Vacuum

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https://en.wikipedia.org/wiki/Vacuum

Vacuum pump and bell jar for vacuum experiments, used in science education during the early 20th century, on display in the Schulhistorische Sammlung ('School Historical Museum'), Bremerhaven, Germany

A vacuum (pl.: vacuums or vacua) is space devoid of matter. The word is derived from the Latin adjective vacuus (neuter vacuum) meaning "vacant" or "void". An approximation to such vacuum is a region with a gaseous pressure much less than atmospheric pressure. Physicists often discuss ideal test results that would occur in a perfect vacuum, which they sometimes simply call "vacuum" or free space, and use the term partial vacuum to refer to an actual imperfect vacuum as one might have in a laboratory or in space. In engineering and applied physics on the other hand, vacuum refers to any space in which the pressure is considerably lower than atmospheric pressure. The Latin term in vacuo is used to describe an object that is surrounded by a vacuum.

The quality of a partial vacuum refers to how closely it approaches a perfect vacuum. Other things equal, lower gas pressure means higher-quality vacuum. For example, a typical vacuum cleaner produces enough suction to reduce air pressure by around 20%. But higher-quality vacuums are possible. Ultra-high vacuum chambers, common in chemistry, physics, and engineering, operate below one trillionth (10⁻¹²) of atmospheric pressure (100 nPa), and can reach around 100 particles/cm³. Outer space is an even higher-quality vacuum, with the equivalent of just a few hydrogen atoms per cubic meter on average in intergalactic space.

Vacuum has been a frequent topic of philosophical debate since ancient Greek times, but was not studied empirically until the 17th century. Clemens Timpler (1605) philosophized about the experimental possibility of producing a vacuum in small tubes. Evangelista Torricelli produced the first laboratory vacuum in 1643, and other experimental techniques were developed as a result of his theories of atmospheric pressure. A Torricellian vacuum is created by filling with mercury a tall glass container closed at one end, and then inverting it in a bowl to contain the mercury (see below).

Vacuum became a valuable industrial tool in the 20th century with the introduction of incandescent light bulbs and vacuum tubes, and a wide array of vacuum technologies has since become available. The development of human spaceflight has raised interest in the impact of vacuum on human health, and on life forms in general.

Etymology

The word vacuum comes from Latin 'an empty space, void', noun use of neuter of vacuus, meaning "empty", related to vacare, meaning "to be empty".

Vacuum is one of the few words in the English language that contains two consecutive instances of the vowel u.

Historical understanding

Historically, there has been much dispute over whether such a thing as a vacuum can exist. Ancient Greek philosophers debated the existence of a vacuum, or void, in the context of atomism, which posited void and atom as the fundamental explanatory elements of physics. Lucretius argued for the existence of vacuum in the first century BC and Hero of Alexandria tried unsuccessfully to create an artificial vacuum in the first century AD.

Following Plato, however, even the abstract concept of a featureless void faced considerable skepticism: it could not be apprehended by the senses, it could not, itself, provide additional explanatory power beyond the physical volume with which it was commensurate and, by definition, it was quite literally nothing at all, which cannot rightly be said to exist. Aristotle believed that no void could occur naturally, because the denser surrounding material continuum would immediately fill any incipient rarity that might give rise to a void. In his Physics, book IV, Aristotle offered numerous arguments against the void: for example, that motion through a medium which offered no impediment could continue ad infinitum, there being no reason that something would come to rest anywhere in particular.

In the medieval Muslim world, the physicist and Islamic scholar Al-Farabi wrote a treatise rejecting the existence of the vacuum in the 10th century. He concluded that air's volume can expand to fill available space, and therefore the concept of a perfect vacuum was incoherent. According to Ahmad Dallal, Abū Rayhān al-Bīrūnī states that "there is no observable evidence that rules out the possibility of vacuum". The suction pump was described by Arab engineer Al-Jazari in the 13th century, and later appeared in Europe from the 15th century.

European scholars such as Roger Bacon, Blasius of Parma and Walter Burley in the 13th and 14th century focused considerable attention on issues concerning the concept of a vacuum. The commonly held view that nature abhorred a vacuum was called horror vacui. There was even speculation that even God could not create a vacuum if he wanted and the 1277 Paris condemnations of Bishop Étienne Tempier, which required there to be no restrictions on the powers of God, led to the conclusion that God could create a vacuum if he so wished. From the 14th century onward increasingly departed from the Aristotelian perspective, scholars widely acknowledged that a supernatural void exists beyond the confines of the cosmos itself by the 17th century. This idea, influenced by Stoic physics, helped to segregate natural and theological concerns.

Almost two thousand years after Plato, René Descartes also proposed a geometrically based alternative theory of atomism, without the problematic nothing–everything dichotomy of void and atom. Although Descartes agreed with the contemporary position, that a vacuum does not occur in nature, the success of his namesake coordinate system and more implicitly, the spatial–corporeal component of his metaphysics would come to define the philosophically modern notion of empty space as a quantified extension of volume. By the ancient definition however, directional information and magnitude were conceptually distinct.

Torricelli's mercury barometer produced one of the first sustained vacuums in a laboratory.

Medieval thought experiments into the idea of a vacuum considered whether a vacuum was present, if only for an instant, between two flat plates when they were rapidly separated. There was much discussion of whether the air moved in quickly enough as the plates were separated, or, as Walter Burley postulated, whether a 'celestial agent' prevented the vacuum arising. Jean Buridan reported in the 14th century that teams of ten horses could not pull open bellows when the port was sealed.

The Crookes tube, used to discover and study cathode rays, was an evolution of the Geissler tube.

The 17th century saw the first attempts to quantify measurements of partial vacuum. Evangelista Torricelli's mercury barometer of 1643 and Blaise Pascal's experiments both demonstrated a partial vacuum.

In 1654, Otto von Guericke invented the first vacuum pump and conducted his famous Magdeburg hemispheres experiment, showing that, owing to atmospheric pressure outside the hemispheres, teams of horses could not separate two hemispheres from which the air had been partially evacuated. Robert Boyle improved Guericke's design and with the help of Robert Hooke further developed vacuum pump technology. Thereafter, research into the partial vacuum lapsed until 1850 when August Toepler invented the Toepler pump and in 1855 when Heinrich Geissler invented the mercury displacement pump, achieving a partial vacuum of about 10 Pa (0.1 Torr). A number of electrical properties become observable at this vacuum level, which renewed interest in further research.

While outer space provides the most rarefied example of a naturally occurring partial vacuum, the heavens were originally thought to be seamlessly filled by a rigid indestructible material called aether. Borrowing somewhat from the pneuma of Stoic physics, aether came to be regarded as the rarefied air from which it took its name, (see Aether (mythology)). Early theories of light posited a ubiquitous terrestrial and celestial medium through which light propagated. Additionally, the concept informed Isaac Newton's explanations of both refraction and of radiant heat. 19th century experiments into this luminiferous aether attempted to detect a minute drag on the Earth's orbit. While the Earth does, in fact, move through a relatively dense medium in comparison to that of interstellar space, the drag is so minuscule that it could not be detected. In 1912, astronomer Henry Pickering commented: "While the interstellar absorbing medium may be simply the ether, [it] is characteristic of a gas, and free gaseous molecules are certainly there". Thereafter, however, luminiferous aether was discarded.

Later, in 1930, Paul Dirac proposed a model of the vacuum as an infinite sea of particles possessing negative energy, called the Dirac sea. This theory helped refine the predictions of his earlier formulated Dirac equation, and successfully predicted the existence of the positron, confirmed two years later. Werner Heisenberg's uncertainty principle, formulated in 1927, predicted a fundamental limit within which instantaneous position and momentum, or energy and time can be measured. This far reaching consequences also threatened whether the "emptiness" of space between particles exists.

Classical field theories

The strictest criterion to define a vacuum is a region of space and time where all the components of the stress–energy tensor are zero. This means that this region is devoid of energy and momentum, and by consequence, it must be empty of particles and other physical fields (such as electromagnetism) that contain energy and momentum.

Gravity

In general relativity, a vanishing stress–energy tensor implies, through Einstein field equations, the vanishing of all the components of the Ricci tensor. Vacuum does not mean that the curvature of space-time is necessarily flat: the gravitational field can still produce curvature in a vacuum in the form of tidal forces and gravitational waves (technically, these phenomena are the components of the Weyl tensor). The black hole (with zero electric charge) is an elegant example of a region completely "filled" with vacuum, but still showing a strong curvature.

Electromagnetism

In classical electromagnetism, the vacuum of free space, or sometimes just free space or perfect vacuum, is a standard reference medium for electromagnetic effects. Some authors refer to this reference medium as classical vacuum, a terminology intended to separate this concept from QED vacuum or QCD vacuum, where vacuum fluctuations can produce transient virtual particle densities and a relative permittivity and relative permeability that are not identically unity.

