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Duplexer

A duplexer is an electronic device that allows bi-directional (duplex) communication over a single path. In radar and radio communications systems, it isolates the receiver from the transmitter while permitting them to share a common antenna. Most radio repeater systems include a duplexer. Duplexers can be based on frequency (often a waveguide filter), polarization (such as an orthomode transducer), or timing (as is typical in radar).

Transmit-receive switch
In radar, a transmit/receive (TR) switch alternately connects the transmitter and receiver to a shared antenna. In the simplest arrangement, the switch consists of a gas-discharge tube across the input terminals of the receiver. When the transmitter is active, the resulting high voltage causes the tube to conduct, shorting together the receiver terminals to protect it, while its complementary, the anti-transmit/receive (ATR) switch, is a similar discharge tube which decouples the transmitter from the antenna while not operating, to prevent it from wasting received energy.

Hybrid
A hybrid, such as a magic T, may be used as a duplexer by terminating the fourth port in a matched load. This arrangement suffers from the disadvantage that half of the transmitter power is lost in the matched load, while thermal noise in the load is delivered to the receiver.

Frequency domain

In radio communications (as opposed to radar), the transmitted and received signals can occupy different frequency bands, and so may be separated by frequency-selective filters. These are effectively a higher-performance version of a diplexer, typically with a narrow split between the two frequencies in question (typically around 2%-5% for a commercial two-way radio system).

With a duplexer the high- and low-frequency signals are traveling in opposite directions at the shared port of the duplexer.

Modern duplexers often use nearby frequency bands, so the frequency separation between the two ports is also much less. For example, the transition between the uplink and downlink bands in the GSM frequency bands may be about one percent (915 MHz to 925 MHz). Significant attenuation (isolation) is needed to prevent the transmitter's output from overloading the receiver's input, so such duplexers employ multi-pole filters. Duplexers are commonly made for use on the 30-50 MHz ("low band"), 136-174 MHz ("high band"), 380-520 MHz ("UHF"), plus the 790–862 MHz ("800"), 896-960 MHz ("900") and 1215-1300 MHz ("1200") bands.

There are two predominant types of duplexer in use - "notch duplexers", which exhibit sharp notches at the "unwanted" frequencies and only pass through a narrow band of wanted frequencies and "bandpass duplexers", which have wide-pass frequency ranges and high out-of-band attenuation.

On shared-antenna sites, the bandpass duplexer variety is greatly preferred because this virtually eliminates interference between transmitters and receivers by removing out-of-band transmit emissions and considerably improving the selectivity of receivers. Most professionally engineered sites ban the use of notch duplexers and insist on bandpass duplexers for this reason.

Note 1: A duplexer must be designed for operation in the frequency band used by the receiver and transmitter, and must be capable of handling the output power of the transmitter.

Note 2: A duplexer must provide adequate rejection of transmitter noise occurring at the receive frequency, and must be designed to operate at, or less than, the frequency separation between the transmitter and receiver.

Note 3: A duplexer must provide sufficient isolation to prevent receiver desensitization.

Source: from Federal Standard 1037C

History
The first duplexers were invented for use on the electric telegraph and were known as duplex rather than duplexer. They were an early form of the hybrid coil. The telegraph companies were keen to have such a device since the ability to have simultaneous traffic in both directions had the potential to save the cost of thousands of miles of telegraph wire. The first of these devices was designed in 1853 by Julius Wilhelm Gintl of the Austrian State Telegraph. Gintl's design was not very successful. Further attempts were made by Carl Frischen of Hanover with an artificial line to balance the real line as well as by Siemens & Halske, who bought and modified Frischen's design. The first truly successful duplex was designed by Joseph Barker Stearns of Boston in 1872. This was further developed into the quadruplex telegraph by Thomas Edison. The device is estimated to have saved Western Union $500,000 per year in construction of new telegraph lines.

The first duplexers for radar, sometimes referred to as Transmit/Receive Switches, were invented by Robert Morris Page and Leo C. Young of the United States Naval Research Laboratory in July 1936.

title: Duplexer


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Duty cycle

A duty cycle or power cycle is the fraction of one period in which a signal or system is active. Duty cycle is commonly expressed as a percentage or a ratio. A period is the time it takes for a signal to complete an on-and-off cycle. As a formula, a duty cycle (%) may be expressed as:

 * $$D = \frac{PW}{T} \times 100\%$$

Equally, a duty cycle (ratio) may be expressed as:

 * $$D = \frac{PW}{T}$$

where $$D$$ is the duty cycle, $$PW$$ is the pulse width (pulse active time), and $$T$$ is the total period of the signal. Thus, a 60% duty cycle means the signal is on 60% of the time but off 40% of the time. The "on time" for a 60% duty cycle could be a fraction of a second, a day, or even a week, depending on the length of the period.

Duty cycles can be used to describe the percent time of an active signal in an electrical device such as the power switch in a switching power supply or the firing of action potentials by a living system such as a neuron.

Some publications use $$\alpha$$ as the symbol for duty cycle.

As a ratio, duty cycle is unitless and may be given as decimal fraction and percentage alike.

An alternative term in use is duty factor.

Electrical and electronics
In electronics, duty cycle is the percentage of the ratio of pulse duration, or pulse width (PW) to the total period (T) of the waveform. It is generally used to represent time duration of a pulse when it is high (1). In digital electronics, signals are used in rectangular waveform which are represented by logic 1 and logic 0. Logic 1 stands for presence of an electric pulse and 0 for absence of an electric pulse. For example, a signal (10101010) has 50% duty cycle, because the pulse remains high for 1/2 of the period or low for 1/2 of the period. Similarly, for pulse (10001000) the duty cycle will be 25% because the pulse remains high only for 1/4 of the period and remains low for 3/4 of the period. Electrical motors typically use less than a 100% duty cycle. For example, if a motor runs for one out of 100 seconds, or 1/100 of the time, then, its duty cycle is 1/100, or 1 percent.

Pulse-width modulation (PWM) is used in a variety of electronic situations, such as power delivery and voltage regulation.

In electronic music, music synthesizers vary the duty cycle of their audio-frequency oscillators to obtain a subtle effect on the tone colors. This technique is known as pulse-width modulation.

In the printer / copier industry, the duty cycle specification refers to the rated throughput (that is, printed pages) of a device per month.

In a welding power supply, the maximum duty cycle is defined as the percentage of time in a 10-minute period that it can be operated continuously before overheating.

Biological systems
The concept of duty cycles is also used to describe the activity of neurons and muscle fibers. In neural circuits for example, a duty cycle specifically refers to the proportion of a cycle period in which a neuron remains active.

Generation
One way to generate fairly accurate square wave signals with 1/n duty factor, where n is an integer, is to vary the duty cycle until the nth-harmonic is significantly suppressed. For audio-band signals, this can even be done "by ear"; for example, a -40 dB reduction in the 3rd harmonic corresponds to setting the duty factor to 1/3 with a precision of 1% and -60 dB reduction corresponds to a precision of 0.1%.

Mark-space ratio
 Mark-space ratio, or mark-to-space ratio, is another term for the same concept, to describe the temporal relationship between two alternating periods of a waveform. However, whereas the duty cycle relates the duration of one period to the duration of the entire cycle, the mark-space ratio relates the durations of the two individual periods:

 * $$\text{mark-space ratio} = \frac{PW_\text{on}}{PW_\text{off}}$$

where $$PW_\text{on}$$ and $$PW_\text{off}$$ are the durations of the two alternating periods.

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

title: Duty cycle


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Dynamic range

Dynamic range (abbreviated DR, DNR, or DYR ) is the ratio between the largest and smallest values that a certain quantity can assume. It is often used in the context of signals, like sound and light. It is measured either as a ratio or as a base-10 (decibel) or base-2 (doublings, bits or stops) logarithmic value of the ratio between the largest and smallest signal values.

Electronically reproduced audio and video is often processed to fit the original material with a wide dynamic range into a narrower recorded dynamic range that can more easily be stored and reproduced; this processing is called dynamic range compression.

Human perception
The human senses of sight and hearing have a relatively high dynamic range. However, a human cannot perform these feats of perception at both extremes of the scale at the same time. The human eye takes time to adjust to different light levels, and its dynamic range in a given scene is actually quite limited due to optical glare. The instantaneous dynamic range of human audio perception is similarly subject to masking so that, for example, a whisper cannot be heard in loud surroundings.

A human is capable of hearing (and usefully discerning) anything from a quiet murmur in a soundproofed room to the loudest heavy metal concert. Such a difference can exceed 100 dB which represents a factor of 100,000 in amplitude and a factor 10,000,000,000 in power. The dynamic range of human hearing is roughly 140 dB, varying with frequency, from the threshold of hearing (around −9 dB SPL at 3 kHz) to the threshold of pain (from 120–140 dB SPL  ). This wide dynamic range cannot be perceived all at once, however; the tensor tympani, stapedius muscle, and outer hair cells all act as mechanical dynamic range compressors to adjust the sensitivity of the ear to different ambient levels.

A human can see objects in starlight or in bright sunlight, even though on a moonless night objects receive one billionth (10−9) of the illumination they would on a bright sunny day; a dynamic range of 90 dB. Change of sensitivity is achieved in part through adjustments of the iris and slow chemical changes, which take some time.

In practice, it is difficult for humans to achieve the full dynamic experience using electronic equipment. For example, a good quality liquid-crystal display (LCD) has a dynamic range limited to around 1000:1, and some of the latest CMOS image sensors now have measured dynamic ranges of about 23,000:1. Paper reflectance can produce a dynamic range of about 100:1. A professional video camera such as the Sony Digital Betacam achieves a dynamic range of greater than 90 dB in audio recording.

