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

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

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

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

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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, and used to run vehicles, without wires. 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 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 1901 with the backing of Wall Street bankers, Tesla began construction of a wireless transmitting station, now called the Wardenclyffe tower, at Shoreham, New York, which was to have been a prototype for this system. It consisted of a powerhouse with a 187 foot tower topped by a dome shaped copper "antenna". However by 1904 Tesla had lost funding and the plant was never completed. In the 110 years since that time, no one has been able to transmit power as Tesla claimed to be able to do. For the rest of his life, 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.

Guglielmo Marconi, Tesla's rival, was responsible for the practical development of radio communication. In the period 1895 - 1904, roughly contemporaneously with Tesla's wireless power research, he performed a series of experiments transmitting radio waves longer and longer distances, culminating in his December 12, 1901 transatlantic radio transmission, showing that radio waves could propagate around the curve of the Earth. Although we now know that this was due to the radio waves reflecting off the ionosphere, this was not known at the time. Some researchers such as Oliver Lodge speculated that radio waves passed through the Earth, or that the effect was not due to radio waves but some other kind of electromagnetic disturbance.

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 systems, telephone systems, and submarine telephone cables, which were all constructed during his lifetime.

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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)

Tesla's research

Tesla's background was in electrical engineering; he had studied at Austrian Polytechnic 1875-1878 , had worked for Edison (1882-1885), invented the AC induction motor (1888), and consulted for Westinghouse Corporation designing an early AC power system, the Niagara Falls hydroelectric plant (1888-1897). Tesla knew firsthand 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 without wires from the power plant directly to the consumer, it would revolutionize electric technology

Tesla's theories

Tesla was secretive about the details of his wireless research; which was understandable as he had accused several people, particularly Marconi, of stealing his ideas. He worked alone, with a few assistants whom he swore to secrecy. However he discussed the general outlines of his scheme in public lectures, articles in magazines, and in many interviews in the press over the years, and

Tesla was adamant that his wireless system did not work by "Hertzian waves" (radio waves).

Earth resonance

Fluid mechanical analogy Tesla used to explain his wireless power transmission concept. The charge in the Earth (bottom) acts like a fluid. His magnifying transmitter (left) and resonant receiver (right) act like tuning forks which are coupled through pistons to the fluid. Vibrations from the transmitter are communicated through the fluid to the receiver and cause resonant vibrations in its tuning fork.

Fluid analogy, from 1919 article in electronics experimenter magazine. The Earth is visualized as a ball filled with incompressible fluid, it's electric charge. Tesla's transmitter acts as a pump, while his receivers are compared to balloons attached to nozzles. Each stroke of the pump causes a pulse of pressure which is transmitted to the balloons.

Tesla's central idea was 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, standing waves of voltage would be generated over the surface of the 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 without a return path. The capacitive plate functions as a charge reservoir, allowing alternating current to pass along the wire. Tesla said 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. He speculated that the Earth itself might serve as the resonator, but that it would take further research to determine its capacitance and resonant frequency. Tesla built his laboratory in Colorado Springs in 1899 to pursue this research.

Then on July 3, 1899 in Colorado Springs he made an observation that he always afterwards claimed confirmed his theory. While using his sensitive coherer radio receiver to listen to 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 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.

Tesla believed that, like the metal balls he used as capacitive resonators on his spark oscillators, the Earth had negligible resistance to these "terrestrial" waves. When excited by a grounded Tesla transformer, they would pass through the entire Earth and reflect from the other side, creating "terrestrial standing waves".

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

Legacy

Kurt Van Voorhies 1991 World Wireless Power Prospects, Proc. of the IECEC

For Amplifier

History

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 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 1914.

Early amplifying devices

The development of electronic voice (audio) communication technologies; telephone and intercom systems around 1880 and amplitude modulated radio around 1900, created a need to somehow make an audio signal "louder". The first amplifying devices were invented by telephone companies. Before amplification, a telephone circuit just consisted of a carbon microphone connected through a long wire to an earphone, with a battery in series. Although these circuits were adequate for calls between neighboring cities, the length of a telephone line was limited to several hundred miles because of power lost in the resistance of the long wires. Beyond that distance the sound in the receiver was too weak to hear. Telephone companies' goal was to develop a "repeater", a device that could be inserted in a telephone line when the signal got weak, to amplify the signal to its original strength. As huge well-funded monopolies, telephone companies created some of the first corporate scientific laboratories, which attacked the problem systematically around 1900.

Carbon amplifiers

US Navy "Ampliphone" carbon amplifier used to amplify the audio signal from a crystal radio, 1915

(left) AT&T type 3A Shreeve carbon repeater cartridge, 1919. (center) Cross section showing windings and carbon cell. (right) Bank of Shreeve repeaters for AT&T Boston-Washington D.C. trunkline, Providence, RI, 1919. The repeaters are the double line of cylindrical components along the top. Each bidirectional wire pair telephone line required two repeaters, one to amplify the signal in each direction.

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, attached to a diaphragm. When a sound wave vibrated the diaphragm, it put varying pressure on the carbon granules, which varied their resistance. A constant voltage from a battery was applied to the microphone. The varying resistance caused the current through the microphone to vary. The audio signal produced was proportional to the DC current through the microphone. 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.

Around the late 1800s researchers used this property to build electromechanical amplifiers by coupling an earphone driver to a carbon microphone. The incoming audio signal to be amplified was applied to an electromagnet that vibrated a diaphragm attached to the carbon microphone. A battery passed a large enough current through the microphone that the audio output signal was larger than the input.

These devices were very unsatisfactory amplifiers. The mass of the acoustic driver system gave them a sharp resonance peak that emphasized some voice frequencies. 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 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.

Arnold mercury-arc amplifier tube, 1912

The best of these devices was probably 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 problems Shreeve discarded 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 or two repeaters could be used in a line before the sound quality became unacceptable, so they were unsuitable for long transcontinental lines.

Arnold mercury arc tube

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.

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.

Vacuum tubes

The first widely used amplifying device was the vacuum tube. Around 1905-6 Lee 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 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.

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, when De Forest and Edwin Armstrong built the first amplifiers and amplifying radio receivers with it. Felix Lowenstein probably built the first Audion amplifier in 1911.

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 problems of the device were caused by the residual air in the tube. 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.

The discovery of the triode's amplifying ability in 1912 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. In 1927, Bell Labs engineer Harold Stephen Black first applied negative feedback to amplifiers. Harry Nyquist and Hendrik Wade Bode invented the Nyquist stability criterion (1932) and Bode plots. Walter R. Evans invented the root locus method of analysis in 1948.

Transistors

Vacuum tubes were bulky, fragile, 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, invented in 1947 by John Bardeen, Walter Brattain, and William Shockley, which brought the "vacuum tube" era to a close. Transistor amplification 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 amplifiers called op-amps.

For Tesla coil

A Tesla coil is an electrical resonant transformer circuit invented by Nikola Tesla around 1891. It is used to produce high-voltage, low-current, high frequency alternating-current electricity. Tesla experimented with a number of different configurations consisting of two, or sometimes three, coupled resonant electric circuits.

Tesla used these coils to conduct innovative experiments in electrical lighting, phosphorescence, X-ray generation, high frequency alternating current phenomena, electrotherapy, and the transmission of electrical energy without wires. Tesla coil circuits were used commercially in sparkgap radio transmitters for wireless telegraphy until the 1920s, and in medical equipment such as electrotherapy and violet ray devices. Today their main use is for entertainment and educational displays, although small coils are still used today as leak detectors for high vacuum systems.

Operation

Homemade Tesla coil in operation, showing brush discharges from the toroid. The high electric field causes the air around the high voltage terminal to ionize and conduct electricity, producing colorful corona discharge, brush discharges and streamer arcs. Tesla coils are used for entertainment at science museums and public events, and for special effects in movies and television.