In the theory of classical electromagnetism, free space has the following properties:

• Electromagnetic radiation travels, when unobstructed, at the speed of light, the defined value 299,792,458 m/s in SI units.
• The superposition principle is always exactly true. For example, the electric potential generated by two charges is the simple addition of the potentials generated by each charge in isolation. The value of the electric field at any point around these two charges is found by calculating the vector sum of the two electric fields from each of the charges acting alone.
• The permittivity and permeability are exactly the electric constant ε₀ and magnetic constant μ₀, respectively (in SI units), or exactly 1 (in Gaussian units).
• The characteristic impedance (η) equals the impedance of free space Z₀ ≈ 376.73 Ω.

The vacuum of classical electromagnetism can be viewed as an idealized electromagnetic medium with the constitutive relations in SI units:

${\displaystyle \mathit{D}(\mathit{r},t)={\epsilon }_0\mathit{E}(\mathit{r},t)}$

${\displaystyle \mathit{H}(\mathit{r},t)=\frac{1}{{\mu }_0}\mathit{B}(\mathit{r},t)}$

relating the electric displacement field D to the electric field E and the magnetic field or H-field H to the magnetic induction or B-field B. Here r is a spatial location and t is time.

Quantum mechanics

Main article: Quantum vacuum state

In quantum mechanics and quantum field theory, the vacuum is defined as the state (that is, the solution to the equations of the theory) with the lowest possible energy (the ground state of the Hilbert space). In quantum electrodynamics this vacuum is referred to as 'QED vacuum' to distinguish it from the vacuum of quantum chromodynamics, denoted as QCD vacuum. QED vacuum is a state with no matter particles (hence the name), and no photons. As described above, this state is impossible to achieve experimentally. (Even if every matter particle could somehow be removed from a volume, it would be impossible to eliminate all the blackbody photons.) Nonetheless, it provides a good model for realizable vacuum, and agrees with a number of experimental observations as described next.

QED vacuum has interesting and complex properties. In QED vacuum, the electric and magnetic fields have zero average values, but their variances are not zero. As a result, QED vacuum contains vacuum fluctuations (virtual particles that hop into and out of existence), and a finite energy called vacuum energy. Vacuum fluctuations are an essential and ubiquitous part of quantum field theory. Some experimentally verified effects of vacuum fluctuations include spontaneous emission and the Lamb shift. Coulomb's law and the electric potential in vacuum near an electric charge are modified.

Theoretically, in QCD multiple vacuum states can coexist. The starting and ending of cosmological inflation is thought to have arisen from transitions between different vacuum states. For theories obtained by quantization of a classical theory, each stationary point of the energy in the configuration space gives rise to a single vacuum. String theory is believed to have a huge number of vacua – the so-called string theory landscape.

Outer space

Main article: Outer space

Structure of the magnetosphere - is not a perfect vacuum, but a tenuous plasma awash with charged particles, free elements such as hydrogen, helium and oxygen, electromagnetic fields.

Outer space has very low density and pressure, and is the closest physical approximation of a perfect vacuum. But no vacuum is truly perfect, not even in interstellar space, where there are still a few hydrogen atoms per cubic meter.

Stars, planets, and moons keep their atmospheres by gravitational attraction, and as such, atmospheres have no clearly delineated boundary: the density of atmospheric gas simply decreases with distance from the object. The Earth's atmospheric pressure drops to about 32 millipascals (4.6×10⁻⁶ psi) at 100 kilometres (62 mi) of altitude, the Kármán line, which is a common definition of the boundary with outer space. Beyond this line, isotropic gas pressure rapidly becomes insignificant when compared to radiation pressure from the Sun and the dynamic pressure of the solar winds, so the definition of pressure becomes difficult to interpret. The thermosphere in this range has large gradients of pressure, temperature and composition, and varies greatly due to space weather. Astrophysicists prefer to use number density to describe these environments, in units of particles per cubic centimetre.

But although it meets the definition of outer space, the atmospheric density within the first few hundred kilometers above the Kármán line is still sufficient to produce significant drag on satellites. Most artificial satellites operate in this region, called low Earth orbit, and must fire their engines every couple of weeks or a few times a year (depending on solar activity). The drag here is low enough that it could theoretically be overcome by radiation pressure on solar sails, a proposed propulsion system for interplanetary travel.

All of the observable universe is filled with large numbers of photons, the so-called cosmic background radiation, and quite likely a correspondingly large number of neutrinos. The current temperature of this radiation is about 3 K (−270.15 °C; −454.27 °F).

Measurement

Main article: Pressure measurement

The quality of a vacuum is indicated by the amount of matter remaining in the system, so that a high quality vacuum is one with very little matter left in it. Vacuum is primarily measured by its absolute pressure, but a complete characterization requires further parameters, such as temperature and chemical composition. One of the most important parameters is the mean free path (MFP) of residual gases, which indicates the average distance that molecules will travel between collisions with each other. As the gas density decreases, the MFP increases, and when the MFP is longer than the chamber, pump, spacecraft, or other objects present, the continuum assumptions of fluid mechanics do not apply. This vacuum state is called high vacuum, and the study of fluid flows in this regime is called particle gas dynamics. The MFP of air at atmospheric pressure is very short, 70 nm, but at 100 mPa (≈10⁻³ Torr) the MFP of room temperature air is roughly 100 mm, which is on the order of everyday objects such as vacuum tubes. The Crookes radiometer turns when the MFP is larger than the size of the vanes.

Vacuum quality is subdivided into ranges according to the technology required to achieve it or measure it. These ranges were defined in ISO 3529-1:2019 as shown in the following table (100 Pa corresponds to 0.75 Torr; Torr is a non-SI unit):

Pressure range	Definition	The reasoning for the definition of the ranges is as follows (typical circumstances):
Prevailing atmospheric pressure (31 kPa to 110 kPa) to 100 Pa	low (rough) vacuum	Pressure can be achieved by simple materials (e.g. regular steel) and positive displacement vacuum pumps; viscous flow regime for gases
<100 Pa to 0.1 Pa	medium (fine) vacuum	Pressure can be achieved by elaborate materials (e.g. stainless steel) and positive displacement vacuum pumps; transitional flow regime for gases
<0.1 Pa to 1×10−6 Pa	high vacuum (HV)	Pressure can be achieved by elaborate materials (e.g. stainless steel), elastomer sealings and high vacuum pumps; molecular flow regime for gases
<1×10−6 Pa to 1×10−9 Pa	ultra-high vacuum (UHV)	Pressure can be achieved by elaborate materials (e.g. low-carbon stainless steel), metal sealings, special surface preparations and cleaning, bake-out and high vacuum pumps; molecular flow regime for gases
below 1×10−9 Pa	extreme-high vacuum (XHV)	Pressure can be achieved by sophisticated materials (e.g. vacuum fired low-carbon stainless steel, aluminium, copper-beryllium, titanium), metal sealings, special surface preparations and cleaning, bake-out and additional getter pumps; molecular flow regime for gases

• Atmospheric pressure is variable but 101.325 and 100 kilopascals (1013.25 and 1000.00 mbar) are common standard or reference pressures.
• Deep space is generally much more empty than any artificial vacuum. It may or may not meet the definition of high vacuum above, depending on what region of space and astronomical bodies are being considered. For example, the MFP of interplanetary space is smaller than the size of the Solar System, but larger than small planets and moons. As a result, solar winds exhibit continuum flow on the scale of the Solar System, but must be considered a bombardment of particles with respect to the Earth and Moon.
• Perfect vacuum is an ideal state of no particles at all. It cannot be achieved in a laboratory, although there may be small volumes which, for a brief moment, happen to have no particles of matter in them. Even if all particles of matter were removed, there would still be photons, as well as dark energy, virtual particles, and other aspects of the quantum vacuum.

Relative versus absolute measurement

Vacuum is measured in units of pressure, typically as a subtraction relative to ambient atmospheric pressure on Earth. But the amount of relative measurable vacuum varies with local conditions. On the surface of Venus, where ground-level atmospheric pressure is much higher than on Earth, much higher relative vacuum readings would be possible. On the surface of the Moon with almost no atmosphere, it would be extremely difficult to create a measurable vacuum relative to the local environment.

Similarly, much higher than normal relative vacuum readings are possible deep in the Earth's ocean. A submarine maintaining an internal pressure of 1 atmosphere submerged to a depth of 10 atmospheres (98 metres; a 9.8-metre column of seawater has the equivalent weight of 1 atm) is effectively a vacuum chamber keeping out the crushing exterior water pressures, though the 1 atm inside the submarine would not normally be considered a vacuum.

Therefore, to properly understand the following discussions of vacuum measurement, it is important that the reader assumes the relative measurements are being done on Earth at sea level, at exactly 1 atmosphere of ambient atmospheric pressure.

