will point in the direction $$\scriptstyle \mathbf{c}$$.
 * When the index finger points in the direction $$\scriptstyle \mathbf{b}$$ and the middle finger points in the direction $$\scriptstyle \mathbf{c}$$, then the thumb will point in the direction $$\scriptstyle \mathbf{a}$$.
 * When the index finger points in the direction $$\scriptstyle \mathbf{c}$$ and the middle finger points in the direction $$\scriptstyle \mathbf{a}$$, then the thumb will point in the direction $$\scriptstyle \mathbf{b}$$.
 * When the index finger points in the direction $$\scriptstyle \mathbf{b}$$ and the middle finger points in the direction $$\scriptstyle \mathbf{a}$$, then the thumb will point in the direction $$\scriptstyle \mathbf{c}$$.

For example, Fleming's right-hand rule (above) for the current induced in a wire by a magnetic field could be given in three different forms
 * If the thumb points in the direction of motion $$\scriptstyle \mathbf{v}$$ of the conductor and the index finger points in the direction of the magnetic field $$\scriptstyle \mathbf{B}$$, then the middle finger will point in the direction of the current $$\scriptstyle \mathbf{I}$$
 * If the index finger points in the direction of motion $$\scriptstyle \mathbf{v}$$ of the conductor and the middle finger points in the direction of the magnetic field $$\scriptstyle \mathbf{B}$$, then the thumb will point in the direction of the current $$\scriptstyle \mathbf{I}$$
 * If the middle finger points in the direction of motion $$\scriptstyle \mathbf{v}$$ of the conductor and the thumb points in the direction of the magnetic field $$\scriptstyle \mathbf{B}$$, then the index finger will point in the direction of the current $$\scriptstyle \mathbf{I}$$

For some applications of the right-hand rule there is a traditional assignment of vectors to fingers (sometimes accompanied by a mnemonic to aid memorization), that is followed by most sources.