A Tesla coil is a radio frequency oscillator that drives a double-tuned resonant transformer to produce high voltages. Tesla's original circuits as well as most modern coils use a simple spark gap to excite oscillations in the tuned transformer. More sophisticated designs use transistor or vacuum tube electronic oscillators to drive the resonant transformer. These are described in later sections.

Tesla coils can produce output voltages from a hundred kilovolts to several million volts for large coils. The alternating current output is in the radio frequency range, usually between 50 kHz and 1 MHz. Although some oscillator-driven coils generate a continuous alternating current, most Tesla coils have a pulsed output; the high voltage consists of a rapid string of pulses of radio frequency alternating current.

The common spark-excited Tesla coil circuit, shown below, consists of these components:

• A high voltage supply transformer (T), to step the AC mains voltage up to a high enough voltage to jump the spark gap. Typical voltages are between 5 and 30 kilovolts (kV).
• A high voltage capacitor (C1) that forms a tuned circuit with the primary winding L1 of the Tesla transformer
• A spark gap (SG) that acts as a switch in the primary circuit
• The Tesla coil (L1, L2), an air core resonant transformer, which generates the high output voltage.
• Optionally, a capacitive electrode (E) in the form of a smooth metal sphere or torus attached to the secondary terminal of the coil. It's large surface area suppresses premature corona discharge and streamer arcs, increasing the Q factor and output voltage.

Resonant transformer

(left) Unipolar coil design widely used in modern entertainment coils. The primary is the flat red spiral winding at bottom, the secondary is the vertical cylindrical coil wound with fine red wire. The high voltage terminal is the aluminum torus attached to the top of the secondary coil.
(right) Bipolar coil. There are two high voltage output terminals, each connected to one end of the secondary, with a spark gap between them. The primary is 12 turns of heavy wire, which is located at the midpoint of the secondary to discourage arcs between the coils.

Unipolar Tesla coil circuit. Capacitor C2 represents the parasitic capacitance of the secondary windings L2, plus the capacitance to ground of the toroid electrode E.

The specialized transformer coil used in the Tesla circuit, called a resonant transformer, oscillation transformer, or RF transformer, functions differently from an ordinary transformer used in AC power circuits. While an ordinary transformer is designed to transfer energy efficiently from primary to secondary winding, the resonant transformer is also designed to store electrical energy. Each winding has a capacitance across it and functions as an LC circuit (resonant circuit, tuned circuit), storing oscillating electrical energy, analogously to a tuning fork. The primary winding (L1) consisting of a relatively few turns of heavy copper wire or tubing, is connected to a capacitor (C1) through the spark gap. The secondary winding (L2) consists of many turns (hundreds to thousands) of fine wire on a hollow cylindrical form inside the primary. The secondary is not connected to an actual capacitor, but it also functions as an LC circuit, the inductance (L2) resonates with (C2), the sum of the parasitic capacitance between the windings of the coil, and the capacitance of the toroidal metal electrode attached to the high voltage terminal. The primary and secondary circuits are tuned so they resonate at the same frequency, they have the same resonant frequency. This allows them to exchange energy, so the oscillating current alternates back and forth between the primary and secondary coils.

The peculiar design of the coil is dictated by the need to achieve low resistive energy losses (high Q factor) at high frequencies, which results in the largest secondary voltages:

• Ordinary power transformers have an iron core to increase the magnetic coupling between the coils. However at high frequencies an iron core causes energy losses due to eddy currents and hysteresis, so it is not used in the Tesla coil.
• Ordinary transformers are designed to be "tightly coupled" with a high mutual inductance (M). Due to the iron core and close proximity of the windings, the coupling coefficient is near one, which means almost all the magnetic field of the primary winding passes through the secondary. The Tesla transformer in contrast is "loosely coupled", the primary winding is larger in diameter and spaced apart from the secondary, so the mutual inductance is lower and the coupling coefficient is only 0.05 to 0.2; meaning only 5% to 20% of the magnetic field of each coil passes through the other. This slows the exchange of energy between the coils, so that the energy alternates between primary and secondary slower. This allows the oscillating energy to stay in the secondary circuit longer before it returns back to the primary and begins dissipating in the spark.
• Each winding is also limited to a single layer of wire, which reduces proximity effect losses, and the primary is made of thick copper tubing or strip with its turns spaced apart, which reduces proximity and skin effect losses.

The output circuit can have two forms:

• Unipolar - One end of the secondary winding is connected to a single high voltage terminal, the other end is grounded. This type is used in modern coils designed for entertainment. The primary winding is located near the bottom, low potential end of the secondary, to minimize arcs between the windings, and the grounded sides of both windings are connected together. Since the ground (Earth) serves as the return path for the high voltage, streamer arcs from the terminal tend to jump to any nearby grounded object.
• Bipolar - Neither end of the secondary winding is grounded, and both are brought out to high voltage terminals. The primary winding is located at the center of the secondary coil, equidistant between the two high potential terminals.

Operation cycle

The circuit operates in a rapid repeating cycle in which the supply transformer (T) charges the primary capacitor (C1) up, which then discharges in a spark through the spark gap, creating a brief pulse of oscillating current in the primary circuit which excites a high oscillating voltage across the secondary:

1. Current from the supply transformer (T) charges the capacitor (C1) to a high voltage.
2. When the voltage across the capacitor reaches the breakdown voltage of the spark gap (SG) a spark starts, reducing the spark gap resistance to a very low value. This completes the primary circuit and current from the capacitor flows through the primary coil (L1). The current flows rapidly back and forth between the plates of the capacitor through the coil, generating radio frequency oscillating current in the primary circuit at the circuit's resonant frequency.
3. The oscillating magnetic field of the primary winding induces an oscillating current in the secondary winding (L2), by Faraday's law of induction. Over a number of cycles, the energy in the primary circuit is transferred to the secondary. The total energy in the tuned circuits is limited to the energy originally stored in the capacitor C1, so as the oscillating voltage in the secondary increases in amplitude ("ring up") the oscillations in the primary decrease to zero ("ring down"). Although the ends of the secondary coil are open, it also acts as a tuned circuit due to the parasitic capacitance (C2) between the ends of the coil plus the capacitance of the toroid electrode. Current flows rapidly back and forth through the secondary coil between its ends. Because of the small capacitance, the oscillating voltage across the secondary coil which appears on the output terminal is much larger than the primary voltage.
4. The secondary current creates a magnetic field that induces voltage back in the primary coil, and over a number of additional cycles the energy is transferred back to the primary. This process repeats, the energy shifting rapidly back and forth between the primary and secondary tuned circuits. The oscillating currents in the primary and secondary gradually die out ("ring down") due to energy dissipated as heat in the spark gap and resistance of the coil.
5. When the current through the spark gap is no longer sufficient to keep the air in the gap ionized, the spark stops ("quenches"), terminating the current in the primary circuit. The oscillating current in the secondary may continue for some time.
6. The current from the supply transformer begins charging the capacitor C1 again and the cycle repeats.

This entire cycle takes place very rapidly, the oscillations dying out in a time of the order of a millisecond. Each spark across the spark gap produces a pulse of damped sinusoidal high voltage at the output terminal of the coil. Each pulse dies out before the next spark occurs, so the coil generates a string of damped waves, not a continuous sinusoidal voltage. The high voltage from the supply transformer that charges the capacitor is a 50 or 60 Hz sine wave. Depending on how the spark gap is set, usually one to several sparks occur at the peak of each half-cycle of the mains current, so there are more than a hundred sparks (and pulses of output voltage) per second. Thus the spark at the spark gap appears continuous, as do the high voltage streamers from the top of the coil.

The supply transformer (T) secondary winding is connected across the primary tuned circuit. It might seem that the transformer would be a leakage path for the RF current, damping the oscillations. However its large inductance gives it a very high impedance at the resonant frequency, so it acts as an open circuit to the oscillating current.