Measurements relative to 1 atm

A glass McLeod gauge, drained of mercury

The SI unit of pressure is the pascal (symbol Pa), but vacuum is often measured in torrs, named for an Italian physicist Torricelli (1608–1647). A torr is equal to the displacement of a millimeter of mercury (mmHg) in a manometer with 1 torr equaling 133.3223684 pascals above absolute zero pressure. Vacuum is often also measured on the barometric scale or as a percentage of atmospheric pressure in bars or atmospheres. Low vacuum is often measured in millimeters of mercury (mmHg) or pascals (Pa) below standard atmospheric pressure. "Below atmospheric" means that the absolute pressure is equal to the current atmospheric pressure.

In other words, most low vacuum gauges that read, for example 50.79 Torr. Many inexpensive low vacuum gauges have a margin of error and may report a vacuum of 0 Torr but in practice this generally requires a two-stage rotary vane or other medium type of vacuum pump to go much beyond (lower than) 1 torr.

Measuring instruments

Many devices are used to measure the pressure in a vacuum, depending on what range of vacuum is needed.

Hydrostatic gauges (such as the mercury column manometer) consist of a vertical column of liquid in a tube whose ends are exposed to different pressures. The column will rise or fall until its weight is in equilibrium with the pressure differential between the two ends of the tube. The simplest design is a closed-end U-shaped tube, one side of which is connected to the region of interest. Any fluid can be used, but mercury is preferred for its high density and low vapour pressure. Simple hydrostatic gauges can measure pressures ranging from 1 torr (100 Pa) to above atmospheric. An important variation is the McLeod gauge which isolates a known volume of vacuum and compresses it to multiply the height variation of the liquid column. The McLeod gauge can measure vacuums as high as 10⁻⁶ torr (0.1 mPa), which is the lowest direct measurement of pressure that is possible with current technology. Other vacuum gauges can measure lower pressures, but only indirectly by measurement of other pressure-controlled properties. These indirect measurements must be calibrated via a direct measurement, most commonly a McLeod gauge.

The kenotometer is a particular type of hydrostatic gauge, typically used in power plants using steam turbines. The kenotometer measures the vacuum in the steam space of the condenser, that is, the exhaust of the last stage of the turbine.

Mechanical or elastic gauges depend on a Bourdon tube, diaphragm, or capsule, usually made of metal, which will change shape in response to the pressure of the region in question. A variation on this idea is the capacitance manometer, in which the diaphragm makes up a part of a capacitor. A change in pressure leads to the flexure of the diaphragm, which results in a change in capacitance. These gauges are effective from 10³ torr to 10⁻⁴ torr, and beyond.

Thermal conductivity gauges rely on the fact that the ability of a gas to conduct heat decreases with pressure. In this type of gauge, a wire filament is heated by running current through it. A thermocouple or Resistance Temperature Detector (RTD) can then be used to measure the temperature of the filament. This temperature is dependent on the rate at which the filament loses heat to the surrounding gas, and therefore on the thermal conductivity. A common variant is the Pirani gauge which uses a single platinum filament as both the heated element and RTD. These gauges are accurate from 10 torr to 10⁻³ torr, but they are sensitive to the chemical composition of the gases being measured.

Ionization gauges are used in ultrahigh vacuum. They come in two types: hot cathode and cold cathode. In the hot cathode version an electrically heated filament produces an electron beam. The electrons travel through the gauge and ionize gas molecules around them. The resulting ions are collected at a negative electrode. The current depends on the number of ions, which depends on the pressure in the gauge. Hot cathode gauges are accurate from 10⁻³ torr to 10⁻¹⁰ torr. The principle behind cold cathode version is the same, except that electrons are produced in a discharge created by a high voltage electrical discharge. Cold cathode gauges are accurate from 10⁻² torr to 10⁻⁹ torr. Ionization gauge calibration is very sensitive to construction geometry, chemical composition of gases being measured, corrosion and surface deposits. Their calibration can be invalidated by activation at atmospheric pressure or low vacuum. The composition of gases at high vacuums will usually be unpredictable, so a mass spectrometer must be used in conjunction with the ionization gauge for accurate measurement.

Uses

Light bulbs contain a partial vacuum, usually backfilled with argon, which protects the tungsten filament

Vacuum is useful in a variety of processes and devices. Its first widespread use was in the incandescent light bulb to protect the filament from chemical degradation. The chemical inertness produced by a vacuum is also useful for electron-beam welding, cold welding, vacuum packing and vacuum frying. Ultra-high vacuum is used in the study of atomically clean substrates, as only a very good vacuum preserves atomic-scale clean surfaces for a reasonably long time (on the order of minutes to days). High to ultra-high vacuum removes the obstruction of air, allowing particle beams to deposit or remove materials without contamination. This is the principle behind chemical vapor deposition, physical vapor deposition, and dry etching which are essential to the fabrication of semiconductors and optical coatings, and to surface science. The reduction of convection provides the thermal insulation of thermos bottles. Deep vacuum lowers the boiling point of liquids and promotes low temperature outgassing which is used in freeze drying, adhesive preparation, distillation, metallurgy, and process purging. The electrical properties of vacuum make electron microscopes and vacuum tubes possible, including cathode-ray tubes. Vacuum interrupters are used in electrical switchgear. Vacuum arc processes are industrially important for production of certain grades of steel or high purity materials. The elimination of air friction is useful for flywheel energy storage and ultracentrifuges.

This shallow water well pump reduces atmospheric air pressure inside the pump chamber. Atmospheric pressure extends down into the well, and forces water up the pipe into the pump to balance the reduced pressure. Above-ground pump chambers are only effective to a depth of approximately 9 meters due to the water column weight balancing the atmospheric pressure.

Vacuum-driven machines

Vacuums are commonly used to produce suction, which has an even wider variety of applications. The Newcomen steam engine used vacuum instead of pressure to drive a piston. In the 19th century, vacuum was used for traction on Isambard Kingdom Brunel's experimental atmospheric railway. Vacuum brakes were once widely used on trains in the UK but, except on heritage railways, they have been replaced by air brakes.

Manifold vacuum can be used to drive accessories on automobiles. The best known application is the vacuum servo, used to provide power assistance for the brakes. Obsolete applications include vacuum-driven windscreen wipers and Autovac fuel pumps. Some aircraft instruments (Attitude Indicator (AI) and the Heading Indicator (HI)) are typically vacuum-powered, as protection against loss of all (electrically powered) instruments, since early aircraft often did not have electrical systems, and since there are two readily available sources of vacuum on a moving aircraft, the engine and an external venturi. Vacuum induction melting uses electromagnetic induction within a vacuum.

Maintaining a vacuum in the condenser is an important aspect of the efficient operation of steam turbines. A steam jet ejector or liquid ring vacuum pump is used for this purpose. The typical vacuum maintained in the condenser steam space at the exhaust of the turbine (also called condenser backpressure) is in the range 5 to 15 kPa (absolute), depending on the type of condenser and the ambient conditions.

Outgassing

Main article: Outgassing

Evaporation and sublimation into a vacuum is called outgassing. All materials, solid or liquid, have a small vapour pressure, and their outgassing becomes important when the vacuum pressure falls below this vapour pressure. Outgassing has the same effect as a leak and will limit the achievable vacuum. Outgassing products may condense on nearby colder surfaces, which can be troublesome if they obscure optical instruments or react with other materials. This is of great concern to space missions, where an obscured telescope or solar cell can ruin an expensive mission.

The most prevalent outgassing product in vacuum systems is water absorbed by chamber materials. It can be reduced by desiccating or baking the chamber, and removing absorbent materials. Outgassed water can condense in the oil of rotary vane pumps and reduce their net speed drastically if gas ballasting is not used. High vacuum systems must be clean and free of organic matter to minimize outgassing.

Ultra-high vacuum systems are usually baked, preferably under vacuum, to temporarily raise the vapour pressure of all outgassing materials and boil them off. Once the bulk of the outgassing materials are boiled off and evacuated, the system may be cooled to lower vapour pressures and minimize residual outgassing during actual operation. Some systems are cooled well below room temperature by liquid nitrogen to shut down residual outgassing and simultaneously cryopump the system.

Pumping and ambient air pressure

Deep wells have the pump chamber down in the well close to the water surface, or in the water. A "sucker rod" extends from the handle down the center of the pipe deep into the well to operate the plunger. The pump handle acts as a heavy counterweight against both the sucker rod weight and the weight of the water column standing on the upper plunger up to ground level.

Main article: Vacuum pump

Fluids cannot generally be pulled, so a vacuum cannot be created by suction. Suction can spread and dilute a vacuum by letting a higher pressure push fluids into it, but the vacuum has to be created first before suction can occur. The easiest way to create an artificial vacuum is to expand the volume of a container. For example, the diaphragm muscle expands the chest cavity, which causes the volume of the lungs to increase. This expansion reduces the pressure and creates a partial vacuum, which is soon filled by air pushed in by atmospheric pressure.