Audio
Audio engineers use dynamic range to describe the ratio of the amplitude of the loudest possible undistorted signal to the noise floor, say of a microphone or loudspeaker. Dynamic range is therefore the signal-to-noise ratio (SNR) for the case where the signal is the loudest possible for the system. For example, if the ceiling of a device is 5 V (rms) and the noise floor is 10 μV (rms) then the dynamic range is 500000:1, or 114 dB:

$$20 \times \log_{10} \left(\frac{\rm 5\,V}{10 \, \mu \mathrm{V}}\right) = 20 \times \log_{10}(500000) = 20 \times 5.7 = 114 \,\mathrm{dB}$$

In digital audio theory the dynamic range is limited by quantization error. The maximum achievable dynamic range for a digital audio system with Q-bit uniform quantization is calculated as the ratio of the largest sine-wave rms to rms noise is:

$$ \mathrm{DR_{ADC}} = 20 \times \log_{10} \left(\frac{2^Q}{1}\right) = \left ( 6.02 \cdot Q \right )\ \mathrm{dB} \,\!$$

However, the usable dynamic range may be greater, as a properly dithered recording device can record signals well below the noise floor.

The 16-bit compact disc has a theoretical undithered dynamic range of about 96 dB; however, the perceived dynamic range of 16-bit audio can be 120 dB or more with noise-shaped dither, taking advantage of the frequency response of the human ear.

Digital audio with undithered 20-bit quantization is theoretically capable of 120 dB dynamic range, while 24-bit digital audio affords 144 dB dynamic range. Most Digital audio workstations process audio with 32-bit floating-point representation which affords even higher dynamic range and so loss of dynamic range is no longer a concern in terms of digital audio processing. Dynamic range limitations typically result from improper gain staging, recording technique including ambient noise and intentional application of dynamic range compression.

Dynamic range in analog audio is the difference between low-level thermal noise in the electronic circuitry and high-level signal saturation resulting in increased distortion and, if pushed higher, clipping. Multiple noise processes determine the noise floor of a system. Noise can be picked up from microphone self-noise, preamp noise, wiring and interconnection noise, media noise, etc.

Early 78 rpm phonograph discs had a dynamic range of up to 40 dB, soon reduced to 30 dB and worse due to wear from repeated play. Vinyl microgroove phonograph records typically yield 55-65 dB, though the first play of the higher-fidelity outer rings can achieve a dynamic range of 70 dB.

German magnetic tape in 1941 was reported to have had a dynamic range of 60 dB, though modern day restoration experts of such tapes note 45-50 dB as the observed dynamic range. Ampex tape recorders in the 1950s achieved 60 dB in practical usage, In the 1960s, improvements in tape formulation processes resulted in 7 dB greater range, and Ray Dolby developed the Dolby A-Type noise reduction system that increased low- and mid-frequency dynamic range on magnetic tape by 10 dB, and high-frequency by 15 dB, using companding (compression and expansion) of four frequency bands. The peak of professional analog magnetic recording tape technology reached 90 dB dynamic range in the midband frequencies at 3% distortion, or about 80 dB in practical broadband applications. The Dolby SR noise reduction system gave a 20 dB further increased range resulting in 110 dB in the midband frequencies at 3% distortion.

Compact Cassette tape performance ranges from 50 to 56 dB depending on tape formulation, with type IV tape tapes giving the greatest dynamic range, and systems such as XDR, dbx and Dolby noise reduction system increasing it further. Specialized bias and record head improvements by Nakamichi and Tandberg combined with Dolby C noise reduction yielded 72 dB dynamic range for the cassette.

A dynamic microphone is able to withstand high sound intensity and can have a dynamic range of up to 140 dB. Condenser microphones are also rugged but their dynamic range may be limited by the overloading of their associated electronic circuitry. Practical considerations of acceptable distortion levels in microphones combined with typical practices in a recording studio result in a useful dynamic range of 125 dB.

In 1981, researchers at Ampex determined that a dynamic range of 118 dB on a dithered digital audio stream was necessary for subjective noise-free playback of music in quiet listening environments.

Since the early 1990s, it has been recommended by several authorities, including the Audio Engineering Society, that measurements of dynamic range be made with an audio signal present, which is then filtered out in the noise floor measurement used in determining dynamic range. This avoids questionable measurements based on the use of blank media, or muting circuits.

The term dynamic range may be confusing in audio production because it has two conflicting definitions, particularly in the understanding of the loudness war phenomenon. Dynamic range may refer to micro-dynamics, related to crest factor,  whereas the European Broadcasting Union, in EBU3342 Loudness Range, defines dynamic range as the difference between the quietest and loudest volume, a matter of macro-dynamics.

Electronics
In electronics dynamic range is used in the following contexts:
 * Specifies the ratio of a maximum level of a parameter, such as power, current, voltage or frequency, to the minimum detectable value of that parameter. (See Audio system measurements.)
 * In a transmission system, the ratio of the overload level (the maximum signal power that the system can tolerate without distortion of the signal) to the noise level of the system.
 * In digital systems or devices, the ratio of maximum and minimum signal levels required to maintain a specified bit error ratio.
 * Optimization of bit width of digital data path (according to the dynamic ranges of signal) can reduce the area, cost, and power consumption of digital circuits and systems while improving their performance. Optimal bit width for a digital data path is the smallest bit width that can satisfy the required signal-to-noise ratio and also avoid overflow.

In audio and electronics applications, the ratio involved is often large enough that it is converted to a logarithm and specified in decibels.

Metrology
In metrology, such as when performed in support of science, engineering or manufacturing objectives, dynamic range refers to the range of values that can be measured by a sensor or metrology instrument. Often this dynamic range of measurement is limited at one end of the range by saturation of a sensing signal sensor or by physical limits that exist on the motion or other response capability of a mechanical indicator. The other end of the dynamic range of measurement is often limited by one or more sources of random noise or uncertainty in signal levels that may be described as defining the sensitivity of the sensor or metrology device. When digital sensors or sensor signal converters are a component of the sensor or metrology device, the dynamic range of measurement will be also related to the number of binary digits (bits) used in a digital numeric representation in which the measured value is linearly related to the digital number. For example, a 12-bit digital sensor or converter can provide a dynamic range in which the ratio of the maximum measured value to the minimum measured value is up to 212 = 4096.

Metrology systems and devices may use several basic methods to increase their basic dynamic range. These methods include averaging and other forms of filtering, correction of receivers characteristics, repetition of measurements, nonlinear transformations to avoid saturation, etc. In more advance forms of metrology, such as multiwavelength digital holography, interferometry measurements made at different scales (different wavelengths) can be combined to retain the same low-end resolution while extending the upper end of the dynamic range of measurement by orders of magnitude.

Music
In music, dynamic range describes the difference between the quietest and loudest volume of an instrument, part or piece of music. In modern recording, this range is often limited through dynamic range compression, which allows for louder volume, but can make the recording sound less exciting or live.

The dynamic range of music as normally perceived in a concert hall does not exceed 80 dB, and human speech is normally perceived over a range of about 40 dB.

Photography
Photographers use dynamic range to describe the luminance range of a scene being photographed, or the limits of luminance range that a given digital camera or film can capture, or the opacity range of developed film images, or the reflectance range of images on photographic papers.

The dynamic range of digital photography is comparable to the capabilities of photographic film and both are comparable to the capabilities of the human eye.

There are photographic techniques that support even higher dynamic range.
 * Graduated neutral density filters are used to decrease the dynamic range of scene luminance that can be captured on photographic film (or on the image sensor of a digital camera): The filter is positioned in front of the lens at the time the exposure is made; the top half is dark and the bottom half is clear. The dark area is placed over a scene's high-intensity region, such as the sky. The result is more even exposure in the focal plane, with increased detail in the shadows and low-light areas. Though this does not increase the fixed dynamic range available at the film or sensor, it stretches usable dynamic range in practice.
 * High-dynamic-range imaging overcomes the limited dynamic range of the sensor by selectively combining multiple exposures of the same scene in order to retain detail in light and dark areas. Tone mapping maps the image differently in shadow and highlights in order to better distribute the lighting range across the image. The same approach has been used in chemical photography to capture an extremely wide dynamic range: A three-layer film with each underlying layer at one hundredth (10−2) the sensitivity of the next higher one has, for example, been used to record nuclear-weapons tests.

Consumer-grade image file formats sometimes restrict dynamic range. The most severe dynamic-range limitation in photography may not involve encoding, but rather reproduction to, say, a paper print or computer screen. In that case, not only local tone mapping but also dynamic range adjustment can be effective in revealing detail throughout light and dark areas: The principle is the same as that of dodging and burning (using different lengths of exposures in different areas when making a photographic print) in the chemical darkroom. The principle is also similar to gain riding or automatic level control in audio work, which serves to keep a signal audible in a noisy listening environment and to avoid peak levels that overload the reproducing equipment, or which are unnaturally or uncomfortably loud.

If a camera sensor is incapable of recording the full dynamic range of a scene, high-dynamic-range (HDR) techniques may be used in postprocessing, which generally involve combining multiple exposures using software.