Oscillation frequency

To produce the largest output voltage, the primary and secondary tuned circuits are adjusted to resonance with each other. Generally the secondary is not adjustable, so the primary circuit is tuned, usually by a moveable tap on the primary coil L₁. The resonant frequency is

$f={1 \over 2\pi }{\sqrt {1 \over L_{1}C_{1}}}={1 \over 2\pi }{\sqrt {1 \over L_{2}C_{2}}}\,$

Therefore the condition for resonance between primary and secondary is

$L_{1}C_{1}=L_{2}C_{2}\,$

The resonant frequency of Tesla coils is in the radio frequency (RF) range, usually between 50 kHz and 1 MHz.

Output voltage

Because of resonance, the output voltage is not proportional to the turns ratio, as in an ordinary transformer. It can be calculated approximately from conservation of energy. At the beginning of the cycle, when the spark starts, all of the energy in the primary circuit $\scriptstyle W_{1}$ is stored in the primary capacitor $\scriptstyle C_{1}$. If $\scriptstyle V_{1}$ is the voltage at which the spark gap breaks down, which is usually made close to the peak output voltage of the supply transformer T, this energy is

$W_{1}={1 \over 2}C_{1}V_{1}^{2}\,$

During the "ring up" this energy is transferred to the secondary circuit. Although some is lost as heat in the spark and other resistances, in most modern coils around 80% of the energy ends up in the secondary. At the peak ($\scriptstyle V_{2}$) of the secondary sinusoidal voltage waveform, all the energy is stored in the capacitance $\scriptstyle C_{2}$ between the ends of the secondary coil

$W_{2}={1 \over 2}C_{2}V_{2}^{2}\,$

Assuming no energy losses, $\scriptstyle W_{2}\;=\;W_{1}$. Substituting into this equation and simplifying, the peak secondary voltage is

$V_{2}={\sqrt {C_{1} \over C_{2}}}V_{1}={\sqrt {L_{2} \over L_{1}}}V_{1}\,$

The second formula above is derived from the first using $\scriptstyle L_{1}C_{1}\;=\;L_{2}C_{2}$. Since the capacitance of the secondary coil is very small compared to the primary capacitor, the primary voltage is stepped up to a high value.

It might seem that the output voltage could be increased indefinitely by reducing $\scriptstyle C_{2}$ and $\scriptstyle L_{1}$. However as the output voltage increases, it reaches the point where the air next to the high voltage terminal ionizes and corona discharges, brush discharges and streamer arcs break out from the secondary coil. The resulting energy loss damps the oscillation, so the above lossless model is no longer accurate, and the voltage does not reach the theoretical maximum above. Most entertainment Tesla coil designs have a spherical or toroidal shaped electrode on the high voltage terminal. Although the "toroid" increases the secondary capacitance, which tends to reduce the peak voltage, its main effect is that its large diameter curved surface reduces the potential gradient (electric field) at the high voltage terminal, increasing the voltage threshold at which corona and streamer arcs form. Suppressing premature air breakdown and energy loss allows the voltage to build to higher values on the peaks of the waveform, creating longer, more spectacular streamers.

Types

The name "Tesla coil" is applied to a range of high voltage resonant transformer circuits.

Tesla coil circuits can be classified by the type of "excitation" they use, what type of circuit is used to apply current to the primary winding of the resonant transformer:

• Spark-excited or Spark Gap Tesla Coil (SGTC) - This type uses sparks across a spark gap to excite the oscillation in the transformer. This pulsed (disruptive) drive creates a pulsed high voltage output. Spark gaps have disadvantages due to the high primary current: they produce a very loud noise when operating and high temperatures which often require a cooling system. The energy dissipated in the spark also reduces the Q factor and the output voltage.
  • Static spark gap - This is the most common type, described above, usually used in entertainment coils. An AC voltage from a high voltage supply transformer charges a capacitor, which discharges through the spark gap. The spark rate is not adjustable but is determined by the line frequency. Multiple sparks may occur on each half-cycle, so the pulses of output voltage may not be equally-spaced.
  • Static triggered spark gap - Commercial and industrial circuits often apply a DC voltage from a capacitor power supply to the spark gap, and use high voltage pulses generated by an oscillator applied to a triggering electrode to trigger the spark. This allows control of the spark rate and voltage. Commercial spark gaps are often enclosed in an insulating gas atmosphere such as sulfur hexafluoride, reducing the length and thus the energy loss in the spark.
  • Rotary spark gap - These use a spark gap consisting of electrodes around the periphery of a wheel rotated by a motor, which pass by a stationary electrode. Tesla used this type on his big coils, and they are used today on large entertainment coils. The rapid separation speed of the electrodes quenches the spark quickly, allowing "first notch" quenching. The wheel is driven by a synchronous motor, so the sparks are synchronized with the AC line frequency, the spark occurring at the same point on the AC waveform on each cycle, so the primary pulses are repeatable.
• Switched or Solid State Tesla Coil (SSTC) - These use power semiconductor devices, usually transistors such as MOSFETs or IGBTs to switch pulses of current from a DC power supply through the primary winding. They provide pulsed (disruptive) excitation without the disadvantages of a spark gap: the loud noise and high temperatures. They allow fine control of the voltage, spark rate and exciting waveform. This type is used in most commercial, industrial, and research applications as well as higher quality entertainment coils.
  • Single resonant solid state Tesla coil - The primary does not have a capacitor and so is not a tuned circuit; the pulses of current to the primary from the switching transistors just excite resonance in the secondary tuned circuit. Single tuned SSTCs are simpler, but don't have as high a Q and cannot produce as high voltage from a given input power as the DTSSC.
  • Dual Resonant Solid State Tesla Coil (DRSSTC) - A double tuned Tesla transformer driven by solid state switching supply. This functions similarly to the double tuned spark excited circuits.
  • Musical Tesla coil - This is a Tesla coil which can be played like a musical instrument, reproducing simple musical tones. The pulse rate applied to the primary is modulated by a solid state "interrupter" circuit, causing the arc discharge from the high voltage terminal to emit sounds. The musical notes usually come from a keyboard or MIDI file applied to the circuit through a MIDI interface. Two modulation techniques have been used: AM (amplitude modulation of the exciting voltage) and PFM (pulse-frequency modulation). As the sound quality is low they are mainly built as novelties.
• Continuous wave - In these the transformer is driven by a feedback oscillator, which applies a sinusoidal current to the transformer. The primary tuned circuit serves as the tank circuit of the oscillator, and the circuit resembles a radio transmitter. Unlike the previous circuits which generate a pulsed output, they generate a continuous sine wave output. Power vacuum tubes are often used as active devices instead of transistors because they are more robust and tolerant of overloads. In general, continuous excitation produces lower output voltages from a given input power than pulsed excitation.

Tesla circuits can also be classified by how many coils (inductors) they contain:

• Two coil or double-resonant circuits - Virtually all present Tesla coils use the two coil resonant transformer, consisting of a primary winding to which current pulses are applied, and a secondary winding that produces the high voltage, invented by Tesla in 1891. The term "Tesla coil" normally refers to these circuits.
• Three coil, triple-resonant, or magnifier circuits - These are circuits with three coils, based on Tesla's "magnifying transmitter" circuit which he began experimenting with sometime before 1898 and installed in his Colorado Springs lab 1899-1900, and patented in 1902. They consist of a two coil air-core step-up transformer similar to the Tesla transformer, with the secondary connected to a third coil not magnetically coupled to the others, called the "extra" or "resonator" coil, which is series-fed and resonates with its own capacitance. The presence of three energy-storing tank circuits gives this circuit more complicated resonant behavior. It is the subject of research, but has been used in few practical applications.