To continue evacuating a chamber indefinitely without requiring infinite growth, a compartment of the vacuum can be repeatedly closed off, exhausted, and expanded again. This is the principle behind positive displacement pumps, like the manual water pump for example. Inside the pump, a mechanism expands a small sealed cavity to create a vacuum. Because of the pressure differential, some fluid from the chamber (or the well, in our example) is pushed into the pump's small cavity. The pump's cavity is then sealed from the chamber, opened to the atmosphere, and squeezed back to a minute size.

A cutaway view of a turbomolecular pump, a momentum transfer pump used to achieve high vacuum

The above explanation is merely a simple introduction to vacuum pumping, and is not representative of the entire range of pumps in use. Many variations of the positive displacement pump have been developed, and many other pump designs rely on fundamentally different principles. Momentum transfer pumps, which bear some similarities to dynamic pumps used at higher pressures, can achieve much higher quality vacuums than positive displacement pumps. Entrapment pumps can capture gases in a solid or absorbed state, often with no moving parts, no seals and no vibration. None of these pumps are universal; each type has important performance limitations. They all share a difficulty in pumping low molecular weight gases, especially hydrogen, helium, and neon.

The lowest pressure that can be attained in a system is also dependent on many things other than the nature of the pumps. Multiple pumps may be connected in series, called stages, to achieve higher vacuums. The choice of seals, chamber geometry, materials, and pump-down procedures will all have an impact. Collectively, these are called vacuum technique. And sometimes, the final pressure is not the only relevant characteristic. Pumping systems differ in oil contamination, vibration, preferential pumping of certain gases, pump-down speeds, intermittent duty cycle, reliability, or tolerance to high leakage rates.

In ultra high vacuum systems, some very "odd" leakage paths and outgassing sources must be considered. The water absorption of aluminium and palladium becomes an unacceptable source of outgassing, and even the adsorptivity of hard metals such as stainless steel or titanium must be considered. Some oils and greases will boil off in extreme vacuums. The permeability of the metallic chamber walls may have to be considered, and the grain direction of the metallic flanges should be parallel to the flange face.

The lowest pressures currently achievable in laboratory are about 1×10⁻¹³ torrs (13 pPa). However, pressures as low as 5×10⁻¹⁷ torrs (6.7 fPa) have been indirectly measured in a 4 K (−269.15 °C; −452.47 °F) cryogenic vacuum system. This corresponds to ≈100 particles/cm³.

Effects on humans and animals

See also: Space exposure and Uncontrolled decompression

This painting, An Experiment on a Bird in the Air Pump by Joseph Wright of Derby, 1768, depicts an experiment performed by Robert Boyle in 1660.

Humans and animals exposed to vacuum will lose consciousness after a few seconds and die of hypoxia within minutes, but the symptoms are not nearly as graphic as commonly depicted in media and popular culture. The reduction in pressure lowers the temperature at which blood and other body fluids boil, but the elastic pressure of blood vessels ensures that this boiling point remains above the internal body temperature of 37 °C. Although the blood will not boil, the formation of gas bubbles in bodily fluids at reduced pressures, known as ebullism, is still a concern. The gas may bloat the body to twice its normal size and slow circulation, but tissues are elastic and porous enough to prevent rupture. Swelling and ebullism can be restrained by containment in a flight suit. Shuttle astronauts wore a fitted elastic garment called the Crew Altitude Protection Suit (CAPS) which prevents ebullism at pressures as low as 2 kPa (15 Torr). Rapid boiling will cool the skin and create frost, particularly in the mouth, but this is not a significant hazard.

Animal experiments show that rapid and complete recovery is normal for exposures shorter than 90 seconds, while longer full-body exposures are fatal and resuscitation has never been successful. A study by NASA on eight chimpanzees found all of them survived two and a half minute exposures to vacuum. There is only a limited amount of data available from human accidents, but it is consistent with animal data. Limbs may be exposed for much longer if breathing is not impaired. Robert Boyle was the first to show in 1660 that vacuum is lethal to small animals.

An experiment indicates that plants are able to survive in a low pressure environment (1.5 kPa) for about 30 minutes.

Cold or oxygen-rich atmospheres can sustain life at pressures much lower than atmospheric, as long as the density of oxygen is similar to that of standard sea-level atmosphere. The colder air temperatures found at altitudes of up to 3 km generally compensate for the lower pressures there. Above this altitude, oxygen enrichment is necessary to prevent altitude sickness in humans that did not undergo prior acclimatization, and spacesuits are necessary to prevent ebullism above 19 km. Most spacesuits use only 20 kPa (150 Torr) of pure oxygen. This pressure is high enough to prevent ebullism, but decompression sickness and gas embolisms can still occur if decompression rates are not managed.

Rapid decompression can be much more dangerous than vacuum exposure itself. Even if the victim does not hold his or her breath, venting through the windpipe may be too slow to prevent the fatal rupture of the delicate alveoli of the lungs. Eardrums and sinuses may be ruptured by rapid decompression, soft tissues may bruise and seep blood, and the stress of shock will accelerate oxygen consumption leading to hypoxia. Injuries caused by rapid decompression are called barotrauma. A pressure drop of 13 kPa (100 Torr), which produces no symptoms if it is gradual, may be fatal if it occurs suddenly.

Some extremophile microorganisms, such as tardigrades, can survive vacuum conditions for periods of days or weeks.

Mathematical descriptions of the electromagnetic field

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Mathematical_descriptions_of_the_electromagnetic_field 

There are various mathematical descriptions of the electromagnetic field that are used in the study of electromagnetism, one of the four fundamental interactions of nature. In this article, several approaches are discussed, although the equations are in terms of electric and magnetic fields, potentials, and charges with currents, generally speaking.

Vector field approach

Main article: Classical electromagnetism

The most common description of the electromagnetic field uses two three-dimensional vector fields called the electric field and the magnetic field. These vector fields each have a value defined at every point of space and time and are thus often regarded as functions of the space and time coordinates. As such, they are often written as E(x, y, z, t) (electric field) and B(x, y, z, t) (magnetic field).

If only the electric field (E) is non-zero, and is constant in time, the field is said to be an electrostatic field. Similarly, if only the magnetic field (B) is non-zero and is constant in time, the field is said to be a magnetostatic field. However, if either the electric or magnetic field has a time-dependence, then both fields must be considered together as a coupled electromagnetic field using Maxwell's equations.

Maxwell's equations in the vector field approach

Main article: Maxwell's equations

The behaviour of electric and magnetic fields, whether in cases of electrostatics, magnetostatics, or electrodynamics (electromagnetic fields), is governed by Maxwell-Heaviside's equations:

Maxwell's equations (vector fields)
${\displaystyle \mathrm{\nabla }\cdot \mathbf{E}=\frac{\rho }{{\epsilon }_0}}$	Gauss's law
${\displaystyle \mathrm{\nabla }\cdot \mathbf{B}=0}$	Gauss's law for magnetism
${\displaystyle \mathrm{\nabla }\times \mathbf{E}=-\frac{\mathrm{\partial }\mathbf{B}}{\mathrm{\partial }t}}$	Faraday's law of induction
${\displaystyle \mathrm{\nabla }\times \mathbf{B}={\mu }_0\mathbf{J}+{\mu }_0{\epsilon }_0\frac{\mathrm{\partial }\mathbf{E}}{\mathrm{\partial }t}}$	Ampère-Maxwell law

where ρ is the charge density, which can (and often does) depend on time and position, ε₀ is the electric constant, μ₀ is the magnetic constant, and J is the current per unit area, also a function of time and position. The equations take this form with the International System of Quantities.

When dealing with only nondispersive isotropic linear materials, Maxwell's equations are often modified to ignore bound charges by replacing the permeability and permittivity of free space with the permeability and permittivity of the linear material in question. For some materials that have more complex responses to electromagnetic fields, these properties can be represented by tensors, with time-dependence related to the material's ability to respond to rapid field changes (dispersion (optics), Green–Kubo relations), and possibly also field dependencies representing nonlinear and/or nonlocal material responses to large amplitude fields (nonlinear optics).

Potential field approach

Many times in the use and calculation of electric and magnetic fields, the approach used first computes an associated potential: the electric potential, ${\displaystyle \phi }$, for the electric field, and the magnetic vector potential, A, for the magnetic field. The electric potential is a scalar field, while the magnetic potential is a vector field. This is why sometimes the electric potential is called the scalar potential and the magnetic potential is called the vector potential. These potentials can be used to find their associated fields as follows: $${\displaystyle \mathbf{E}=-\mathrm{\nabla }\phi -\frac{\mathrm{\partial }\mathbf{A}}{\mathrm{\partial }t}}$$ $${\displaystyle \mathbf{B}=\mathrm{\nabla }\times \mathbf{A}}$$

Maxwell's equations in potential formulation

These relations can be substituted into Maxwell's equations to express the latter in terms of the potentials. Faraday's law and Gauss's law for magnetism (the homogeneous equations) turn out to be identically true for any potentials. This is because of the way the fields are expressed as gradients and curls of the scalar and vector potentials. The homogeneous equations in terms of these potentials involve the divergence of the curl ${\displaystyle \mathrm{\nabla }\cdot \mathrm{\nabla }\times \mathbf{A}}$ and the curl of the gradient ${\displaystyle \mathrm{\nabla }\times \mathrm{\nabla }\phi }$, which are always zero. The other two of Maxwell's equations (the inhomogeneous equations) are the ones that describe the dynamics in the potential formulation.