External list

 * Audible dynamic range (online test)

title: Dynamic range


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Echo (mythology)

In Greek mythology, Echo (Ἠχώ, Ēkhō, "echo", from ἦχος (ēchos), "sound" ) was an Oread who resided on Mount Cithaeron. Zeus loved consorting with beautiful nymphs and often visited them on Earth. Eventually, Zeus's wife, Hera, became suspicious, and came from Mount Olympus in an attempt to catch Zeus with the nymphs. Echo, by trying to protect Zeus (as he had ordered her to do), endured Hera's wrath, and Hera made her only able to speak the last words spoken to her. So when Echo met Narcissus and fell in love with him, she was unable to tell him how she felt and was forced to watch him as he fell in love with himself.

Metamorphoses
In Metamorphoses (8 AD), the poet Ovid tells of Juno (Hera in Greek mythology) and the jealousy she felt over her husband Jupiter's (Zeus in Greek mythology) many affairs. Though vigilant, whenever she was about to catch him, Echo distracted her with lengthy conversations. When at last Juno realized the truth, she cursed Echo. From that moment on, the once loquacious nymph could only repeat the most recently spoken words of another person.

Sometime after being cursed, Echo spied a young man, Narcissus, while he was out hunting deer with his companions. She immediately fell in love with him and, infatuated, followed quietly. The more she looked at the young man, the more she longed for him. Though she wished with all her heart to call out to Narcissus, Juno's curse prevented her.

During the hunt, Narcissus became separated from his companions and called out, ‘is anyone there,’ and heard the nymph repeat his words. Startled, Narcissus answered the voice, ‘come here,’ only to be told the same. When Narcissus saw that nobody had emerged from the glade, he concluded that the owner of the voice must be running away from him and called out again. Finally, he shouted, "This way, we must come together." Taking this to be a reciprocation of her love, Echo concurred ecstatically, "We must come together!"

In her delight, Echo rushed to Narcissus ready to throw her arms around her beloved. Narcissus, however, was appalled and, spurning her, exclaimed, ‘Hands off! May I die before you enjoy my body.’ All Echo could whisper in reply was, ‘enjoy my body’ and having done so she fled, scorned, humiliated, and shamed.

Despite the harshness of his rejection, Echo's love for Narcissus only grew. When Narcissus died, wasting away before his own reflection, consumed by a love that could not be, Echo mourned over his body. When Narcissus, looking one last time into the pool uttered, "Oh marvellous boy, I loved you in vain, farewell", Echo too chorused, "Farewell."

Eventually, Echo, too, began to waste away. Her beauty faded, her skin shrivelled, and her bones turned to stone. Today, all that remains of Echo is the sound of her voice.

Daphnis and Chloe

The tale of Daphnis and Chloe is a 2nd-century romance by Greek author Longus. At one point in the novel, Daphnis and Chloe are staring out at the boats gliding across the sea. Chloe, having never heard an echo before, is confused on hearing the fisherman's song repeated in a nearby valley. Daphnis promises to tell her the story of Echo in exchange for ten more kisses.

Daphnis’ rendition differs radically from Ovid's account. According to Daphnis, Echo was raised among the Nymphæ because her mother was a nymph. Her father, however, was merely a man and hence Echo was not herself a nymph but mortal. Echo spent her days dancing with the Nymphæ and singing with the Muses who taught her all manner of musical instruments. Pan then grew angry with her, envious of her musical virtuosity and covetous of her virginity, which she would yield neither to men nor gods. Pan drove the men of the fields mad, and, like wild animals, they tore Echo apart and scattered the still singing fragments of her body across the earth.

Showing favour to the Nymphæ, Gaia hid the shreds of Echo within herself providing shelter for her music and at the Muses’ command, Echo's body will still sing, imitating with perfect likeness the sound of any earthly thing. Daphnis recounts that Pan himself often hears his very own pipes and, giving chase across the mountains, looks in vain for the secret student he can never find.

Other
Both the Homeric and Orphic Hymns to Pan reiterate Longus’ tale of Pan chasing Echo's secret voice across the mountains.

Codex 190 of Photius' Bibliotheca states that Pan's unrequited love for Echo was placed there by Aphrodite, angry at his verdict in a beauty contest.

Nonnus’ Dionysiaca contains a number of references to Echo. In Nonnus’ account, though Pan frequently chased Echo, he never won her affection. Book VI also makes reference to Echo in the context of the Great Deluge. Nonnus states that the waters rose so far that even high on the hills Echo was forced to swim. Having escaped the advances of Pan, she feared now the lust of Poseidon.

Whereas Nonnus is adamant that Pan never wins Echo, in Apuleius' The Golden Ass Pan is described with Echo in his arms, teaching the nymph to repeat all manner of songs. Similarly in the Suda, Echo is described as bearing Pan a child, Iynx. Other fragments mention a second daughter, Iambe.

The Lay of Narcissus

The Lay of Narcissus, one of many titles by which the work is known, is Norman-French verse narrative written towards the end of the 12th century. In the four manuscripts that remain, an unknown author borrows from the Echo and Narcissus of Ovid to create a story better suited to the needs of his time.

This medieval account alters the characters of both Echo and Narcissus. In Ovid's account Echo is a beautiful nymph residing with the Muses, and Narcissus is a haughty prince. In The Lay of Narcissus, Echo is replaced by the princess Dané. Conversely, Narcissus loses the royal status he bore in Ovid's account: in this rendition he is no more than a commoner, a vassal of Dané's father, the King.

In the Lay, Dané is pierced by the arrows of Amor and falls madly in love with Narcissus. Though aware that she should first consult her father, she nonetheless shares her feelings with Narcissus. Despite her emphasising her royal lineage, Narcissus spurns her just as he spurns and flees from all women.

Humiliated, Dané calls out to Amor, and, in response, the god curses Narcissus. In a classic example of poetic justice, Narcissus is forced to suffer the same pain he inflicted on others, namely the pain of unrequited love. The vehicle of this justice is a pool of water in which Narcissus falls in love with his own reflection, which he at first mistakes for a woman. Deranged by lust, Dané searches for Narcissus, naked but for a cloak, and finds him at the point of death. Devastated, Dané repents ever calling to Amor. Dané expresses her love for the last time, pulls close to her beloved and dies in his arms. The poet warns men and women alike not to disdain suitors lest they suffer a similar fate.

While Ovid's story is still recognisable, many of the details have changed considerably. Almost all references to pagan deities are gone, save Amor who is little more than a personification of love. Narcissus is demoted to the status of a commoner while Echo is elevated to the status of princess. Allusions to Narcissus’ homosexuality are expunged. While Ovid talks of Narcissus' disdain for both male and female suitors, the Lay only mentions his hatred of women. Similarly, in the Lay, Narcissus mistakes his reflection for that of a woman, whereas no mention is made of this in Ovid's account. Finally, the tale is overtly moralized with messages about courtly love. Such exhortations were entirely absent from the Metamorphoses rendition.

The Romance of the Rose

The Romance of the Rose is a medieval French poem, the first section of which was written by Guillaume de Lorris in around 1230. The poem was completed by Jean de Meun in around 1275. Part of a much larger narrative, the tale of Echo and Narcissus is relayed when the central figure stumbles across the pool wherein Narcissus first glimpsed his own reflection.

In this rendition, Echo is not a nymph, or a princess, but a noble lady. She fell madly in love with Narcissus, so much so that she declared that she would die should he fail to love her in turn. Narcissus refuses, not because he despises all women, but merely because he is haughty and excessively proud of his own beauty.

Guillaume relays that on hearing Narcissus’ rejection, Echo's grief and anger were so great that she died at once. However, in a similar vein to the Lay of Narcissus, just before she dies, Echo calls out to Deus. She asks that Narcissus might one day be tormented by unrequited love as she had been, and, in so doing, understand how the spurned suffer.

As in the classical myth, Narcissus comes across a pool following a hunt. Though Echo prayed to Deus, and the tale notes that he answered her prayer, it is Amor who waits for Narcissus by the water. Amor causes Narcissus to fall for his own reflection, leading quickly to his death. The tale makes clear that this is not merely justice for Echo, but also punishment for Narcissus’ slight against love itself.

The tale concludes with an exhortation to all men warning them that, should they scorn their lovers, God will repay the offence.

Guillaume's rendition builds on the themes of courtly love emphasised in the Lay and moves further away from Ovid's initial account. The curse of Athena is absent entirely, and the tale is overtly moralised. Unlike in the Lay, however, this moral message is aimed solely at women; this despite the fact that the offending behaviour is perpetrated by Narcissus not Echo.

title: Echo (mythology)


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Effective data transfer rate

In telecommunication, effective data transfer rate is the average number of units of data, such as bits, characters, blocks, or frames, transferred per unit time from a source and accepted as valid by a sink.

Note: The effective data transfer rate is usually expressed in bits, characters, blocks, or frames per second. The effective data transfer rate may be averaged over a period of seconds, minutes, or hours.

title: Effective data transfer rate


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Effective height

In telecommunication, the effective height of an antenna is the height of the antenna's center of radiation above the ground.

In low-frequency applications involving loaded or nonloaded vertical antennas, the effective height is the moment of the current distribution in the vertical section, divided by the input current. For an antenna with a symmetrical current distribution, the center of radiation is the center of the distribution. For an antenna with asymmetrical current distribution, the center of radiation is the center of current moments when viewed from points near the direction of maximum radiation.