History

Henry Rowland's 1889 resonant transformer, a predecessor to the Tesla coil.

(above) Drawing from Tesla's May 20, 1891 lecture at Columbia College, New York.
(left) First drawing of Tesla coil circuit from Tesla's April 1891 patent.

Nikola Tesla patented the Tesla coil circuit April 25, 1891. and first publicly demonstrated it May 20, 1891 in his lecture "Experiments with Alternate Currents of Very High Frequency and Their Application to Methods of Artificial Illumination" before the American Institute of Electrical Engineers at Columbia College, New York. Although Tesla patented many similar circuits during this period, this was the first that contained all the elements of the Tesla coil: high voltage primary transformer, capacitor, spark gap, and air core "oscillation transformer".

Invention

One of Tesla's coils at his New York lab in 1892 with a conical secondary.

Prototype "magnifying transmitter" in Tesla's New York lab around 1898 producing 2.5 million volts. The round "spiderweb" secondary coil is visible in background

Compact coil designed by Tesla for use as an ozone generator for water treatment

During the mid 1800s Industrial Revolution, industry had exploited direct current and low frequency alternating current, but little was known about high frequencies above 20 kHz, what are now called radio frequencies. In 1887, four years previously, Heinrich Hertz had discovered Hertzian waves (radio waves), electromagnetic waves which oscillated at very high frequencies. This attracted much attention, and a number of researchers began experimenting with high frequency currents.

Tesla's background was in the new field of alternating current power systems, so he understood transformers and resonance. In 1888 he set up a laboratory at 33 South Fifth Avenue, New York, where he researched high frequencies, initially repeating Hertz's experiments.

Tesla first developed alternators as sources of high frequency current, but by 1890 found they were limited to about 20 kHz. In search of higher frequencies he turned to spark-excited resonant circuits. Tesla's innovation was in applying resonance to transformers. Inductors and transformers functioned differently at high frequencies than at the low frequencies used in power systems; the iron core in conventional transformers caused energy losses due to eddy currents and hysteresis. Tesla and Elihu Thomson independently invented a new type of transformer without an iron core, the "oscillation transformer". The high Q RF transformer designs pioneered by Tesla have been used in radio transmitters ever since.

In 1891 Tesla was trying to develop a "wireless" lighting system, with a type of "phosphorescent" gas discharge light bulb that would glow in an electrostatic field from a high voltage, high frequency power source. He investigated many resonant transformer circuits. He found that enormous voltages could be generated by resonant transformers in which the primary winding with few turns is in a "closed" tuned circuit with a capacitor and spark gap, powered by a step-up transformer, and the secondary winding, with many turns, is an "open" circuit. High voltages are produced when the primary and secondary circuits have the same resonant frequency.

Tesla was not the first to invent this circuit. Henry Rowland built a spark-excited resonant transformer circuit (above) in 1889 and Elihu Thomson had experimented with similar circuits in 1890, including one which could produce 64 inch sparks, and other sources confirm Tesla was not the first. Tesla did not perform detailed mathematical analyses of the circuit, relying instead on trial and error and his intuitive understanding of resonance. He even realized that the secondary coil functioned as a quarter-wave resonator; he specified the length of the wire in the secondary coil must be a quarter wavelength at the resonant frequency. The first mathematical analyses of the circuit were done by Anton Oberbeck (1895) and Paul Drude (1904).

Tesla's demonstrations

Tesla demonstrating wireless power transmission at his 1891 lecture at Columbia College. The two metal sheets are connected to a Tesla coil oscillator, which applies a high radio frequency oscillating voltage. The oscillating electric field between the sheets ionizes the low pressure gas in the two long Geissler tubes he is holding, causing them to glow by fluorescence, similar to neon lights, without wires.

A consummate showman and self-promoter, Tesla used the Tesla coil in dramatic public lectures to demonstrate the mysteries of the new science of high voltage, high frequency electricity. In lectures at Columbia College May 20, 1891, scientific societies in Britain and France during a 1892 European speaking tour, the Franklin Institute, Philadelphia in February 1893, and the National Electric Light Association, St. Louis in March 1893, he impressed audiences with colorful million volt brush discharges and streamers, passed high voltage discharges through his body without harm, heated iron by induction heating, showed RF current could pass through insulators and be conducted by a single wire without a return path, and powered light bulbs and motors without wires. These lectures introduced the "Tesla oscillator" to the public, and made Tesla internationally famous.

Magnifying transmitter

Circuit of magnifying transmitter at Tesla's Colorado Springs laboratory. C2 represents the parasitic capacitance between the windings of coil L3.

Tesla's wireless power research required increasingly high voltages, and he had reached the limit of the voltages he could generate within the space of his New York lab. Between 1899-1900 he built a laboratory in Colorado Springs and performed experiments on wireless power transmission there. The Colorado Springs laboratory had one of the largest Tesla coils ever built, which Tesla called a "magnifying transmitter" as it was intended to transmit power to a distant receiver. With an input power of 300 kilowatts it could produce potentials of the order of 12 million volts, at frequencies of 50-150 kHz, creating huge "lightning" bolts reportedly up to 135 feet long. When Tesla first turned it on, the resulting overload set fire to the alternator of the Colorado Springs power company, destroying it. In the magnifying transmitter, Tesla used a modified design (see circuit, right) different from his previous double-tuned circuits, which he had been experimenting with since 1898 and patented in 1902, In addition to the primary (L1) and secondary (L2) coils it had a third coil (L3) which he called the "extra" coil, not magnetically coupled to the others, attached to the top terminal of the secondary. When driven by the secondary it produced high voltage by resonance, being adjusted to resonate with its own parasitic capacitance (C2) at the frequency of the primary circuit. The use of a series-fed resonator coil to generate high voltages was independently discovered by Paul Marie Oudin in 1893 and employed in his Oudin coil.

The Colorado Springs apparatus consisted of a 53 foot diameter Tesla transformer around the periphery of the lab, with a single-turn primary (L1) buried in the ground and a secondary (L2) of 50 turns of heavy wire wound on an 8 foot high circular "fence". The primary was connected to a bank of oil capacitors (C1) to make a tuned circuit, with a rotary spark gap (SG), powered by 20 to 40 kilovolts from a powerful utility step-up transformer (T). The top of the secondary was connected to the 100-turn 10 ft diameter "extra" or "resonator" coil (L3) in the center of the room. It's high voltage end was connected to a telescoping 143 foot "antenna" rod with a 30 inch metal ball on top which could project through the roof of the lab.

Famous image of magnifying transmitter in operation with Tesla sitting next to it. This is a "trick" photo, a double exposure; Tesla was not in the room when the coil was operating.

Coil in operation at 12 million volts. The 10 ft diameter "resonator" coil is shown. The streamer discharge is 65 feet across.

Discharge of same coil with a metal sphere capacitive terminal.

Primary circuit, showing oil capacitor bank (boxes, foreground), 40 kV supply transformer and spark gap (rear), and part of secondary winding (wall, left)

The huge "magnifying transmitter" coil at Tesla's Colorado Springs laboratory, 1899-1900, photos taken by photographer Dickenson Alley December 1899.

Wireless power

Main article: World Wireless System

Light bulb (bottom) powered wirelessly by "receiver" coil tuned to resonance with the huge "magnifying transmitter" coil at Tesla's Colorado Springs lab, 1899.

Tesla invented the Tesla coil as part of his effort to achieve wireless power transmission, his lifelong dream. In the period 1891 to 1900 he used it to perform the first experiments in wireless power, transmitting RF power by capacitive coupling between elevated metal terminals and by inductive coupling between coils of wire. In his early 1890s demonstrations such as those before the American Institute of Electrical Engineers and at the 1893 Columbian Exposition in Chicago he lit light bulbs from across a stage. He found he could increase the distance by using a receiving LC circuit tuned to resonance with the Tesla coil's LC circuit, transferring energy by resonant inductive coupling. At his Colorado Springs laboratory during 1899-1900, by using voltages of the order of 20 megavolts generated by his enormous magnifying transmitter coil, he was able to light three incandescent lamps at a distance of about 100 feet (30 m). Today the resonant inductive coupling pioneered by Tesla is a familiar concept in electronics, widely used in short range wireless power transmission systems such as cellphone charging pads.