Maxwell's equations (potential formulation)

${\displaystyle {\mathrm{\nabla }}^2\phi +\frac{\mathrm{\partial }}{\mathrm{\partial }t}\left(\mathrm{\nabla }\cdot \mathbf{A}\right)=-\frac{\rho }{{\epsilon }_0}}$

${\displaystyle \left({\mathrm{\nabla }}^2\mathbf{A}-\frac{1}{c^2}\frac{{\mathrm{\partial }}^2\mathbf{A}}{\mathrm{\partial }{t}^2}\right)-\mathrm{\nabla }\left(\mathrm{\nabla }\cdot \mathbf{A}+\frac{1}{c^2}\frac{\mathrm{\partial }\phi }{\mathrm{\partial }t}\right)=-{\mu }_0\mathbf{J}}$

These equations taken together are as powerful and complete as Maxwell's equations. Moreover, the problem has been reduced somewhat, as the electric and magnetic fields together had six components to solve for. In the potential formulation, there are only four components: the electric potential and the three components of the vector potential. However, the equations are messier than Maxwell's equations using the electric and magnetic fields.

Gauge freedom

These equations can be simplified by taking advantage of the fact that the electric and magnetic fields are physically meaningful quantities that can be measured; the potentials are not. There is a freedom to constrain the form of the potentials provided that this does not affect the resultant electric and magnetic fields, called gauge freedom. Specifically for these equations, for any choice of a twice-differentiable scalar function of position and time λ, if (φ, A) is a solution for a given system, then so is another potential (φ′, A′) given by: $${\displaystyle {\phi }^{\prime }=\phi -\frac{\mathrm{\partial }\lambda }{\mathrm{\partial }t}}$$ $${\displaystyle {\mathbf{A}}^{\prime }=\mathbf{A}+\mathrm{\nabla }\lambda }$$

This freedom can be used to simplify the potential formulation. Either of two such scalar functions is typically chosen: the Coulomb gauge and the Lorenz gauge.

Coulomb gauge

Main article: Gauge fixing § Coulomb gauge

The Coulomb gauge is chosen in such a way that ${\displaystyle \mathrm{\nabla }\cdot {\mathbf{A}}^{\prime }=0}$, which corresponds to the case of magnetostatics. In terms of λ, this means that it must satisfy the equation $${\displaystyle {\mathrm{\nabla }}^2\lambda =-\mathrm{\nabla }\cdot \mathbf{A}.}$$

This choice of function results in the following formulation of Maxwell's equations: $${\displaystyle {\mathrm{\nabla }}^2{\phi }^{\prime }=-\frac{\rho }{{\epsilon }_0}}$$ $${\displaystyle {\mathrm{\nabla }}^2{\mathbf{A}}^{\prime }-{\mu }_0{\epsilon }_0\frac{{\mathrm{\partial }}^2{\mathbf{A}}^{\prime }}{\mathrm{\partial }{t}^2}=-{\mu }_0\mathbf{J}+{\mu }_0{\epsilon }_0\mathrm{\nabla }\left(\frac{\mathrm{\partial }{\phi }^{\prime }}{\mathrm{\partial }t}\right)}$$

Several features about Maxwell's equations in the Coulomb gauge are as follows. Firstly, solving for the electric potential is very easy, as the equation is a version of Poisson's equation. Secondly, solving for the magnetic vector potential is particularly difficult. This is the big disadvantage of this gauge. The third thing to note, and something that is not immediately obvious, is that the electric potential changes instantly everywhere in response to a change in conditions in one locality.

For instance, if a charge is moved in New York at 1 pm local time, then a hypothetical observer in Australia who could measure the electric potential directly would measure a change in the potential at 1 pm New York time. This seemingly violates causality in special relativity, i.e. the impossibility of information, signals, or anything travelling faster than the speed of light. The resolution to this apparent problem lies in the fact that, as previously stated, no observers can measure the potentials; they measure the electric and magnetic fields. So, the combination of ∇φ and ∂A/∂t used in determining the electric field restores the speed limit imposed by special relativity for the electric field, making all observable quantities consistent with relativity.

Lorenz gauge condition

Main article: Lorenz gauge condition

A gauge that is often used is the Lorenz gauge condition. In this, the scalar function λ is chosen such that $${\displaystyle \mathrm{\nabla }\cdot {\mathbf{A}}^{\prime }=-{\mu }_0{\epsilon }_0\frac{\mathrm{\partial }{\phi }^{\prime }}{\mathrm{\partial }t},}$$ meaning that λ must satisfy the equation $${\displaystyle {\mathrm{\nabla }}^2\lambda -{\mu }_0{\epsilon }_0\frac{{\mathrm{\partial }}^2\lambda }{\mathrm{\partial }{t}^2}=-\mathrm{\nabla }\cdot \mathbf{A}-{\mu }_0{\epsilon }_0\frac{\mathrm{\partial }\phi }{\mathrm{\partial }t}.}$$

The Lorenz gauge results in the following form of Maxwell's equations: $${\displaystyle {\mathrm{\nabla }}^2{\phi }^{\prime }-{\mu }_0{\epsilon }_0\frac{{\mathrm{\partial }}^2{\phi }^{\prime }}{\mathrm{\partial }{t}^2}=-{◻}^2{\phi }^{\prime }=-\frac{\rho }{{\epsilon }_0}}$$ $${\displaystyle {\mathrm{\nabla }}^2{\mathbf{A}}^{\prime }-{\mu }_0{\epsilon }_0\frac{{\mathrm{\partial }}^2{\mathbf{A}}^{\prime }}{\mathrm{\partial }{t}^2}=-{◻}^2{\mathbf{A}}^{\prime }=-{\mu }_0\mathbf{J}}$$

The operator ${\displaystyle {◻}^2}$ is called the d'Alembertian (some authors denote this by only the square ${\displaystyle ◻}$). These equations are inhomogeneous versions of the wave equation, with the terms on the right side of the equation serving as the source functions for the wave. As with any wave equation, these equations lead to two types of solution: advanced potentials (which are related to the configuration of the sources at future points in time), and retarded potentials (which are related to the past configurations of the sources); the former are usually disregarded where the field is to analyzed from a causality perspective.

As pointed out above, the Lorenz gauge is no more valid than any other gauge since the potentials cannot be directly measured, however the Lorenz gauge has the advantage of the equations being Lorentz invariant.

Extension to quantum electrodynamics

Canonical quantization of the electromagnetic fields proceeds by elevating the scalar and vector potentials; φ(x), A(x), from fields to field operators. Substituting 1/c² = ε₀μ₀ into the previous Lorenz gauge equations gives:

$${\displaystyle {\mathrm{\nabla }}^2\mathbf{A}-\frac{1}{c^2}\frac{{\mathrm{\partial }}^2\mathbf{A}}{\mathrm{\partial }{t}^2}=-{\mu }_0\mathbf{J}}$$ $${\displaystyle {\mathrm{\nabla }}^2\phi -\frac{1}{c^2}\frac{{\mathrm{\partial }}^2\phi }{\mathrm{\partial }{t}^2}=-\frac{\rho }{{\epsilon }_0}}$$

Here, J and ρ are the current and charge density of the matter field. If the matter field is taken so as to describe the interaction of electromagnetic fields with the Dirac electron given by the four-component Dirac spinor field ψ, the current and charge densities have form: $${\displaystyle \mathbf{J}=-e{\psi }^{†}\mathit{\alpha }\psi \rho =-e{\psi }^{†}\psi ,}$$ where α are the first three Dirac matrices. Using this, we can re-write Maxwell's equations as:

Maxwell's equations (QED)

${\displaystyle {\mathrm{\nabla }}^2\mathbf{A}-\frac{1}{c^2}\frac{{\mathrm{\partial }}^2\mathbf{A}}{\mathrm{\partial }{t}^2}={\mu }_{0}e{\psi }^{†}\mathit{\alpha }\psi }$

${\displaystyle {\mathrm{\nabla }}^2\phi -\frac{1}{c^2}\frac{{\mathrm{\partial }}^2\phi }{\mathrm{\partial }{t}^2}=\frac{1}{{\epsilon }_0}e{\psi }^{†}\psi }$

which is the form used in quantum electrodynamics.

Geometric algebra formulations

Analogous to the tensor formulation, two objects, one for the electromagnetic field and one for the current density, are introduced. In geometric algebra (GA) these are multivectors, which sometimes follow Ricci calculus.

Algebra of physical space

In the Algebra of physical space (APS), also known as the Clifford algebra ${\displaystyle C{\ell }_{3,0}(\mathbb{R})}$, the field and current are represented by multivectors.