In antenna theory, it is often used interchangeably with antenna effective length, where it is defined as the ratio of the open circuit voltage across the antenna's terminals to the incident electric field of a radio signal.

title: Effective height


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Effective input noise temperature

In telecommunications, effective input noise temperature is the source noise temperature in a two-port network or amplifier that will result in the same output noise power, when connected to a noise-free network or amplifier, as that of the actual network or amplifier connected to a noise-free source. If F is the noise factor numeric and 290K the standard noise temperature, then the effective noise temperature is given by T n = 290(F − 1).

title: Effective input noise temperature


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Effective mode volume

For an optical fiber, the effective mode volume is the square of the product of the diameter of the near-field pattern and the sine of the radiation angle of the far-field pattern. The diameter of the near-field radiation pattern is defined here as the full width at half maximum and the radiation angle at half maximum radiant intensity. Effective mode volume is proportional to the breadth of the relative distribution of power amongst the modes in a multimode fiber. It is not truly a spatial volume but rather an "optical volume" equal to the product of area and solid angle. The power divided by the effective mode volume is proportional to the radiance of the light emitted by the fiber.

title: Effective mode volume


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Effective power

Effective power may refer to:

 * Active power or real power, a concept in AC power
 * A measurement of horsepower
 * "effective power" (bug), a bug with the iPhone's SpringBoard app

title: Effective power


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Effective transmission rate

In telecommunications, effective transmission rate (average rate of transmission, effective speed of transmission) is the rate at which information is processed by a transmission facility.

 * The effective transmission rate is calculated as (a) the measured number of units of data, such as bits, characters, blocks, or frames, transmitted during a significant measurement time interval divided by (b) the measurement time interval.
 * The effective transmission rate is usually expressed as a number of units of data per unit time, such as bits per second or characters per second.

title: Effective transmission rate


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Efficiency factor

Efficiency factor is a ratio of some measure of performance to an expected value.

Data communication
In data communications, the factor is the ratio of the time to transmit a text automatically at a specified modulation rate to the time actually required to receive the same text at a specified maximum error rate. All of the communication facilities are assumed to be in the normal condition of adjustment and operation. The practical conditions of measurement should be specified, especially the duration of the measurement.

Telegraph communications may have different temporal efficiency factors for the two directions of transmission.

Industrial engineering
In industrial engineering, the efficiency factor is the relationship between the allowance time and the time taken, in the form of percentage.

Efficiency factors are used in performance rating and remuneration calculation exercises. The efficiency factor is an extremely simple to use and readily comprehensible index, the prerequisite being exact time management for maintaining the allowed times.

title: Efficiency factor


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Electrical length

In electrical engineering, electrical length is a dimensionless parameter equal to the physical length of an electrical conductor such as a cable or wire, divided by the wavelength of alternating current at a given frequency traveling through the conductor. In other words, it is the length of the conductor measured in wavelengths. It can alternately be expressed as an angle, in radians or degrees, equal to the phase shift the alternating current experiences traveling through the conductor.

Electrical length is defined for a conductor operating at a specific frequency or narrow band of frequencies. It is determined by the construction of the cable, so different cables of the same length operating at the same frequency can have different electrical lengths. A conductor is called electrically long if it has an electrical length much greater than one; that is it is much longer than the wavelength of the alternating current passing through it, and electrically short if it is much shorter than a wavelength. Electrical lengthening and electrical shortening means adding reactance (capacitance or inductance) to an antenna or conductor to increase or decrease the electrical length, usually for the purpose of making it resonant at a different resonant frequency.

This concept is used throughout electronics, and particularly in radio frequency circuit design, transmission line and antenna theory and design. Electrical length determines when wave effects (phase shift along conductors) become important in a circuit. Ordinary lumped element electric circuits only work well for alternating currents at frequencies for which the circuit is electrically small (electrical length much less than one). For frequencies high enough that the wavelength approaches the size of the circuit (the electrical length approaches one) the lumped element model on which circuit theory is based becomes inaccurate, and transmission line techniques must be used.

Definition
Electrical length is defined for conductors carrying alternating current (AC) at a single frequency or narrow band of frequencies. An alternating electric current of a single frequency $$f$$ is an oscillating sine wave which repeats with a period of $$T=1/f$$. This current flows through a given conductor such as a wire or cable at a particular phase velocity $$v_p$$. It takes time for later portions of the wave to reach a given point on the conductor so the spatial distribution of current and voltage along the conductor at any time is a moving sine wave. After a time equal to the period $$T$$ a complete cycle of the wave has passed a given point and the wave repeats; during this time a point of constant phase on the wave has traveled a distance of
 * $$\lambda = v_p T = v_p/f$$

so $$\lambda$$ (Greek lambda) is the wavelength of the wave along the conductor, the distance between successive crests of the wave.

The electrical length $$G$$ of a conductor with a physical length of $$l$$ at a given frequency $$f$$ is the number of wavelengths or fractions of a wavelength of the wave along the conductor; in other words the conductor's length measured in wavelengths $$ The phase velocity $$v_p$$ at which electrical signals travel along a transmission line or other cable depends on the construction of the line. Therefore, the wavelength $$\lambda$$ corresponding to a given frequency varies in different types of lines, thus at a given frequency different conductors of the same physical length can have different electrical lengths.

Phase shift definition
In radio frequency applications, when a delay is introduced due to a conductor, it is often the phase shift $$\phi$$, the difference in phase of the sinusoidal wave between the two ends of the conductor, that is of importance. The length of a sinusoidal wave is commonly expressed as an angle, in units of degrees (with 360° in a wavelength) or radians (with 2π radians in a wavelength). So alternately the electrical length can be expressed as an angle which is the phase shift of the wave between the ends of the conductor
 * $$\phi = 360^\circ{l \over \lambda} \, \text{degrees}$$
 * $$\quad = 2\pi{l \over \lambda} \, \text{radians}$$

Significance
The electrical length of a conductor determines when wave effects (phase shift along the conductor) are important. If the electrical length $$G$$ is much less than one, that is the physical length of a conductor is much shorter than the wavelength, say less than one tenth of the wavelength ($$l < \lambda/10$$) it is called electrically short. In this case the voltage and current are approximately constant along the conductor, so it acts as a simple connector which transfers alternating current with negligible phase shift. In circuit theory the connecting wires between components are usually assumed to be electrically short, so the lumped element circuit model is only valid for alternating current when the circuit is electrically small, much smaller than a wavelength. When the electrical length approaches or is greater than one, a conductor will have significant reactance, inductance or capacitance, depending on its length. So simple circuit theory is inadequate and transmission line techniques (the distributed-element model) must be used.

Velocity factor
In a vacuum an electromagnetic wave (radio wave) travels at the speed of light $$v_p = c = $$ 2.9979×108 meters per second, and very close to this speed in air, so the free space wavelength of the wave is $$\lambda_\text{0} = c/f$$. (in this article free space variables are distinguished by a subscript 0) Thus a physical length $$l$$ of a radio wave in space or air has an electrical length of
 * $$G_\text{0} = {l \over \lambda_\text{0}} = {lf \over c}$$ wavelengths.

In the SI system of units, empty space has a permittivity of $$\epsilon_\text{0} =$$ 8.854×10−12 F/m (farads per metre) and a magnetic permeability of $$\mu_\text{0} =$$ 1.257×10−6 H/m (henries per meter). These universal constants determine the speed of light
 * $$c = {1 \over \sqrt{\epsilon_\text{0}\mu_\text{0}}}$$

In an electrical cable, for a cycle of the alternating current to move a given distance along the line, it takes time to charge the capacitance between the conductors, and the rate of change of the current is slowed by the series inductance of the wires. This determines the phase velocity $$v_p$$ at which the wave moves along the line. In cables and transmission lines an electrical signal travels at a rate determined by the effective shunt capacitance $$C$$ and series inductance $$L$$ per unit length of the transmission line
 * $$v_p = {1 \over \sqrt{LC} }$$

Some transmission lines consist only of bare metal conductors, if they are far away from other high permittivity materials their signals propagate at very close to the speed of light, $$c$$. In most transmission lines the material construction of the line slows the velocity of the signal so it travels at a reduced phase velocity
 * $$v_p = \kappa c$$

where $$\kappa$$ (kappa) is a dimensionless number between 0 and 1 called the velocity factor (VF), characteristic of the type of line, equal to the ratio of signal velocity in the line to the speed of light.

Most transmission lines contain a dielectric material (insulator) filling some or all of the space in between the conductors. The permittivity $$\epsilon$$ or dielectric constant of that material increases the distributed capacitance $$C$$ in the cable, which reduces the velocity factor below unity. If there is a material with high magnetic permeability ($$\mu$$) in the line such as steel or ferrite which increases the distributed inductance $$L$$, it can also reduce $$\kappa$$, but this is almost never the case. If all the space around the transmission line conductors containing the near fields was filled with a material of permittivity $$\epsilon$$ and permeability $$\mu$$, the phase velocity on the line would be $$ The effective permittivity $$\epsilon$$ and permeability $$\mu$$ per unit length of the line are frequently given as dimensionless constants; relative permittivity: $$\epsilon_\text{r}$$ and relative permeability: $$\mu_\text{r}$$ equal to the ratio of these parameters compared to the universal constants $$\epsilon_\text{0}$$ and $$\mu_\text{0}$$
 * $$\epsilon_\text{r} = {\epsilon \over \epsilon_\text{0}} \qquad \mu_\text{r} = {\mu \over \mu_\text{0}}$$

so the phase velocity is
 * $$v_\text{p} = {1 \over \sqrt{\epsilon\mu}} = {1 \over \sqrt{\epsilon_\text{0}\epsilon_\text{r}\mu_\text{0}\mu_\text{r}}} = c{1 \over \sqrt{\epsilon_\text{r}\mu_\text{r}}}$$

So the velocity factor of the line is
 * $$\kappa = {v_p \over c} = {1 \over \sqrt{\epsilon_\text{r}\mu_\text{r}}}$$

In many lines, for example twin lead, only a fraction of the space surrounding the line containing the fields is occupied by a solid dielectric. With only part of the electromagnetic field effected by the dielectric, there is less reduction of the wave velocity. In this case an effective permittivity $$\epsilon_\text{eff}$$ can be calculated which if it filled all the space around the line would give the same phase velocity. This is computed as a weighted average of the relative permittivity of free space, unity, and that of the dielectric: $$\epsilon_\text{eff}= (1-F) + F \epsilon_\text{r}$$ where the fill factor $F$ expresses the effective proportion of space around the line occupied by dielectric.