The inductive and capacitive coupling used in Tesla's experiments are "near-field" effects, meaning that the energy transferred decreases with the sixth power of the distance between transmitter and receiver, so they cannot be used for long-distance transmission. However, Tesla was obsessed with developing a long range wireless power transmission system which could transmit power from power plants directly into homes and factories without wires, described in a visionary June, 1900 article in Century Magazine; "The Problem of Increasing Human Energy". He claimed to be able to transmit power on a worldwide scale, using a method that involved conduction through the Earth and atmosphere. Tesla believed that the entire Earth could act as an electrical resonator, and that by driving current pulses into the Earth at its resonant frequency from a grounded Tesla coil with an elevated capacitance, the potential of the Earth could be made to oscillate, creating global standing waves, and this alternating current could be received with a capacitive antenna tuned to resonance with it at any point on Earth. Another of his ideas was that transmitting and receiving terminals could be suspended in the air by balloons at 30,000 feet (9,100 m) altitude, where the air pressure is lower. At this altitude, Tesla thought, an ionized layer would allow electricity to be sent at high voltages (millions of volts) over long distances.

Wardenclyffe Tower

Main article: Wardenclyffe tower

(left) Wardenclyffe Tower wireless station, a large Tesla coil intended as a prototype wireless power transmitter, built by Tesla at Shoreham, NY, 1901-1904.
(right) Circuit of magnifying transmitter Tesla reportedly used in Wardenclyffe plant.

In 1901, convinced his wireless theories were correct, Tesla with financing from banker J. P. Morgan began construction of a high-voltage wireless station, now called the Wardenclyffe Tower, at Shoreham, New York. Although it was built as a transatlantic wireless telegraphy (radio) station, Tesla also intended it to transmit electric power, as a prototype transmitter for his proposed "World Wireless System" that was to transmit both information and power worldwide. Essentially an enormous Tesla coil, it consisted of a powerhouse with a 187 foot tower topped by a 68 foot diameter copper dome capacitive electrode. Due to Tesla's secrecy there is little information on exactly how it was supposed to work.

By 1904 his investors had pulled out and the facility was never completed; it was torn down in 1916. Although Tesla seems to have believed his 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 above. The only report of long-distance power transmission by Tesla is a claim, not found in reliable sources, that in 1899 he wirelessly lit 200 light bulbs at a distance of 26 miles (42 km). 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 from Tesla's first biographer, John J. O'Neill, who said he pieced it together from "fragmentary material... in a number of publications". In the 110 years since Tesla's experiments, efforts by others to achieve long distance power transmission using Tesla coils have failed, and the scientific consensus is his World Wireless system would not have worked. Contemporary scientists point out that while Tesla's coils 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. Long range wireless power transmission was only achieved in the 1960s with the development of microwave technology. Tesla's world power transmission scheme remains today what it was in Tesla's time: a bold, fascinating dream.

Use in radio

(left) Spark-gap transmitter, showing series spark gaps (horizontal cylindrical objects), Leyden jar capacitors (vertical cylinders, rear), and resonant transformer (top).
(right) Spark transmitter circuit from Marconi's 1900 patent. It's similarity to a Tesla coil can be seen; the main difference is the addition of a variable inductor (g) to tune the antenna (f) to resonance.

"[The Tesla coil] was invented not for wireless but for making vacuum lamps glow without external electrodes, and it later played a principal part in other hands in the operation of big spark stations." --William H. Eccles, 1933

One of the largest applications of the Tesla coil circuit was in early radio transmitters called spark gap transmitters. The first radio wave generators, invented by Heinrich Hertz in 1887, were spark gaps connected directly to antennas, powered by Ruhmkorff coils. Due to their high damping, these "untuned" transmitters had extremely wide bandwidth, so they interfered with one another; with multiple transmitters there would be no way for a receiver to select one signal over another.

In 1892 William Crookes, a friend of Tesla, had given an influential lecture on the future uses of radio in which he suggested using resonance in the transmitter and receiver. Different transmitters could be "tuned" to transmit on different frequencies, and a receiver could receive a particular transmission by "tuning" its resonant circuit to the same frequency as the transmitter's resonant circuit, allowing selective signalling.

When a wire antenna was connected to its high voltage terminal, the Tesla coil circuit could function as such a "tuned" radio transmitter. In his March 1893 St. Louis lecture, Tesla demonstrated a wireless system that, although it was intended for power transmission, had all the elements of a "tuned" radio communication system. A grounded Tesla transformer attached to an elevated wire antenna transmitted radio waves, which were received across the room by a wire antenna attached to a receiver consisting of a second grounded resonant transformer tuned to the transmitter's frequency, which lighted a Geissler tube. This system, patented by Tesla September 2, 1897, was the first use of the "four tuned circuit" concept later claimed by Marconi, and is often cited by supporters as the basis for claims that Tesla "invented" radio. However, although he performed many experiments with radio wave transmission, Tesla was mainly interested in wireless power and never developed a practical radio communication system.

Practical wireless telegraphy communication systems were developed by Guglielmo Marconi beginning in 1895, initially using "untuned" circuits. By 1897 the advantages of "tuned" systems were recognized, and resonant circuits, capacitors and inductors, were incorporated in transmitters and receivers. The Tesla "closed primary, open secondary" resonant transformer circuit proved a superior transmitter, because the loosely-coupled transformer partially isolated the oscillating primary circuit from the energy-radiating antenna circuit, reducing the damping, allowing it to produce long "ringing" waves which had a narrower bandwidth. Versions of the circuit were incorporated into radio systems patented by Marconi, John Stone Stone and Oliver Lodge, and were widely used in radio for twenty years.

Although their damping had been reduced as much as possible, spark transmitters still produced damped waves which had a wide bandwidth, creating interference with other transmitters. Around 1920 they became obsolete, superseded by vacuum tube transmitters which generated continuous waves at a single frequency, which could also be modulated to carry sound. Tesla's resonant transformer continued in use in vacuum tube transmitters, and is widely used to this day.

During the "spark era" the radio engineering profession gave credit to Tesla, his circuit became known as the "Tesla coil" or "Tesla transformer". However Tesla did not benefit financially, due to competing patent claims. Marconi had claimed rights to the "closed primary open secondary" circuit in his controversial 1900 "four circuit" wireless patent. Tesla accused Marconi of infringing his patents, and sued him. Marconi's patent became the focus of a long-running litigation by several parties. Due to the Marconi Company's deep pockets the patent stood for 43 years, but in 1943 the US Supreme Court invalidated it. The ruling cited the prior patents of Tesla, Oliver Lodge, and John Stone Stone, but did not decide which of these parties had rights to the circuit. Of course by this time the issue was moot; Tesla was dead and spark transmitters had long been obsolete.

Although there is some disagreement over the role Tesla himself played in the invention of radio, sources agree on the importance of his circuit in early radio transmitters. From a modern perspective, most spark transmitters could be regarded as Tesla coils.

Use in medicine

Small Tesla coil for electrotherapy, 1905. The Tesla transformer is immersed in a tank of oil for insulation to prevent arcs.

Effluvation treatment with an Oudin coil (left), a high voltage transformer very similar to a Tesla coil, 1915

Tesla electrotherapy coil manufactured by Adolphe Gaiffe, early 1900s. The primary capacitor is in the box; the spark gap is mounted on top.