The field multivector, known as the Riemann–Silberstein vector, is $${\displaystyle \mathbf{F}=\mathbf{E}+Ic\mathbf{B}=E^k{\sigma }_k+Ic{B}^k{\sigma }_k,}$$ and the four-current multivector is $${\displaystyle c\rho -\mathbf{J}=c\rho -J^k{\sigma }_k}$$ using an orthonormal basis ${\displaystyle \{{\sigma }_k\}}$. Similarly, the unit pseudoscalar is ${\displaystyle I={\sigma }_1{\sigma }_2{\sigma }_3}$, due to the fact that the basis used is orthonormal. These basis vectors share the algebra of the Pauli matrices, but are usually not equated with them, as they are different objects with different interpretations.

After defining the derivative $${\displaystyle \mathbf{\nabla }={\sigma }^k{\mathrm{\partial }}_k,}$$

Maxwell's equations are reduced to the single equation

Maxwell's equations (APS formulation)

${\displaystyle \left(\frac{1}{c}{\displaystyle \frac{\mathrm{\partial }}{\mathrm{\partial }t}}+\mathbf{\nabla }\right)\mathbf{F}={\mu }_{0}c(c\rho -\mathbf{J}).}$

In three dimensions, the derivative has a special structure allowing the introduction of a cross product: $${\displaystyle \mathbf{\nabla }\mathbf{F}=\mathbf{\nabla }\cdot \mathbf{F}+\mathbf{\nabla }\wedge \mathbf{F}=\mathbf{\nabla }\cdot \mathbf{F}+I\mathbf{\nabla }\times \mathbf{F}}$$ from which it is easily seen that Gauss's law is the scalar part, the Ampère–Maxwell law is the vector part, Faraday's law is the pseudovector part, and Gauss's law for magnetism is the pseudoscalar part of the equation. After expanding and rearranging, this can be written as $${\displaystyle \left(\mathbf{\nabla }\cdot \mathbf{E}-\frac{\rho }{{\epsilon }_0}\right)-c\left(\mathbf{\nabla }\times \mathbf{B}-{\mu }_0{\epsilon }_0\frac{\mathrm{\partial }\mathbf{E}}{\mathrm{\partial }t}-{\mu }_0\mathbf{J}\right)+I\left(\mathbf{\nabla }\times \mathbf{E}+\frac{\mathrm{\partial }\mathbf{B}}{\mathrm{\partial }t}\right)+Ic\left(\mathbf{\nabla }\cdot \mathbf{B}\right)=0}$$

Spacetime algebra

Further information: Spacetime algebra § Classical electromagnetism

We can identify APS as a subalgebra of the spacetime algebra (STA) ${\displaystyle C{\ell }_{1,3}(\mathbb{R})}$, defining ${\displaystyle {\sigma }_k={\gamma }_k{\gamma }_0}$ and ${\displaystyle I={\gamma }_0{\gamma }_1{\gamma }_2{\gamma }_3}$. The ${\displaystyle {\gamma }_{\mu }}$s have the same algebraic properties of the gamma matrices but their matrix representation is not needed. The derivative is now $${\displaystyle \mathrm{\nabla }={\gamma }^{\mu }{\mathrm{\partial }}_{\mu }.}$$

The Riemann–Silberstein becomes a bivector $${\displaystyle F=\mathbf{E}+Ic\mathbf{B}=E^1{\gamma }_1{\gamma }_0+E^2{\gamma }_2{\gamma }_0+E^3{\gamma }_3{\gamma }_0-c(B^1{\gamma }_2{\gamma }_3+B^2{\gamma }_3{\gamma }_1+B^3{\gamma }_1{\gamma }_2),}$$ and the charge and current density become a vector $${\displaystyle J=J^{\mu }{\gamma }_{\mu }=c\rho {\gamma }_0+J^k{\gamma }_k={\gamma }_0(c\rho -J^k{\sigma }_k).}$$

Owing to the identity $${\displaystyle {\gamma }_0\mathrm{\nabla }={\gamma }_0{\gamma }^0{\mathrm{\partial }}_0+{\gamma }_0{\gamma }^k{\mathrm{\partial }}_k={\mathrm{\partial }}_0+{\sigma }^k{\mathrm{\partial }}_k=\frac{1}{c}{\displaystyle \frac{\mathrm{\partial }}{\mathrm{\partial }t}}+\mathbf{\nabla },}$$

Maxwell's equations reduce to the single equation

Maxwell's equations (STA formulation)

${\displaystyle \mathrm{\nabla }F={\mu }_{0}cJ.}$

Differential forms approach

See also: Differential form and Exterior algebra

In what follows, cgs-Gaussian units, not SI units are used. (To convert to SI, see here.) By Einstein notation, we implicitly take the sum over all values of the indices that can vary within the dimension.

Field 2-form

In free space, where ε = ε₀ and μ = μ₀ are constant everywhere, Maxwell's equations simplify considerably once the language of differential geometry and differential forms is used. The electric and magnetic fields are now jointly described by a 2-form F in a 4-dimensional spacetime manifold. The Faraday tensor ${\displaystyle F_{\mu \nu }}$ (electromagnetic tensor) can be written as a 2-form in Minkowski space with metric signature (− + + +) as $${\displaystyle \begin{array}{rl}\mathbf{F}& \equiv \frac{1}{2}{F}_{\mu \nu }\mathrm{d}{x}^{\mu }\wedge \mathrm{d}{x}^{\nu }\\ & =B_x\mathrm{d}y\wedge \mathrm{d}z+B_y\mathrm{d}z\wedge \mathrm{d}x+B_z\mathrm{d}x\wedge \mathrm{d}y+E_x\mathrm{d}x\wedge \mathrm{d}t+E_y\mathrm{d}y\wedge \mathrm{d}t+E_z\mathrm{d}z\wedge \mathrm{d}t\end{array}}$$ which is the exterior derivative of the electromagnetic four-potential ${\displaystyle \mathbf{A}:}$ $${\displaystyle \mathbf{A}=-\varphi \mathrm{d}t+A_x\mathrm{d}x+A_y\mathrm{d}y+A_z\mathrm{d}z.}$$

The source free equations can be written by the action of the exterior derivative on this 2-form. But for the equations with source terms (Gauss's law and the Ampère-Maxwell equation), the Hodge dual of this 2-form is needed. The Hodge star operator takes a p-form to a (n − p)-form, where n is the number of dimensions. Here, it takes the 2-form (F) and gives another 2-form (in four dimensions, n − p = 4 − 2 = 2). For the basis cotangent vectors, the Hodge dual is given as (see Hodge star operator § Four dimensions) $${\displaystyle \star (\mathrm{d}x\wedge \mathrm{d}y)=-\mathrm{d}z\wedge \mathrm{d}t,\star (\mathrm{d}x\wedge \mathrm{d}t)=\mathrm{d}y\wedge \mathrm{d}z,}$$ and so on. Using these relations, the dual of the Faraday 2-form is the Maxwell tensor, $${\displaystyle \star \mathbf{F}=-B_x\mathrm{d}x\wedge \mathrm{d}t-B_y\mathrm{d}y\wedge \mathrm{d}t-B_z\mathrm{d}z\wedge \mathrm{d}t+E_x\mathrm{d}y\wedge \mathrm{d}z+E_y\mathrm{d}z\wedge \mathrm{d}x+E_z\mathrm{d}x\wedge \mathrm{d}y}$$

Current 3-form, dual current 1-form

Here, the 3-form J is called the electric current form or current 3-form: $${\displaystyle \mathbf{J}=\rho \mathrm{d}x\wedge \mathrm{d}y\wedge \mathrm{d}z-j_x\mathrm{d}t\wedge \mathrm{d}y\wedge \mathrm{d}z-j_y\mathrm{d}t\wedge \mathrm{d}z\wedge \mathrm{d}x-j_z\mathrm{d}t\wedge \mathrm{d}x\wedge \mathrm{d}y.}$$

That F is a closed form, and the exterior derivative of its Hodge dual is the current 3-form, express Maxwell's equations:

Maxwell's equations

${\displaystyle \mathrm{d}\mathbf{F}=0}$

${\displaystyle \mathrm{d}\star \mathbf{F}=\mathbf{J}}$

Here d denotes the exterior derivative – a natural coordinate- and metric-independent differential operator acting on forms, and the (dual) Hodge star operator ${\displaystyle \star }$ is a linear transformation from the space of 2-forms to the space of (4 − 2)-forms defined by the metric in Minkowski space (in four dimensions even by any metric conformal to this metric). The fields are in natural units where 1/(4πε₀) = 1.