In most transmission lines there are no materials with high magnetic permeability, so $$\mu = \mu_\text{0}$$ and $$\mu_\text{r} = 1$$ and so $$ Since the electromagnetic waves travel slower in the line than in free space, the wavelength of the wave in the transmission line $$\lambda$$ is shorter than the free space wavelength by the factor kappa: $$\lambda = v_\text{p}/f = \kappa c/f = \kappa\lambda_\text{0}$$. Therefore, more wavelengths fit in a transmission line of a given length $$l$$ than in the same length of wave in free space, so the electrical length of a transmission line is longer than the electrical length of a wave of the same frequency in free space $$

Transmission lines
Ordinary electrical cable suffices to carry alternating current when the cable is electrically short; the electrical length of the cable is small compared to one, that is when the physical length of the cable is small compared to a wavelength, say $$l < \lambda/10$$.

As frequency gets high enough that the length of the cable becomes a significant fraction of a wavelength, $$l > \lambda/10$$, ordinary wires and cables become poor conductors of AC. Impedance discontinuities at the source, load, connectors and switches begin to reflect the electromagnetic current waves back toward the source, creating bottlenecks so not all the power reaches the load. Ordinary wires act as antennas, radiating the power into space as radio waves, and in radio receivers can also pick up radio frequency interference (RFI).

To mitigate these problems, at these frequencies transmission line is used instead. A transmission line is a specialized cable designed for carrying electric current of radio frequency. The distinguishing feature of a transmission line is that it is constructed to have a constant characteristic impedance along its length and through connectors and switches, to prevent reflections. This also means AC current travels at a constant phase velocity along its length, while in ordinary cable phase velocity may vary. The velocity factor $$\kappa$$ depends on the details of construction, and is different for each type of transmission line. However the approximate velocity factor for the major types of transmission lines is given in the table.

Electrical length is widely used with a graphical aid called the Smith chart to solve transmission line calculations. A Smith chart has a scale around the circumference of the circular chart graduated in wavelengths and degrees, which represents the electrical length of the transmission line from the point of measurement to the source or load.

The equation for the voltage as a function of time along a transmission line with a matched load, so there is no reflected power, is
 * $$v(x, t) = V_\text{p} \cos(\omega t - \beta x)$$

where
 * $$V_\text{p}$$ is the peak voltage along the line
 * $$\omega = 2\pi f = 2\pi/T$$ is the angular frequency of the alternating current in radians per second
 * $$\beta = 2\pi/\lambda$$ is the wavenumber, equal to the number of radians of the wave in one meter
 * $$x$$ is the distance along the line
 * $$t$$ is time

In a matched transmission line, the current is in phase with the voltage, and their ratio is the characteristic impedance $$Z_\text{0}$$ of the line
 * $$i(x, t) = {v(x, t) \over Z_\text{0}} = {V_\text{p} \over Z_\text{0}} \cos(\omega t - \beta x) = {V_\text{p} \over Z_\text{0}} \cos \omega(t - x/\kappa c)$$

Antennas

An important class of radio antenna is the thin element antenna in which the radiating elements are conductive wires or rods. These include monopole antennas and dipole antennas, as well as antennas based on them such as the whip antenna, T antenna, mast radiator, Yagi, log periodic, and turnstile antennas. These are resonant antennas, in which the radio frequency electric currents travel back and forth in the antenna conductors, reflecting from the ends.

If the antenna rods are not too thick (have a large enough length to diameter ratio), the current along them is close to a sine wave, so the concept of electrical length also applies to these. The current is in the form of two oppositely directed sinusoidal traveling waves which reflect from the ends, which interfere to form standing waves. The electrical length of an antenna, like a transmission line, is its length in wavelengths of the current on the antenna at the operating frequency. An antenna's resonant frequency, radiation pattern, and driving point impedance depend not on its physical length but on its electrical length. A thin antenna element is resonant at frequencies at which the standing current wave has a node (zero) at the ends (and in monopoles an antinode (maximum) at the ground plane). A dipole antenna is resonant at frequencies at which its electrical length is a half wavelength ($$\lambda/2, \phi = 180^\circ \;\text{or}\; \pi \;\text{radians}$$) or a multiple of it. A monopole antenna is resonant at frequencies at which its electrical length is a quarter wavelength ($$\lambda/4, \phi = 90^\circ \;\text{or}\; \pi/2 \;\text{radians}$$) or a multiple of it.

Resonant frequency is important because at frequencies at which the antenna is resonant the input impedance it presents to its feedline is purely resistive. If the resistance of the antenna is matched to the characteristic resistance of the feedline, it absorbs all the power supplied to it, while at other frequencies it has reactance and reflects some power back down the line toward the transmitter, causing standing waves (high SWR) on the feedline. Since only a portion of the power is radiated this causes inefficiency, and can possibly overheat the line or transmitter. Therefore, transmitting antennas are usually designed to be resonant at the transmitting frequency; and if they cannot be made the right length they are electrically lengthened or shortened to be resonant (see below).

End effects
A thin-element antenna can be thought of as a transmission line with the conductors separated,  so the near-field electric and magnetic fields extend further into space than in a transmission line, in which the fields are mainly confined to the vicinity of the conductors. Near the ends of the antenna elements the electric field is not perpendicular to the conductor axis as in a transmission line but spreads out in a fan shape (fringing field). As a result, the end sections of the antenna have increased capacitance, storing more charge, so the current waveform departs from a sine wave there, decreasing faster toward the ends. When approximated as a sine wave, the current does not quite go to zero at the ends; the nodes of the current standing wave, instead of being at the ends of the element, occur somewhat beyond the ends. Thus the electrical length of the antenna is longer than its physical length.

The electrical length of an antenna element also depends on the length-to-diameter ratio of the conductor. As the ratio of the diameter to wavelength increases, the capacitance increases, so the node occurs farther beyond the end, and the electrical length of the element increases. When the elements get too thick, the current waveform becomes significantly different from a sine wave, so the entire concept of electrical length is no longer applicable, and the behavior of the antenna must be calculated by electromagnetic simulation computer programs like NEC.

As with a transmission line, an antenna's electrical length is increased by anything that adds shunt capacitance or series inductance to it, such as the presence of high permittivity dielectric material around it. In microstrip antennas which are fabricated as metal strips on printed circuit boards, the dielectric constant of the substrate board increases the electrical length of the antenna. Proximity to the Earth or a ground plane, a dielectric coating on the conductor, nearby grounded towers, metal structural members, guy lines and the capacitance of insulators supporting the antenna also increase the electrical length.

These factors, called "end effects", cause the electrical length of an antenna element to be somewhat longer than the length of the same wave in free space. In other words, the physical length of the antenna at resonance will be somewhat shorter than the resonant length in free space (one-half wavelength for a dipole, one-quarter wavelength for a monopole). As a rough generalization, for a typical dipole antenna, the physical resonant length is about 5% shorter than the free space resonant length.

Electrical lengthening and shortening
In many circumstances for practical reasons it is inconvenient or impossible to use an antenna of resonant length. An antenna of nonresonant length at the operating frequency can be made resonant by adding a reactance, a capacitance or inductance, either in the antenna itself or in a matching network between the antenna and its feedline. A nonresonant antenna appears at its feedpoint electrically equivalent to a resistance in series with a reactance. Adding an equal but opposite type of reactance in series with the feedline will cancel the antenna's reactance; the combination of the antenna and reactance will act as a series resonant circuit, so at its operating frequency its input impedance will be purely resistive, allowing it to be fed power efficiently at a low SWR without reflections.

In a common application, an antenna which is electrically short, shorter than its fundamental resonant length, a monopole antenna with an electrical length shorter than a quarter-wavelength ($$\lambda/4$$), or a dipole antenna shorter than a half-wavelength ($$\lambda/2$$) will have capacitive reactance. Adding an inductor (coil of wire), called a loading coil, at the feedpoint in series with the antenna, with inductive reactance equal to the antenna's capacitive reactance at the operating frequency, will cancel the capacitance of the antenna, so the combination of the antenna and coil will be resonant at the operating frequency. Since adding inductance is equivalent to increasing the electrical length, this technique is called electrically lengthening the antenna. This is the usual technique for matching an electrically short transmitting antenna to its feedline, so it can be fed power efficiently. However, an electrically short antenna that has been loaded in this way still has the same radiation pattern; it does not radiate as much power, and therefore has lower gain than a full-sized antenna.

Conversely, an antenna longer than resonant length at its operating frequency, such as a monopole longer than a quarter wavelength but shorter than a half wavelength, will have inductive reactance. This can be cancelled by adding a capacitor of equal but opposite reactance at the feed point to make the antenna resonant. This is called electrically shortening the antenna.