Treatment of cancer with an Oudin coil, 1910.

Combined Tesla/D'Arsonval/Oudin electrotherapy and x-ray outfit 1907

A violet ray wand, a handheld Tesla coil sold as a quack home medical device until about 1940. Said to cure everything from carbuncles to lumbago.

The three circuits used in electrotherapy apparatus in the early 20th century: (1) Tesla coil, (2) D'Arsonval coil, (3) Oudin coil. In medical coils for safety two capacitors (Leyden jars) were used, one in each branch of the primary circuit, to completely isolate the patient's body from the potentially lethal currents of the supply transformer, in case of an electrical fault.

Tesla had observed as early as 1891 that high frequency currents above 10 kHz applied to the body did not cause the sensation of electric shock, and in fact currents that would be lethal at lower frequencies could be passed through the body without apparent harm. He was also one of the first to observe the heating effect of high frequency currents on the body, later used in diathermy. During his highly-publicized early 1890s demonstrations he passed hundreds of thousands of volts through his body. Tesla wrote a few papers on the medical uses of high frequency currents but did not do further work in the field.

Several other researchers were also applying high frequency currents to medicine at this time. Elihu Thomson, the co-inventor of the Tesla coil, also experimented with applying its currents to the body, so in electrotherapy the Tesla coil became known as the "Tesla-Thomson apparatus". In France, from 1889 physician and pioneering biophysicist Jacques d'Arsonval had been documenting the physiological effects of high frequency alternating current on the body. During his 1892 European trip Tesla met with D'Arsonval and was pleased to find the doctor was using apparatus similar to his own. D'Arsonval's resonant spark circuits did not produce as high voltage as the Tesla coil. In 1893 French physician Paul Marie Oudin added a "resonator" coil to the D'Arsonval circuit to create the high voltage Oudin coil, a circuit very similar to the Tesla coil, which was widely used for treating patients in Europe.

During this period, medical ethics were looser and doctors could experiment on their patients. By the turn of the century, application of "high frequency" currents to the body had become part of a Victorian era medical field, part legitimate experimental medicine and part quack medicine, called electrotherapy. Manufacturers produced apparatus to generate "Tesla currents", "D'Arsonval currents", and "Oudin currents" for physicians, often combined with x-ray machines. In electrotherapy, a pointed electrode attached to the high voltage terminal of the coil was held near the patient, and the luminous brush discharges from it (called "effluves") were applied to parts of the body to treat various medical conditions. In order to apply current directly to the skin, or tissues inside the mouth, anus or vagina, a "vacuum electrode" consisting of a metal electrode sealed inside a partially evacuated glass tube was used, which produced a dramatic violet glow. These "violet ray" wands were also sold as a quack home medical device. The popularity of electrotherapy peaked after World War 1, but by the 1920s authorities began to crack down on fraudulent medical treatments, and electrotherapy largely became obsolete. A part of the field that survived was diathermy, the application of high frequency current to heat body tissue. By 1930 "long wave" (0.5~2 MHz) Tesla coil diathermy machines were being replaced by "short wave" (10~100 MHz) vacuum tube diathermy machines, but Tesla coils continued to be used in both diathermy and quack medical devices like violet ray until World War 2.

During the 1920s and 30s all unipolar (single terminal) high voltage medical coils came to be called Oudin coils, so today's unipolar Tesla coils are sometimes referred to as "Oudin coils".

Use in show business

"Electrice" sideshow performer being "electrocuted" 1914

"Electrice" lighting a candle with brush discharge from her fingers

Evangelist Irwin Moon shooting "lightning bolts" from fingers, 1938.

Demonstrating 10 inch (25 cm) brush discharge from hand, 1913

RF current from Tesla coil lights the bulb's filament as it passes through to charge the performer's body, which acts as a capacitor plate.

Turn-of-the-century sideshow performers did stunts with Tesla coils that would be considered foolhardy today. DON'T TRY THESE AT HOME

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Magician David Blaine's Vegas act in which he is bombarded by streamers from 7 Tesla coils. He wears a grounded suit to avoid electrocution

The Tesla coil's spectacular displays of sparks, and the fact that its currents could pass through the human body without causing electric shock, made it popular in the entertainment business. In the early 20th century it appeared in traveling carnivals, freak shows and circus and carnival sideshows, which often had an act in which a performer would pass high voltages through his body Performers such as "Dr. Resisto", "The Human Dynamo", "Madamoiselle Electra", "The Great Volta", and "Electrice", would have their body connected to the high voltage terminal of a hidden Tesla coil, causing sparks to shoot from their fingertips and other parts of their body, and fluorescent light bulbs to light up when held in their hand or even brought near them. They could also light candles or cigarettes with their fingers. These acts could cause painful burns; to prevent them performers sometimes wore metal thimbles on their fingertips (Rev. Moon, center image above, is using them). They were also extremely dangerous and could be lethal if the Tesla coil was misadjusted. The high voltage coils were often salvaged from old electrotherapy apparatus. In carny lingo this was called an "electric chair act" because it often included a spark-laced "electrocution" of the performer in an electric chair, exploiting public fascination with this exotic new method of capital punishment, which had become the United States' dominant method of execution around 1900. Today entertainers still perform high voltage acts with Tesla coils, usually with better protection against the high voltages.

Tesla coils were also used as dramatic props in early mystery and science fiction motion pictures, starting in the silent era. The crackling, writhing sparks emanating from the electrode of a giant Tesla coil became Hollywood's iconic symbol of the "mad scientist", recognized throughout the world. This was probably because the eccentric Nikola Tesla himself, with his famous high voltage demonstrations and his mysterious Colorado Springs laboratory, was one of the prototypes from which the "mad scientist" stock character originated. Some early films in which Tesla coils appeared were Wolves of Kultur (1918), The Power God (1926), Metropolis (1927), Frankenstein (1931) and its many sequels and knockoffs such as Son of Frankenstein (1939), The Mask of Fu Manchu (1932), Chandu the Magician (1932), The Lost City (1935), and The Clutching Hand (1936).

Million volt coil at Griffith Observatory, Los Angeles, one of the oldest working educational Tesla coils.

The Tesla coils for many of these movies were constructed by Kenneth Strickfaden (1896-1984) who, beginning with his spectacular effects in the 1931 Frankenstein, became Hollywood's preeminent electrical special effects wizard. His large "Meg Senior" Tesla coil seen in many of these movies consisted of a 6 foot 1000 turn conical secondary and a 10 turn primary, connected to a capacitor through a rotary spark gap, powered by a 20 kV transformer. It could produce 6 foot sparks. Some of his last gigs were the reassembly of the original 1931 Frankenstein high voltage apparatus for the Mel Brooks satire Young Frankenstein (1974), and construction of a million volt Tesla coil which produced 12 foot sparks for a 1976 stage show by the rock band Kiss. When he was not on a movie job, Strickfaden would take his high voltage apparatus on the road as a "science show" to high schools and colleges.

Starting with Tesla's 1890s demonstrations, Tesla coils have been exhibited in educational venues to promote interest in science and technology. One of the best known and oldest working coils is the Griffith Observatory coil in Los Angeles. It was originally one of a pair of coils built in 1910 by Earle Ovington, a friend of Tesla and manufacturer of high voltage electrotherapy coils. An employee of the Edison Electric Illuminating Co. heard about the coils and offered to pay Ovington $1000 to display them at the December electrical trade show at Madison Square Garden in New York City, providing the coil could produce sparks at least 10 feet long. Called the Million Volt Oscillator, the twin coils were installed on the balcony at the show and every hour the lights were dimmed while the coils put on a display of 10 foot arcs. Ovington gave the coils to Dr. Frederick Finch Strong, a leading figure in the alternative health field of electrotherapy. In 1937 Strong donated the coils to the Griffith Observatory museum. The museum didn't have room to display both coils and one was sold. The other was lovingly restored by Kenneth Strickfaden and has been in daily operation ever since. It consists of a 48 in. high conical secondary coil topped by a 12 in. diameter copper ball electrode, with a 9-turn primary of 2 in. copper strip, a glass plate capacitor and rotary spark gap.