Since d² = 0, the 3-form J satisfies the conservation of current (continuity equation): $${\displaystyle \mathrm{d}\mathbf{J}={\mathrm{d}}^2\star \mathbf{F}=0.}$$ The current 3-form can be integrated over a 3-dimensional space-time region. The physical interpretation of this integral is the charge in that region if it is spacelike, or the amount of charge that flows through a surface in a certain amount of time if that region is a spacelike surface cross a timelike interval. As the exterior derivative is defined on any manifold, the differential form version of the Bianchi identity makes sense for any 4-dimensional manifold, whereas the source equation is defined if the manifold is oriented and has a Lorentz metric. In particular the differential form version of the Maxwell equations are a convenient and intuitive formulation of the Maxwell equations in general relativity.

Note: In much of the literature, the notations ${\displaystyle \mathbf{J}}$ and ${\displaystyle \star \mathbf{J}}$ are switched, so that ${\displaystyle \mathbf{J}}$ is a 1-form called the current and ${\displaystyle \star \mathbf{J}}$ is a 3-form called the dual current.

Linear macroscopic influence of matter

In a linear, macroscopic theory, the influence of matter on the electromagnetic field is described through more general linear transformation in the space of 2-forms. We call $${\displaystyle C:{\mathrm{\Lambda }}^2\ni \mathbf{F}\mapsto \mathbf{G}\in {\mathrm{\Lambda }}^{(4-2)}}$$ the constitutive transformation. The role of this transformation is comparable to the Hodge duality transformation. The Maxwell equations in the presence of matter then become: $${\displaystyle \mathrm{d}\mathbf{F}=0}$$ $${\displaystyle \mathrm{d}\mathbf{G}=\mathbf{J}}$$ where the current 3-form J still satisfies the continuity equation dJ = 0.

When the fields are expressed as linear combinations (of exterior products) of basis forms θⁱ, $${\displaystyle \mathbf{F}=\frac{1}{2}{F}_{pq}{\theta }^p\wedge {\theta }^q.}$$ the constitutive relation takes the form $${\displaystyle G_{pq}=C_{pq}^{mn}{F}_{mn}}$$ where the field coefficient functions and the constitutive coefficients are anticommutative for swapping of each one's indices. In particular, the Hodge star operator that was used in the above case is obtained by taking $${\displaystyle C_{pq}^{mn}=\frac{1}{2}{g}^{ma}{g}^{nb}{\epsilon }_{abpq}\sqrt{-g}}$$ in terms of tensor index notation with respect to a (not necessarily orthonormal) basis $\left\{\frac{\mathrm{\partial }}{\mathrm{\partial }{x}_1},\dots ,\frac{\mathrm{\partial }}{\mathrm{\partial }{x}_n}\right\}$ in a tangent space ${\displaystyle V=T_{p}M}$ and its dual basis ${\displaystyle \{d{x}_1,\dots ,d{x}_n\}}$ in ${\displaystyle V^{\ast }=T_p^{\ast }M}$, having the gram metric matrix $(g_{ij})=\left(⟨\frac{\mathrm{\partial }}{\mathrm{\partial }{x}_i},\frac{\mathrm{\partial }}{\mathrm{\partial }{x}_j}⟩\right)$ and its inverse matrix ${\displaystyle (g^{ij})=(⟨d{x}^i,d{x}^j⟩)}$, and ${\displaystyle {\epsilon }_{abpq}}$ is the Levi-Civita symbol with ${\displaystyle {\epsilon }_{1234}=1}$. Up to scaling, this is the only invariant tensor of this type that can be defined with the metric.

In this formulation, electromagnetism generalises immediately to any 4-dimensional oriented manifold or with small adaptations any manifold.

Alternative metric signature

In the particle physicist's sign convention for the metric signature (+ − − −), the potential 1-form is $${\displaystyle \mathbf{A}=\varphi \mathrm{d}t-A_x\mathrm{d}x-A_y\mathrm{d}y-A_z\mathrm{d}z.}$$

The Faraday curvature 2-form becomes $${\displaystyle \begin{array}{rl}\mathbf{F}\equiv & \frac{1}{2}{F}_{\mu \nu }\mathrm{d}{x}^{\mu }\wedge \mathrm{d}{x}^{\nu }\\ =& E_x\mathrm{d}t\wedge \mathrm{d}x+E_y\mathrm{d}t\wedge \mathrm{d}y+E_z\mathrm{d}t\wedge \mathrm{d}z-B_x\mathrm{d}y\wedge \mathrm{d}z-B_y\mathrm{d}z\wedge \mathrm{d}x-B_z\mathrm{d}x\wedge \mathrm{d}y\end{array}}$$ and the Maxwell tensor becomes $${\displaystyle \star \mathbf{F}=-E_x\mathrm{d}y\wedge \mathrm{d}z-E_y\mathrm{d}z\wedge \mathrm{d}x-E_z\mathrm{d}x\wedge \mathrm{d}y-B_x\mathrm{d}t\wedge \mathrm{d}x-B_y\mathrm{d}t\wedge \mathrm{d}y-B_z\mathrm{d}t\wedge \mathrm{d}z.}$$

The current 3-form J is $${\displaystyle \mathbf{J}=-\rho \mathrm{d}x\wedge \mathrm{d}y\wedge \mathrm{d}z+j_x\mathrm{d}t\wedge \mathrm{d}y\wedge \mathrm{d}z+j_y\mathrm{d}t\wedge \mathrm{d}z\wedge \mathrm{d}x+j_z\mathrm{d}t\wedge \mathrm{d}x\wedge \mathrm{d}y}$$ and the corresponding dual 1-form is $${\displaystyle \star \mathbf{J}=-\rho \mathrm{d}t+j_x\mathrm{d}x+j_y\mathrm{d}y+j_z\mathrm{d}z.}$$

The current norm is now positive and equals $${\displaystyle \mathbf{J}\wedge \star \mathbf{J}=[{\rho }^2+(j_x{)}^2+(j_y{)}^2+(j_z{)}^2]\star (1)}$$ with the canonical volume form ${\displaystyle \star (1)=\mathrm{d}t\wedge \mathrm{d}x\wedge \mathrm{d}y\wedge \mathrm{d}z}$.

Curved spacetime

Main article: Maxwell's equations in curved spacetime

Traditional formulation

Matter and energy generate curvature of spacetime. This is the subject of general relativity. Curvature of spacetime affects electrodynamics. An electromagnetic field having energy and momentum also generates curvature in spacetime. Maxwell's equations in curved spacetime can be obtained by replacing the derivatives in the equations in flat spacetime with covariant derivatives. (Whether this is the appropriate generalization requires separate investigation.) The sourced and source-free equations become (cgs-Gaussian units): $${\displaystyle \frac{4\pi }{c}{j}^{\beta }={\mathrm{\partial }}_{\alpha }{F}^{\alpha \beta }+{{\mathrm{\Gamma }}^{\alpha }}_{\mu \alpha }{F}^{\mu \beta }+{{\mathrm{\Gamma }}^{\beta }}_{\mu \alpha }{F}^{\alpha \mu }\stackrel{\mathrm{d}\mathrm{e}\mathrm{f}}{=}{\mathrm{\nabla }}_{\alpha }{F}^{\alpha \beta }\stackrel{\mathrm{d}\mathrm{e}\mathrm{f}}{=}{F^{\alpha \beta }}_{;\alpha }}$$ and $${\displaystyle 0={\mathrm{\partial }}_{\gamma }{F}_{\alpha \beta }+{\mathrm{\partial }}_{\beta }{F}_{\gamma \alpha }+{\mathrm{\partial }}_{\alpha }{F}_{\beta \gamma }={\mathrm{\nabla }}_{\gamma }{F}_{\alpha \beta }+{\mathrm{\nabla }}_{\beta }{F}_{\gamma \alpha }+{\mathrm{\nabla }}_{\alpha }{F}_{\beta \gamma }.}$$

Here, $${\displaystyle {{\mathrm{\Gamma }}^{\alpha }}_{\mu \beta }}$$ is a Christoffel symbol that characterizes the curvature of spacetime and ∇_α is the covariant derivative.

Formulation in terms of differential forms

The formulation of the Maxwell equations in terms of differential forms can be used without change in general relativity. The equivalence of the more traditional general relativistic formulation using the covariant derivative with the differential form formulation can be seen as follows. Choose local coordinates x^α that gives a basis of 1-forms dx^α in every point of the open set where the coordinates are defined. Using this basis and cgs-Gaussian units we define

• The antisymmetric field tensor F_αβ, corresponding to the field 2-form F $${\displaystyle \mathbf{F}=\frac{1}{2}{F}_{\alpha \beta }\mathrm{d}{x}^{\alpha }\wedge \mathrm{d}{x}^{\beta }.}$$
• The current-vector infinitesimal 3-form J $${\displaystyle \mathbf{J}=\frac{4\pi }{c}\left(\frac{1}{6}{j}^{\alpha }\sqrt{-g}{\epsilon }_{\alpha \beta \gamma \delta }\mathrm{d}{x}^{\beta }\wedge \mathrm{d}{x}^{\gamma }\wedge \mathrm{d}{x}^{\delta }.\right)}$$

The epsilon tensor contracted with the differential 3-form produces 6 times the number of terms required.