Scaling properties of antennas
Two antennas that are similar (scaled copies of each other), fed with different frequencies, will have the same radiation resistance and radiation pattern and fed with equal power will radiate the same power density in any direction if they have the same electrical length at the operating frequency; that is, if their lengths are in the same proportion as the wavelengths.
 * $${l_\text{1} \over l_\text{2}} = {\lambda_\text{1} \over \lambda_\text{2}} = {f_\text{2} \over f_\text{1}}$$

This means the length of antenna required for a given antenna gain scales with the wavelength (inversely with the frequency), or equivalently the aperture scales with the square of the wavelength.

Electrically short antennas
An electrically short conductor, much shorter than one wavelength, makes an inefficient radiator of electromagnetic waves. As the length of an antenna is made shorter than its fundamental resonant length (a half-wavelength for a dipole antenna and a quarter-wavelength for a monopole), the radiation resistance the antenna presents to the feedline decreases with the square of the electrical length, that is the ratio of physical length to wavelength, $$(l/\lambda)^2$$. As a result, other resistances in the antenna, the ohmic resistance of metal antenna elements, the ground system if present, and the loading coil, dissipate an increasing fraction of transmitter power as heat. A monopole antenna with an electrical length below .05$$\lambda$$ or 18° has a radiation resistance of less than one ohm, making it very hard to drive.

A second disadvantage is that since the capacitive reactance of the antenna and inductive reactance of the required loading coil do not decrease, the Q factor of the antenna increases; it acts electrically like a high Q tuned circuit. As a result, the bandwidth of the antenna decreases with the square of electrical length, reducing the data rate that can be transmitted. At VLF frequencies even the huge toploaded wire antennas that must be used have bandwidths of only ~10 hertz, limiting the data rate that can be transmitted.

Regimes of electromagnetics
The field of electromagnetics is the study of electric fields, magnetic fields, electric charge, electric currents and electromagnetic waves. Classic electromagnetism is based on the solution of Maxwell's equations. These equations are mathematically difficult to solve in all generality, so approximate methods have been developed that apply to situations in which the electrical length of the apparatus is very short ($$G \ll 1$$) or very long ($$G \gg 1$$). Electromagnetics is divided into three regimes or fields of study depending on the electrical length of the apparatus, that is the physical length $$l$$ of the apparatus compared to the wavelength $$\lambda = c/f$$ of the waves:    Completely different apparatus is used to conduct and process electromagnetic waves in these different wavelength ranges Historically, electric circuit theory and optics developed as separate branches of physics until at the end of the 19th century James Clerk Maxwell's electromagnetic theory and Heinrich Hertz's discovery that light was electromagnetic waves unified these fields as branches of electromagnetism.
 * $$\lambda \gg l$$ Circuit theory: When the wavelength of the electrical oscillations is much larger than the physical size of the circuit ($$ G \ll 1$$), say $$\lambda > 50l$$, the action occurs in the near field.  The phase of the oscillations and therefore the current and voltage can be approximated as constant along the length of connecting wires.  Also little energy is radiated in the form of electromagnetic waves, the power radiated by a conductor as an antenna is proportional to the electrical length squared $$(l/\lambda)^2 = G^2$$.   So the electrical energy remains in the wires and components as quasistatic near-field electric and magnetic fields.  Therefore, the approximation of the lumped element model can be used, and electric currents  oscillating at these frequencies can be processed by electric circuits consisting of lumped circuit elements such as resistors, capacitors, inductors, transformers, transistors, and integrated circuits linked by ordinary wires.  Mathematically Maxwell's equations reduce to circuit theory (Kirchhoff's circuit laws).
 * $$\lambda \approx l$$, Distributed-element model (microwave theory): When the wavelength of the waves is of the same order of magnitude as the size of the equipment ($$ G \approx 1$$), as it is in the microwave part of the spectrum, full solutions of Maxwell's equations must be used.  At these frequencies, wires are replaced by transmission lines and waveguide and lumped elements are replaced by resonant stubs, irises, and cavity resonators.  Often only a single mode (wave pattern) is propagating through the apparatus, which simplifies the mathematics.  A modification of circuit theory called the distributed-element model can often be used, in which extended objects are regarded as electrical circuits with capacitance, inductance and resistance distributed along their length.   A graphical aid called the Smith chart is often used to analyze transmission lines.
 * $$\lambda \ll l$$, Optics: When the wavelength of the electromagnetic wave is much smaller than the physical size of the equipment that manipulates it ($$ G \gg 1$$), say $$\lambda < l/50$$, most of the path of the waves is in the far field. In the far field, the electric and magnetic fields cannot be separated but propagate together as an electromagnetic wave.  Unlike in the case of microwaves, unless coherent light sources like lasers are used, the number of modes propagating is usually large. Since little of the energy is stored in the quasistatic (induction) electric or magnetic fields at the surface boundaries between media (called evanescent fields in optics), the concepts of voltage, current, capacitance, and inductance have little meaning and are not used, and the medium is characterized by its index of refraction $$\nu = c/v_\text{p} = \sqrt{\epsilon_\text{r}\mu_\text{r}}$$, absorption, permittivity $$\epsilon$$, permeability $$\mu$$, and dispersion.  At these frequencies electromagnetic waves are manipulated by optical elements such as lenses, mirrors, prisms, optical filters and diffraction gratings.   Maxwell's equations can be approximated by the equations of geometrical optics or physical optics.

title: Electrical length


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Electric field

An electric field (sometimes called E-field ) is the physical field that surrounds electrically charged particles. Charged particles exert attractive forces on each other when their charges are opposite, and repulse each other when their charges are the same. Because these forces are exerted mutually, two charges must be present for the forces to take place. The electric field of a single charge (or group of charges) describes their capacity to exert such forces on another charged object. These forces are described by Coulomb's law, which says that the greater the magnitude of the charges, the greater the force, and the greater the distance between them, the weaker the force. Thus, we may informally say that the greater the charge of an object, the stronger its electric field. Similarly, an electric field is stronger nearer charged objects and weaker further away. Electric fields originate from electric charges and time-varying electric currents. Electric fields and magnetic fields are both manifestations of the electromagnetic field, Electromagnetism is one of the four fundamental interactions of nature.

Electric fields are important in many areas of physics, and are exploited in electrical technology. For example, in atomic physics and chemistry, the interaction in the electric field between the atomic nucleus and electrons is the force that holds these particles together in atoms. Similarly, the interaction in the electric field between atoms is the force responsible for chemical bonding that result in molecules.

The electric field is defined as a vector field that associates to each point in space the force per unit of charge exerted on an infinitesimal test charge at rest at that point. The SI unit for the electric field is the volt per meter (V/m), which is equal to the newton per coulomb (N/C).

Description
The electric field is defined at each point in space as the force that would be experienced by a infinitesimally small stationary test charge at that point divided by the charge. The electric field is defined in terms of force, and force is a vector (i.e. having both magnitude and direction), so it follows that an electric field may be described by a vector field. The electric field acts between two charges similarly to the way that the gravitational field acts between two masses, as they both obey an inverse-square law with distance. This is the basis for Coulomb's law, which states that, for stationary charges, the electric field varies with the source charge and varies inversely with the square of the distance from the source. This means that if the source charge were doubled, the electric field would double, and if you move twice as far away from the source, the field at that point would be only one-quarter its original strength.

The electric field can be visualized with a set of lines whose direction at each point is the same as those of the field, a concept introduced by Michael Faraday, whose term 'lines of force' is still sometimes used. This illustration has the useful property that the strength of the field is proportional to the density of the lines. Field lines due to stationary charges have several important properties, including always originating from positive charges and terminating at negative charges, they enter all good conductors at right angles, and they never cross or close in on themselves. The field lines are a representative concept; the field actually permeates all the intervening space between the lines. More or fewer lines may be drawn depending on the precision to which it is desired to represent the field. The study of electric fields created by stationary charges is called electrostatics.

Faraday's law describes the relationship between a time-varying magnetic field and the electric field. One way of stating Faraday's law is that the curl of the electric field is equal to the negative time derivative of the magnetic field. In the absence of time-varying magnetic field, the electric field is therefore called conservative (i.e. curl-free). This implies there are two kinds of electric fields: electrostatic fields and fields arising from time-varying magnetic fields. While the curl-free nature of the static electric field allows for a simpler treatment using electrostatics, time-varying magnetic fields are generally treated as a component of a unified electromagnetic field. The study of magnetic and electric fields that change over time is called electrodynamics.

Mathematical formulation
Electric fields are caused by electric charges, described by Gauss's law, and time varying magnetic fields, described by Faraday's law of induction. Together, these laws are enough to define the behavior of the electric field. However, since the magnetic field is described as a function of electric field, the equations of both fields are coupled and together form Maxwell's equations that describe both fields as a function of charges and currents.

Electrostatics
In the special case of a steady state (stationary charges and currents), the Maxwell-Faraday inductive effect disappears. The resulting two equations (Gauss's law $$\nabla \cdot \mathbf{E} = \frac{\rho}{\varepsilon_0}$$ and Faraday's law with no induction term $$\nabla \times \mathbf{E} = 0$$), taken together, are equivalent to Coulomb's law, which states that a particle with electric charge $$q_1$$ at position $$\mathbf r_1$$ exerts a force on a particle with charge $$q_0$$ at position $$\mathbf r_0$$ of: $$\mathbf{F}_{01} = \frac{q_1q_0}{4\pi\varepsilon_0} {\hat\mathbf r_{01}\over {|\mathbf r_{01}|}^2} = \frac{q_1q_0}{4\pi\varepsilon_0} {\mathbf r_{01}\over {|\mathbf r_{01}|}^3}$$ where
 * $$ \mathbf{F}_{01} $$ is the force on charged particle $$ q_0 $$ caused by charged particle $$ q_1 $$.
 * $E$ is the permittivity of free space.
 * $$ \hat \mathbf{r}_{01} $$ is a unit vector directed from $$ \mathbf r_1 $$ to $$ \mathbf r_0 $$.
 * $$ \mathbf{r}_{01} $$ is the displacement vector from $$ \mathbf r_1 $$ to $$ \mathbf r_0 $$.