Later uses

Breit and Tuve's 5 MV Tesla coil used as particle accelerator, 1928

The Tesla coil was used in radio transmitters, electrotherapy devices, x-ray machines and induction heating equipment until the 1920s and 1930s, when vacuum tube oscillators replaced it. The vacuum tube was a much better current control device than the noisy, hot, ozone-producing spark, and could produce continuous waves. After this, industrial use of the Tesla coil was mainly limited to a few specialized applications which were suited to its unique characteristics, such as high voltage insulation testing.

In 1926, pioneering accelerator physicists Merle Tuve and Gregory Breit built a 5 million volt Tesla coil as a linear particle accelerator. The bipolar coil consisted of a pyrex tube a meter long wound with 8000 turns of fine wire, with round corona caps on each end, and a 5 turn spiral primary coil surrounding it at the center. It was operated in a tank of insulating oil pressurized to 500 psi which allowed it to reach a potential of 5.2 megavolts. Although it was used for a short period in 1929-30 it was not a success because the particles acceleration had to be completed within the brief period of a half cycle of the RF voltage.

In 1970 Robert K. Golka built a replica of Tesla's huge Colorado Springs magnifying transmitter in a shed at Wendover Air Force Base, Utah, using data he found in Tesla's lab notes archived at the Nikola Tesla Museum in Beograd, Serbia. This was one of the first experiments with the magnifier circuit since Tesla's time. The coil generated 22 million volts. Golka used it to try to duplicate Tesla's reported synthesis of ball lightning.

Theory

The first mathematical analyses of the Tesla coil circuit were done by Anton Oberbeck (1895) and Paul Drude (1903). Drude showed that maximum voltage was produced with a coupling coefficient of 0.6 and a unity tuning ratio (resonance between primary and secondary). Phung et al (1991) generalized this result, showing that maximum voltage Finkelstein et al (1966) used another metric to optimize the circuit, determining the conditions for a complete energy transfer from the primary to the secondary.

Lumped-element analysis

All traditional analyses of the Tesla coil use the lumped element model, in which the components are modeled by discrete circuit elements. The Tesla circuit has four energy storage devices (L1, C1, L2, C2) so an exact analysis requires a fourth order differential equation, which is extremely laborious to solve in closed form. Therefore virtually all analyses use the simplifying assumption of no resistance ($\scriptstyle R1\;=\;R2\;=\;0$) which results in a 2nd order equation. This is justified since coils are designed with minimum damping and have high Q_factor and results in little error in the values most desired, the peak output voltage and frequency.

Maximizing streamer length

The purpose of entertainment Tesla coils is to produce spectacular displays of sparks; the design goal of these circuits is usually to produce the longest streamer arcs. The length of air gap that a DC spark can jump is simply 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. 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

The secondary as a helical slow-wave resonator

At the high frequency at which it operates, the secondary winding of the Tesla coil 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 a resonator, 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. 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 top load the current doesn't go to zero). 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 been generally recognized for years that the lumped element equations are just a starting point for the design of a Tesla coil, 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 for the secondary, which 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 only even numbered overtones, while unipolar coils have 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 consequence is that when the coil is out of resonance, so the operating frequency is higher than the secondary's resonant frequency, the voltage maximum (antinode) does not occur at the top of the coil, but somewhere below the top, on the winding. This also can cause the commonly-seen problem of discharges and arcs breaking out from the winding.

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For Talk:Nikola Tesla

John J. O'Neill, Prodigal Genius: The Life of Nikola Tesla

• (3)"Tesla created the modern era; he was unquestionably one of the world's greatest geniuses"
• (5)"He created the race of robots... he invented the radar 40 years before its use in World War 2, he gave us our modern neon and other forms of gaseous-tube lighting; he gave us our fluorescent lighting..."
• (7)"He discovered the secret of transmitting power to the utmost ends of the Earth without wires, and demonstrated his system by which useful amounts of power could be drawn from the earth anywhere merely by making a connection to the ground." "...he created the modern radio system; he planned our broadcasting methods of today..."
• (8)"...he invented the tube, fluorescent, and wireless lamps which we now consider such up-to-the-minute developments..."

James O'Neill, Prodigal Genius: The life of Nikola Tesla

• (84) "He had received $1,000,000 from the Westinghouse co. for his first crop of inventions"
• (135)"...wireless experimenters took over [Hertz's short wavelengths] without asking a question over what wavelengths should be used for wireless communication... all except Tesla. ... Having experimented... he knew that short wavelengths were totally unsuitable for communication purposes. He knew that the useful wavelengths ranged from 100 meters to many thousands of meters.
• (137) "...he developed the cone-shaped coil...he carried still further by designing the flat helix, or pancake-shaped coil" "Tesla did not invent the idea of electrical resonance.... However, there were no practical circuits in which resonance could manifest itself until Tesla developed alternating currents..."
• (160) Tesla's mechanical oscillator almost brings down the building

@Bob K31416, I agree with your remarks, it is important to bring it up, a good effort. But you can see from Michael Cambridge's response that many of the people who argue nationality on this page just don't care about Wikipedia verifiability rules. You [1] and I [2], [3] and many others [4] have raised the NOR issue before, without result. For many partisans, the Talk page itself is their goal and their forum. Notice that during this latest round of nationality debates, no one is edit-warring or trying to change the article. If they can present their arguments here on the Talk page, and find someone to argue with, that is enough to keep them going indefinitely.

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• This article barely mentions Tesla's greatest project and lifelong ambition, wireless power transmission. The lengthy section "Colorado Springs" which covers his most intensive wireless power experiments 1899-1900, doesn't mention wireless power. It vaguely describes "high voltage and high frequency experiments" without saying what Tesla was doing, what his goals were. Tesla himself, in innumerable later public lectures and newspaper interviews, as well as all his biographers, describe what he was doing: researching his theories that power could be transmitted by setting up resonant oscillations in the Earth. Although they turned out to be wrong, these should be in the article. The Wardenclyffe section also barely mentions that it was a prototype wireless power transmitter, although in later interviews Tesla describes it that way, and not as a radio transmitter, The article has a long section on "Radio", although sources are unanimous that for Tesla it was a tepid secondary interest to wireless power. There is more in the article about minor topics like "death rays" and communication from Martians than there is about Tesla's greatest dream.

• In the 1890s Tesla was the most well-known inventor and engineer in the country. He was said to be more famous than Edison. This was the Industrial Revolution, when inventors and engineers could become rich, as did Tesla's rivals Westinghouse, Edison, Steinmetz, Marconi. Why did Tesla end up destitute, living off charity, feeding pigeons in the park?

• The article also doesn't say anything about Tesla's legacy or current popularity.

For Right-hand rule

The right-hand rule is a relationship between three vectors (directions) in space, which is visualized by holding the right hand in certain shapes. A similar rule, the right-hand grip rule, is a relation between the direction of circulation around a closed loop and a vector. The right-hand rule defines the direction of the cross product in vector analysis. Therefore it appears in many mathematical and physical laws and formulas which involve this mathematical operation. It defines the direction of the magnetic field created by an electric current in Ampere's circuital law, and the direction of the force on a moving charged particle in a magnetic field due to the Lorentz force law. It gives the direction of many other vector quantities in physics, such as torque and Coriolis force. It also defines the direction of a right-handed screw thread or helix.

It is also used in other areas of mathematics. It defines the orientation of a curve, and the direction of the vector operation curl. A right-handed coordinate system is a 3 dimensional coordinate system in which the three axes, $\scriptstyle \mathbf {x} ,\;\mathbf {y} ,\;\mathbf {z}$ form a right handed triple.