Here g is as usual the determinant of the matrix representing the metric tensor, g_αβ. A small computation that uses the symmetry of the Christoffel symbols (i.e., the torsion-freeness of the Levi-Civita connection) and the covariant constantness of the Hodge star operator then shows that in this coordinate neighborhood we have:

• the Bianchi identity $${\displaystyle \mathrm{d}\mathbf{F}=2({\mathrm{\partial }}_{\gamma }{F}_{\alpha \beta }+{\mathrm{\partial }}_{\beta }{F}_{\gamma \alpha }+{\mathrm{\partial }}_{\alpha }{F}_{\beta \gamma })\mathrm{d}{x}^{\alpha }\wedge \mathrm{d}{x}^{\beta }\wedge \mathrm{d}{x}^{\gamma }=0,}$$
• the source equation $${\displaystyle \mathrm{d}\star \mathbf{F}=\frac{1}{6}{F^{\alpha \beta }}_{;\alpha }\sqrt{-g}{\epsilon }_{\beta \gamma \delta \alpha }\mathrm{d}{x}^{\gamma }\wedge \mathrm{d}{x}^{\delta }\wedge \mathrm{d}{x}^{\alpha }=\mathbf{J},}$$
• the continuity equation $${\displaystyle \mathrm{d}\mathbf{J}=\frac{4\pi }{c}{j^{\alpha }}_{;\alpha }\sqrt{-g}{\epsilon }_{\alpha \beta \gamma \delta }\mathrm{d}{x}^{\alpha }\wedge \mathrm{d}{x}^{\beta }\wedge \mathrm{d}{x}^{\gamma }\wedge \mathrm{d}{x}^{\delta }=0.}$$

Classical electrodynamics as the curvature of a line bundle

An elegant and intuitive way to formulate Maxwell's equations is to use complex line bundles or a principal ${\displaystyle \mathrm{U}(1)}$-bundle, on the fibers of which U(1) acts regularly. The principal U(1)-connection ∇ on the line bundle has a curvature F = ∇², which is a two-form that automatically satisfies dF = 0 and can be interpreted as a field strength. If the line bundle is trivial with flat reference connection d we can write ∇ = d + A and F = dA with A the 1-form composed of the electric potential and the magnetic vector potential.

In quantum mechanics, the connection itself is used to define the dynamics of the system. This formulation allows a natural description of the Aharonov–Bohm effect. In this experiment, a static magnetic field runs through a long magnetic wire (e.g., an iron wire magnetized longitudinally). Outside of this wire the magnetic induction is zero, in contrast to the vector potential, which essentially depends on the magnetic flux through the cross-section of the wire and does not vanish outside. Since there is no electric field either, the Maxwell tensor F = 0 throughout the space-time region outside the tube, during the experiment. This means by definition that the connection ∇ is flat there.

In mentioned Aharonov–Bohm effect, however, the connection depends on the magnetic field through the tube since the holonomy along a non-contractible curve encircling the tube is the magnetic flux through the tube in the proper units. This can be detected quantum-mechanically with a double-slit electron diffraction experiment on an electron wave traveling around the tube. The holonomy corresponds to an extra phase shift, which leads to a shift in the diffraction pattern.

Discussion and other approaches

Following are the reasons for using each of such formulations.

Potential formulation

In advanced classical mechanics it is often useful, and in quantum mechanics frequently essential, to express Maxwell's equations in a potential formulation involving the electric potential (also called scalar potential) φ, and the magnetic potential (a vector potential) A. For example, the analysis of radio antennas makes full use of Maxwell's vector and scalar potentials to separate the variables, a common technique used in formulating the solutions of differential equations. The potentials can be introduced by using the Poincaré lemma on the homogeneous equations to solve them in a universal way (this assumes that we consider a topologically simple, e.g. contractible space). The potentials are defined as in the table above. Alternatively, these equations define E and B in terms of the electric and magnetic potentials that then satisfy the homogeneous equations for E and B as identities. Substitution gives the non-homogeneous Maxwell equations in potential form.

Many different choices of A and φ are consistent with given observable electric and magnetic fields E and B, so the potentials seem to contain more, (classically) unobservable information. The non uniqueness of the potentials is well understood, however. For every scalar function of position and time λ(x, t), the potentials can be changed by a gauge transformation as $${\displaystyle {\phi }^{\prime }=\phi -\frac{\mathrm{\partial }\lambda }{\mathrm{\partial }t},{\mathbf{A}}^{\prime }=\mathbf{A}+\mathrm{\nabla }\lambda }$$ without changing the electric and magnetic field. Two pairs of gauge transformed potentials (φ, A) and (φ′, A′) are called gauge equivalent, and the freedom to select any pair of potentials in its gauge equivalence class is called gauge freedom. Again by the Poincaré lemma (and under its assumptions), gauge freedom is the only source of indeterminacy, so the field formulation is equivalent to the potential formulation if we consider the potential equations as equations for gauge equivalence classes.

The potential equations can be simplified using a procedure called gauge fixing. Since the potentials are only defined up to gauge equivalence, we are free to impose additional equations on the potentials, as long as for every pair of potentials there is a gauge equivalent pair that satisfies the additional equations (i.e. if the gauge fixing equations define a slice to the gauge action). The gauge-fixed potentials still have a gauge freedom under all gauge transformations that leave the gauge fixing equations invariant. Inspection of the potential equations suggests two natural choices. In the Coulomb gauge, we impose ∇ ⋅ A = 0, which is mostly used in the case of magneto statics when we can neglect the c⁻²∂²A/∂t² term. In the Lorenz gauge (named after the Dane Ludvig Lorenz), we impose $${\displaystyle \mathrm{\nabla }\cdot \mathbf{A}+\frac{1}{c^2}\frac{\mathrm{\partial }\phi }{\mathrm{\partial }t}=0.}$$ The Lorenz gauge condition has the advantage of being Lorentz invariant and leading to Lorentz-invariant equations for the potentials.

Manifestly covariant (tensor) approach

Main articles: Classical electromagnetism and special relativity and Covariant formulation of classical electromagnetism

Maxwell's equations are exactly consistent with special relativity—i.e., if they are valid in one inertial reference frame, then they are automatically valid in every other inertial reference frame. In fact, Maxwell's equations were crucial in the historical development of special relativity. However, in the usual formulation of Maxwell's equations, their consistency with special relativity is not obvious; it can only be proven by a laborious calculation.

For example, consider a conductor moving in the field of a magnet. In the frame of the magnet, that conductor experiences a magnetic force. But in the frame of a conductor moving relative to the magnet, the conductor experiences a force due to an electric field. The motion is exactly consistent in these two different reference frames, but it mathematically arises in quite different ways.

For this reason and others, it is often useful to rewrite Maxwell's equations in a way that is "manifestly covariant"—i.e. obviously consistent with special relativity, even with just a glance at the equations—using covariant and contravariant four-vectors and tensors. This can be done using the EM tensor F, or the 4-potential A, with the 4-current J.

Differential forms approach

Gauss's law for magnetism and the Faraday–Maxwell law can be grouped together since the equations are homogeneous, and be seen as geometric identities expressing the field F (a 2-form), which can be derived from the 4-potential A. Gauss's law for electricity and the Ampere–Maxwell law could be seen as the dynamical equations of motion of the fields, obtained via the Lagrangian principle of least action, from the "interaction term" AJ (introduced through gauge covariant derivatives), coupling the field to matter. For the field formulation of Maxwell's equations in terms of a principle of extremal action, see electromagnetic tensor.

Often, the time derivative in the Faraday–Maxwell equation motivates calling this equation "dynamical", which is somewhat misleading in the sense of the preceding analysis. This is rather an artifact of breaking relativistic covariance by choosing a preferred time direction. To have physical degrees of freedom propagated by these field equations, one must include a kinetic term F ⋆F for A, and take into account the non-physical degrees of freedom that can be removed by gauge transformation A ↦ A − dα. See also gauge fixing and Faddeev–Popov ghosts.

Geometric calculus approach

This formulation uses the algebra that spacetime generates through the introduction of a distributive, associative (but not commutative) product called the geometric product. Elements and operations of the algebra can generally be associated with geometric meaning. The members of the algebra may be decomposed by grade (as in the formalism of differential forms) and the (geometric) product of a vector with a k-vector decomposes into a (k − 1)-vector and a (k + 1)-vector. The (k − 1)-vector component can be identified with the inner product and the (k + 1)-vector component with the outer product. It is of algebraic convenience that the geometric product is invertible, while the inner and outer products are not. As such, powerful techniques such as Green's functions can be used. The derivatives that appear in Maxwell's equations are vectors and electromagnetic fields are represented by the Faraday bivector F. This formulation is as general as that of differential forms for manifolds with a metric tensor, as then these are naturally identified with r-forms and there are corresponding operations. Maxwell's equations reduce to one equation in this formalism. This equation can be separated into parts as is done above for comparative reasons.