Note that $$\varepsilon_0$$ must be replaced with $$\varepsilon$$, permittivity, when charges are in non-empty media. When the charges $$q_0$$ and $$q_1$$ have the same sign this force is positive, directed away from the other charge, indicating the particles repel each other. When the charges have unlike signs the force is negative, indicating the particles attract. To make it easy to calculate the Coulomb force on any charge at position $$\mathbf r_0$$ this expression can be divided by $$q_0$$ leaving an expression that only depends on the other charge (the source charge) $$\mathbf{E}_{1} (\mathbf r_0) = \frac{ \mathbf{F}_{01} } {q_0} = \frac{q_1}{4\pi\varepsilon_0} {\hat\mathbf r_{01}\over {|\mathbf r_{01}|}^2} = \frac{q_1}{4\pi\varepsilon_0} {\mathbf r_{01}\over {|\mathbf r_{01}|}^3}$$ where
 * $$\mathbf{E}_{1} (\mathbf r_0) $$ is the component of the electric field at $$ q_0 $$ due to $$ q_1 $$.

This is the electric field at point $$\mathbf r_0$$ due to the point charge $$q_1$$; it is a vector-valued function equal to the Coulomb force per unit charge that a positive point charge would experience at the position $$\mathbf r_0$$. Since this formula gives the electric field magnitude and direction at any point $$\mathbf r_0$$ in space (except at the location of the charge itself, $$\mathbf r_1$$, where it becomes infinite) it defines a vector field. From the above formula it can be seen that the electric field due to a point charge is everywhere directed away from the charge if it is positive, and toward the charge if it is negative, and its magnitude decreases with the inverse square of the distance from the charge.

The Coulomb force on a charge of magnitude $$q$$ at any point in space is equal to the product of the charge and the electric field at that point $$\mathbf{F} = q\mathbf{E} .$$ The SI unit of the electric field is the newton per coulomb (N/C), or volt per meter (V/m); in terms of the SI base units it is kg⋅m⋅s−3⋅A−1.

Superposition principle
Due to the linearity of Maxwell's equations, electric fields satisfy the superposition principle, which states that the total electric field, at a point, due to a collection of charges is equal to the vector sum of the electric fields at that point due to the individual charges. This principle is useful in calculating the field created by multiple point charges. If charges $$q_1, q_2, \dots, q_n$$ are stationary in space at points $$\mathbf r_1,\mathbf r_2,\dots,\mathbf r_n$$, in the absence of currents, the superposition principle says that the resulting field is the sum of fields generated by each particle as described by Coulomb's law: $$\begin{align} \mathbf{E}(\mathbf{r}) = \mathbf{E}_1(\mathbf{r}) + \mathbf{E}_2(\mathbf{r}) + \dots + \mathbf{E}_n(\mathbf{r}) = {1\over4\pi\varepsilon_0} \sum_{i=1}^n q_i {\hat\mathbf r_i\over {|\mathbf r_i|}^2} = {1\over 4\pi\varepsilon_0} \sum_{i=1}^n q_i {\mathbf r_i\over {|\mathbf r_i|}^3} \end{align}$$ where
 * $$\hat\mathbf r_i$$ is the unit vector in the direction from point $$\mathbf r_i$$ to point $$\mathbf r$$
 * $$\mathbf r_i$$ is the displacement vector from point $$\mathbf r_i$$ to point $$\mathbf r$$.

Continuous charge distributions
The superposition principle allows for the calculation of the electric field due to a distribution of charge density $$\rho(\mathbf r)$$. By considering the charge $$\rho(\mathbf r')dv$$ in each small volume of space $$dv$$ at point $$\mathbf r'$$ as a point charge, the resulting electric field, $$d\mathbf{E}(\mathbf r)$$, at point $$\mathbf r$$ can be calculated as $$d\mathbf{E}(\mathbf r) = \frac{\rho(\mathbf r')}{4\pi\varepsilon_0}{\hat\mathbf r'\over {|\mathbf r'|}^2} dv = \frac{\rho(\mathbf r')}{4\pi\varepsilon_0} {\mathbf r'\over {|\mathbf r'|}^3} dv $$ where The total field is found by summing the contributions from all the increments of volume by integrating the charge density over the volume $$V$$: $$\mathbf{E}(\mathbf r) = \frac{1}{4\pi\varepsilon_0} \iiint_V \, \rho(\mathbf r') {\mathbf r'\over {|\mathbf r'|}^3} dv$$
 * $$\hat \mathbf{r}'$$ is the unit vector pointing from $$\mathbf r'$$ to $$\mathbf r$$.
 * $$\mathbf r'$$ is the displacement vector from $$\mathbf r'$$ to $$\mathbf r$$.

Similar equations follow for a surface charge with surface charge density $$\sigma(\mathbf r')$$ on surface $$ S $$ $$\mathbf{E}(\mathbf r) = \frac{1}{4\pi\varepsilon_0} \iint_S \, \sigma(\mathbf r') {\mathbf r'\over {|\mathbf r'|}^3} da,$$ and for line charges with linear charge density $$\lambda(\mathbf r')$$ on line $$ L $$ $$\mathbf{E}(\mathbf r) = \frac{1}{4\pi\varepsilon_0} \int_L \,\lambda(\mathbf r') {\mathbf r'\over {|\mathbf r'|}^3} d \ell.$$

Electric potential
If a system is static, such that magnetic fields are not time-varying, then by Faraday's law, the electric field is curl-free. In this case, one can define an electric potential, that is, a function $$\varphi$$ such that $ \mathbf{E} = -\nabla \varphi $. This is analogous to the gravitational potential. The difference between the electric potential at two points in space is called the potential difference (or voltage) between the two points.

In general, however, the electric field cannot be described independently of the magnetic field. Given the magnetic vector potential, $ε0$, defined so that $\mathbf{B} = \nabla \times \mathbf{A}$, one can still define an electric potential $$ \varphi$$ such that: $$ \mathbf{E} = - \nabla \varphi - \frac { \partial \mathbf{A} } { \partial t } ,$$ where $$\nabla \varphi$$ is the gradient of the electric potential and $$\frac { \partial \mathbf{A} } { \partial t }$$ is the partial derivative of $A$ with respect to time.

Faraday's law of induction can be recovered by taking the curl of that equation $$\nabla \times \mathbf{E} = -\frac{\partial (\nabla \times \mathbf{A})} {\partial t} = -\frac{\partial \mathbf{B}} {\partial t} ,$$ which justifies, a posteriori, the previous form for $A$.

Continuous vs. discrete charge representation
The equations of electromagnetism are best described in a continuous description. However, charges are sometimes best described as discrete points; for example, some models may describe electrons as point sources where charge density is infinite on an infinitesimal section of space.

A charge $$q$$ located at $$\mathbf{r}_0$$ can be described mathematically as a charge density $\rho(\mathbf{r}) = q\delta(\mathbf{r} - \mathbf{r}_0)$, where the Dirac delta function (in three dimensions) is used. Conversely, a charge distribution can be approximated by many small point charges.

Electrostatic fields

Electrostatic fields are electric fields that do not change with time. Such fields are present when systems of charged matter are stationary, or when electric currents are unchanging. In that case, Coulomb's law fully describes the field.

Parallels between electrostatic and gravitational fields
Coulomb's law, which describes the interaction of electric charges: $$\mathbf{F} = q \left(\frac{Q}{4\pi\varepsilon_0} \frac{\mathbf{\hat{r}}}{|\mathbf{r}|^2}\right) = q \mathbf{E}$$ is similar to Newton's law of universal gravitation: $$\mathbf{F} = m\left(-GM\frac{\mathbf{\hat{r}}}{|\mathbf{r}|^2}\right) = m\mathbf{g}$$ (where $\mathbf{\hat{r}} = \mathbf{\frac{r}{|r|}}$ ).

This suggests similarities between the electric field $E$ and the gravitational field $E$, or their associated potentials. Mass is sometimes called "gravitational charge".

Electrostatic and gravitational forces both are central, conservative and obey an inverse-square law.

Uniform fields
A uniform field is one in which the electric field is constant at every point. It can be approximated by placing two conducting plates parallel to each other and maintaining a voltage (potential difference) between them; it is only an approximation because of boundary effects (near the edge of the planes, the electric field is distorted because the plane does not continue). Assuming infinite planes, the magnitude of the electric field $g$ is: $$ E = - \frac{\Delta V}{d} ,$$ where $E$ is the potential difference between the plates and $ΔV$ is the distance separating the plates. The negative sign arises as positive charges repel, so a positive charge will experience a force away from the positively charged plate, in the opposite direction to that in which the voltage increases. In micro- and nano-applications, for instance in relation to semiconductors, a typical magnitude of an electric field is in the order of $V⋅m^{−1}$, achieved by applying a voltage of the order of 1 volt between conductors spaced 1 μm apart.