Definitions

Right-hand rule

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 $\scriptstyle (\mathbf {a} ,\;\mathbf {b} ,\;\mathbf {c} )$ are three noncoplanar vectors

• The index finger is pointed in the direction of $\scriptstyle \mathbf {a}$
• The middle finger is pointed in the direction of $\scriptstyle \mathbf {b}$
• If the thumb points in the direction of $\scriptstyle \mathbf {c}$ then the three vectors obey the right-hand rule; the ordered triple $\scriptstyle (\mathbf {a} ,\;\mathbf {b} ,\;\mathbf {c} )$ is called a right-hand system. If the thumb points opposite to the direction of $\scriptstyle \mathbf {c}$ 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 $\scriptstyle \mathbf {a}$ and $\scriptstyle \mathbf {b}$ is switched, the thumb will point in the opposite direction, so if $\scriptstyle (\mathbf {a} ,\;\mathbf {b} ,\;\mathbf {c} )$ is a right-hand system, then $\scriptstyle (\mathbf {b} ,\;\mathbf {a} ,\;\mathbf {c} )$ is a left-hand system; $\scriptstyle (\mathbf {b} ,\;\mathbf {a} ,\;\mathbf {-c} )$ is a right-hand system.

The right-hand rule appears in problems where the third vector $\scriptstyle \mathbf {c}$ 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

Right hand grip rule

This is a relation between a direction of circulation around a loop, and the direction of a vector passing through the loop.

• 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.

One example would be a rotation axis and the direction of rotation.

Combining 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 $\scriptstyle (\mathbf {a} ,\;\mathbf {b} ,\;\mathbf {c} )$

• The right hand is oriented so when the hand is open the fingers point to $\scriptstyle \mathbf {a}$ but when the fingers are curled toward the palm they rotate toward $\scriptstyle \mathbf {b}$.
• If the thumb points in the direction of $\scriptstyle \mathbf {c}$, then $\scriptstyle (\mathbf {a} ,\;\mathbf {b} ,\;\mathbf {c} )$ obeys the right-hand rule.

Cross product

Right hand rule

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 $\scriptstyle \mathbf {a} \times \mathbf {b}$ of two vectors $\scriptstyle \mathbf {a}$ and $\scriptstyle \mathbf {b}$ is defined to be a vector that is perpendicular to the plane passing through both vectors and has a magnitude equal to $\scriptstyle |\mathbf {a} ||\mathbf {b} |\sin \theta$, where $\scriptstyle \theta$ 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

• The index finger of the right hand is pointed in the direction of $\scriptstyle \mathbf {a}$.
• The middle finger is pointed in the direction of $\scriptstyle \mathbf {b}$.
• Then the thumb points to the side of the plane that the vector $\scriptstyle \mathbf {a} \times \mathbf {b}$ is on.

Thus the vector triple $\scriptstyle (\mathbf {a} ,\;\mathbf {b} ,\;\mathbf {a} \times \mathbf {b} )$ obeys the right-hand rule.

Due to the right-hand rule, the order of the two arguments in the cross product matters. If $\scriptstyle \mathbf {a}$ and $\scriptstyle \mathbf {b}$ 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 $\scriptstyle \mathbf {B}$ and angular momentum $\scriptstyle \mathbf {L}$, 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 $\scriptstyle \mathbf {\Omega }$ and angular momentum $\scriptstyle \mathbf {L}$ vectors is related to the right-hand rule through the cross product. If the $\scriptstyle \mathbf {\Gamma }$ and $\scriptstyle \mathbf {n}$ are an oriented circle and an axis that are related by the right-hand grip rule, and $\scriptstyle \mathbf {r}$ is a radius vector from the axis to the circle and $\scriptstyle \mathbf {s}$ points in the direction of rotation, then $(\scriptstyle \mathbf {r} ,\;\mathbf {s} ,\;\mathbf {n} )$ 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 $\scriptstyle \mathbf {B}$ and angular momentum $\scriptstyle \mathbf {L}$ 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. Which type a screw is can be 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 $\scriptstyle \mathbf {I}$ in a wire generates a magnetic field $\scriptstyle \mathbf {B}$ 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 $\scriptstyle \mathbf {I}$ (conventional current, flow of positive charge)
• Then the fingers will point in the direction that the magnetic field $\scriptstyle \mathbf {B}$ circles the wire.

One of the sources is the Biot-Savart law, an equation which gives the magnetic field due to the current $\scriptstyle \mathbf {dI}$ through a small section of the circuit:

$B={\mu _{0} \over 4\pi }\int {\mathbf {dI} \times \mathbf {r} \over r^{3}}\,$

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

Faraday's law of induction (Fleming's right hand rule)

Main article: Fleming's right hand rule

When a wire attached to a circuit is moved through a magnetic field $\scriptstyle \mathbf {B}$ in a direction $\scriptstyle \mathbf {v}$, the field induces a current $\scriptstyle \mathbf {I}$ 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 $\scriptstyle \mathbf {v}$ of the wire
• The middle finger is pointed in the direction of the magnetic field $\scriptstyle \mathbf {B}$
• Then the thumb will point in the direction of the current $\scriptstyle \mathbf {I}$ 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 $\scriptstyle \mathbf {v}$ 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 $\scriptstyle \mathbf {F}$ along the wire causes the charges to move along the wire, inducing the current $\scriptstyle \mathbf {I}$. Since the equation above has the cross product in it, the three vectors $\scriptstyle (\mathbf {v} ,\;\mathbf {B} ,\;\mathbf {I} )$ are related by the right-hand rule.

Lorentz force law

The Lorentz force law says that a magnetic field $\scriptstyle \mathbf {B}$ will exert a force on a charged particle moving through it with a velocity $\scriptstyle \mathbf {v}$, as long as the field is not parallel to $\scriptstyle \mathbf {v}$. 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

• The index finger of the right hand is pointed in the direction of the particle velocity $\scriptstyle \mathbf {v}$
• The middle finger is pointed in the direction of the magnetic field $\scriptstyle \mathbf {B}$
• Then the thumb will point in the direction of the force on the particle $\scriptstyle \mathbf {F}$.

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:

$\mathbf {F} =q\mathbf {v} \times \mathbf {B} \,$

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

Coriolis force

The Coriolis force $\scriptstyle {\boldsymbol {F_{C}}}$ 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 $\scriptstyle \mathbf {v}$ in the rotating coordinate system, and the angular velocity vector $\scriptstyle {\boldsymbol {\Omega }}$ 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

• When the index finger of the right hand points in the $\scriptstyle \mathbf {v}$ direction
• And the middle finger points in the $\scriptstyle {\boldsymbol {\Omega }}$ direction
• Then the thumb will point in the direction of the Coriolis force $\scriptstyle {\boldsymbol {F_{C}}}$.

Thus on a surface rotating counterclockwise, for example, $\scriptstyle {\boldsymbol {\Omega }}$ is directed up, and the Coriolis force is always to the right as one faces in the direction of motion $\scriptstyle \mathbf {v}$.

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 $\scriptstyle {\boldsymbol {\Omega }}$ points out of the the North Pole. In the Northern Hemisphere a parcel of air moving north has a velocity $\scriptstyle \mathbf {v}$ toward the axis. Applying the right-hand rule, the direction of the Coriolis force $\scriptstyle \mathbf {F_{C}}$ 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, $\scriptstyle \mathbf {F_{C}}$ 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 $\scriptstyle {\boldsymbol {F_{C}}}$ must be directed everywhere outward from the center of the cyclone. Since $\scriptstyle {\boldsymbol {F_{C}}}$ 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

• 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 $\scriptstyle \mathbf {b}$.

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 $\scriptstyle \mathbf {b}$ 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.