Electromagnetic fields

Electromagnetic fields are electric and magnetic fields, which may change with time, for instance when charges are in motion. Moving charges produce a magnetic field in accordance with Ampère's circuital law (with Maxwell's addition), which, along with Maxwell's other equations, defines the magnetic field, $$\mathbf{B}$$, in terms of its curl: $$\nabla \times \mathbf{B} = \mu_0\left(\mathbf{J} + \varepsilon_0 \frac{\partial \mathbf{E}} {\partial t} \right) ,$$ where $$\mathbf{J}$$ is the current density, $$\mu_0$$ is the vacuum permeability, and $$\varepsilon_0$$ is the vacuum permittivity.

That is, both electric currents (i.e. charges in uniform motion) and the (partial) time derivative of the electric field directly contributes to the magnetic field. In addition, the Maxwell–Faraday equation states $$\nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}} {\partial t}.$$ These represent two of Maxwell's four equations and they intricately link the electric and magnetic fields together, resulting in the electromagnetic field. The equations represent a set of four coupled multi-dimensional partial differential equations which, when solved for a system, describe the combined behavior of the electromagnetic fields. In general, the force experienced by a test charge in an electromagnetic field is given by the Lorentz force law: $$\mathbf{F} = q\mathbf{E} + q\mathbf{v} \times \mathbf{B} .$$

Energy in the electric field
The total energy per unit volume stored by the electromagnetic field is $$ u_\text{EM} = \frac{\varepsilon}{2} |\mathbf{E}|^2 + \frac{1}{2\mu} |\mathbf{B}|^2 $$ where $ε$ is the permittivity of the medium in which the field exists, $$\mu$$ its magnetic permeability, and $d$ and $E$ are the electric and magnetic field vectors.

As $B$ and $E$ fields are coupled, it would be misleading to split this expression into "electric" and "magnetic" contributions. In particular, an electrostatic field in any given frame of reference in general transforms into a field with a magnetic component in a relatively moving frame. Accordingly, decomposing the electromagnetic field into an electric and magnetic component is frame-specific, and similarly for the associated energy.

The total energy $B$ stored in the electromagnetic field in a given volume $UEM$ is $$U_\text{EM} = \frac{1}{2} \int_{V} \left( \varepsilon |\mathbf{E}|^2 + \frac{1}{\mu} |\mathbf{B}|^2 \right) dV \, .$$

Definitive equation of vector fields
In the presence of matter, it is helpful to extend the notion of the electric field into three vector fields: $$\mathbf{D} = \varepsilon_0 \mathbf{E} + \mathbf{P}$$ where $V$ is the electric polarization – the volume density of electric dipole moments, and $P$ is the electric displacement field. Since $D$ and $E$ are defined separately, this equation can be used to define $P$. The physical interpretation of $D$ is not as clear as $D$ (effectively the field applied to the material) or $E$ (induced field due to the dipoles in the material), but still serves as a convenient mathematical simplification, since Maxwell's equations can be simplified in terms of free charges and currents.

Constitutive relation
The $P$ and $E$ fields are related by the permittivity of the material, $D$.

For linear, homogeneous, isotropic materials $ε$ and $E$ are proportional and constant throughout the region, there is no position dependence: $$\mathbf{D}(\mathbf{r}) = \varepsilon\mathbf{E}(\mathbf{r}) .$$

For inhomogeneous materials, there is a position dependence throughout the material: $$\mathbf{D}(\mathbf{r}) = \varepsilon (\mathbf{r})\mathbf{E}(\mathbf{r})$$

For anisotropic materials the $D$ and $E$ fields are not parallel, and so $D$ and $E$ are related by the permittivity tensor (a 2nd order tensor field), in component form: $$D_i = \varepsilon_{ij} E_j$$

For non-linear media, $D$ and $E$ are not proportional. Materials can have varying extents of linearity, homogeneity and isotropy.

Point charge in uniform motion
The invariance of the form of Maxwell's equations under Lorentz transformation can be used to derive the electric field of a uniformly moving point charge. The charge of a particle is considered frame invariant, as supported by experimental evidence. Alternatively the electric field of uniformly moving point charges can be derived from the Lorentz transformation of four-force experienced by test charges in the source's rest frame given by Coulomb's law and assigning electric field and magnetic field by their definition given by the form of Lorentz force. However the following equation is only applicable when no acceleration is involved in the particle's history where Coulomb's law can be considered or symmetry arguments can be used for solving Maxwell's equations in a simple manner. The electric field of such a uniformly moving point charge is hence given by: $$\mathbf{E} = \frac q {4 \pi \varepsilon_0 r^3}  \frac {1- \beta^2} {(1- \beta^2 \sin^2 \theta)^{3/2}} \mathbf{r} ,$$ where $$q$$ is the charge of the point source, $$\mathbf{r}$$ is the position vector from the point source to the point in space, $$\beta$$ is the ratio of observed speed of the charge particle to the speed of light and $$\theta$$ is the angle between $$\mathbf{r}$$ and the observed velocity of the charged particle.

The above equation reduces to that given by Coulomb's law for non-relativistic speeds of the point charge. Spherical symmetry is not satisfied due to breaking of symmetry in the problem by specification of direction of velocity for calculation of field. To illustrate this, field lines of moving charges are sometimes represented as unequally spaced radial lines which would appear equally spaced in a co-moving reference frame.

Propagation of disturbances in electric fields
Special theory of relativity imposes the principle of locality, that requires cause and effect to be time-like separated events where the causal efficacy does not travel faster than the speed of light. Maxwell's laws are found to confirm to this view since the general solutions of fields are given in terms of retarded time which indicate that electromagnetic disturbances travel at the speed of light. Advanced time, which also provides a solution for Maxwell's law are ignored as an unphysical solution.For the motion of a charged particle, considering for example the case of a moving particle with the above described electric field coming to an abrupt stop, the electric fields at points far from it do not immediately revert to that classically given for a stationary charge. On stopping, the field around the stationary points begin to revert to the expected state and this effect propagates outwards at the speed of light while the electric field lines far away from this will continue to point radially towards an assumed moving charge. This virtual particle will never be outside the range of propagation of the disturbance in electromagnetic field, since charged particles are restricted to have speeds slower than that of light, which makes it impossible to construct a Gaussian surface in this region that violates Gauss's law. Another technical difficulty that supports this is that charged particles travelling faster than or equal to speed of light no longer have a unique retarded time. Since electric field lines are continuous, an electromagnetic pulse of radiation is generated that connects at the boundary of this disturbance travelling outwards at the speed of light. In general, any accelerating point charge radiates electromagnetic waves however, non-radiating acceleration is possible in a systems of charges.

Arbitrarily moving point charge
For arbitrarily moving point charges, propagation of potential fields such as Lorenz gauge fields at the speed of light needs to be accounted for by using Liénard–Wiechert potential. Since the potentials satisfy Maxwell's equations, the fields derived for point charge also satisfy Maxwell's equations. The electric field is expressed as: $$\mathbf{E}(\mathbf{r}, \mathbf{t}) = \frac{1}{4 \pi \varepsilon_0} \left(\frac{q(\mathbf{n}_s - \boldsymbol{\beta}_s)}{\gamma^2 (1 - \mathbf{n}_s \cdot \boldsymbol{\beta}_s)^3 |\mathbf{r} - \mathbf{r}_s|^2} + \frac{q \mathbf{n}_s \times \big((\mathbf{n}_s - \boldsymbol{\beta}_s) \times \dot{\boldsymbol{\beta}_s}\big)}{c(1 - \mathbf{n}_s \cdot \boldsymbol{\beta}_s)^3 |\mathbf{r} - \mathbf{r}_s|} \right)_{t = t_r}$$ where $$q$$ is the charge of the point source, ${t_r}$ is retarded time or the time at which the source's contribution of the electric field originated, ${r}_s(t)$  is the position vector of the particle, ${n}_s(\mathbf{r},t)$  is a unit vector pointing from charged particle to the point in space, $ \boldsymbol{\beta}_s(t)$  is the velocity of the particle divided by the speed of light, and $\gamma(t)$  is the corresponding Lorentz factor. The retarded time is given as solution of: $$t_r=\mathbf{t}-\frac{|\mathbf{r} - \mathbf{r}_s(t_r)|}{c}$$

The uniqueness of solution for ${t_r}$ for given $$\mathbf{t}$$, $$\mathbf{r}$$ and $$r_s(t)$$ is valid for charged particles moving slower than speed of light. Electromagnetic radiation of accelerating charges is known to be caused by the acceleration dependent term in the electric field from which relativistic correction for Larmor formula is obtained.

There exist yet another set of solutions for Maxwell's equation of the same form but for advanced time ${t_a}$ instead of retarded time given as a solution of:

$$t_a=\mathbf{t}+\frac{|\mathbf{r} - \mathbf{r}_s(t_a)|}{c}$$

Since the physical interpretation of this indicates that the electric field at a point is governed by the particle's state at a point of time in the future, it is considered as an unphysical solution and hence neglected. However, there have been theories exploring the advanced time solutions of Maxwell's equations, such as Feynman Wheeler absorber theory.

The above equation, although consistent with that of uniformly moving point charges as well as its non-relativistic limit, are not corrected for quantum-mechanical effects.

Common formulæ
Electric field infinitely close to a conducting surface in electrostatic equilibrium having charge density $$\sigma$$ at that point is $\frac{\sigma}{\varepsilon_0} \hat\mathbf x$ since charges are only formed on the surface and the surface at the infinitesimal scale resembles an infinite 2D plane. In the absence of external fields, spherical conductors exhibit a uniform charge distribution on the surface and hence have the same electric field as that of uniform spherical surface distribution.

title: Electric field