Definition.

From Coulomb's law, a particle with electric charge formula_1 at position formula_2 exerts a force on a particle with charge formula_3 at position formula_4 of When the charges formula_3 and formula_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 formula_4 this expression can be divided by formula_3 leaving an expression that only depends on the other charge (the "source" charge) This is the "electric field" at point formula_4 due to the point charge formula_1; it is a vector equal to the Coulomb force per unit charge that a positive point charge would experience at the position formula_4. Since this formula gives the electric field magnitude and direction at any point formula_4 in space (except at the location of the charge itself, formula_2, 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.  If there are multiple charges, the resultant Coulomb force on a charge can be found by summing the vectors of the forces due to each charge. This shows the electric field obeys the "superposition principle": the total electric field at a point due to a collection of charges is just equal to the vector sum of the electric fields at that point due to the individual charges. This is the definition of the electric field due to the point "source charges" formula_24. It diverges and becomes infinite at the locations of the charges themselves, and so is not defined there.  The Coulomb force on a charge of magnitude formula_25 at any point in space is equal to the product of the charge and the electric field at that point The units of the electric field in the SI system are newtons per coulomb (N/C), or volts per meter (V/m); in terms of the SI base units they are kg⋅m⋅s⋅A  The electric field due to a continuous distribution of charge formula_27 in space (where formula_28 is the charge density in coulombs per cubic meter) can be calculated by considering the charge formula_29 in each small volume of space formula_30 at point formula_31 as a point charge, and calculating its electric field formula_32 at point formula_23 where formula_35 is the unit vector pointing from formula_31 to formula_23, then adding up the contributions from all the increments of volume by integrating over the volume of the charge distribution formula_38

Sources.

Causes and description.

Electric fields are caused by electric charges, described by Gauss's law, or varying magnetic fields, described by Faraday's law of induction. Together, these laws are enough to define the behavior of the electric field as a function of and magnetic 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.  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 formula_40 and Faraday's law with no induction term formula_41), taken together, are equivalent to Coulomb's law, written as formula_42 for a charge density formula_43 (formula_44 is position in space). Notice that formula_45, the vacuum electric permittivity, must be substituted with formula_46, permittivity, when charges are in non-empty media.

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 formula_25 located at formula_48 can be described mathematically as a charge density formula_49, where the Dirac delta function (in three dimensions) is used. Conversely, a charge distribution can be approximated by many small point charges.

Superposition principle.

Electric fields satisfy the superposition principle, because Maxwell's equations are linear. As a result, if formula_50 and formula_51 are the electric fields resulting from distribution of charges formula_52 and formula_53, a distribution of charges formula_54 will create an electric field formula_55; for instance, Coulomb's law is linear in charge density as well.  This principle is useful to calculate the field created by multiple point charges. If charges formula_56 are stationary in space at formula_57, in the absence of currents, the superposition principle proves that the resulting field is the sum of fields generated by each particle as described by Coulomb's law:  This suggests similarities between the electric field E and the gravitational field g, 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, electric field is distorted because the plane does not continue). Assuming infinite planes, the magnitude of the electric field "E" is:  where Δ"V" is the potential difference between the plates and "d" 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, achieved by applying a voltage of the order of 1 volt between conductors spaced 1 μm apart.

Electrodynamic fields.

Electrodynamic fields are electric fields which do change with time, for instance when charges are in motion.  The electric field cannot be described independently of the magnetic field in that case. If A is the magnetic vector potential, defined so that formula_60, one can still define an electric potential formula_61 such that:  One can recover Faraday's law of induction by taking the curl of that equation  which justifies, a posteriori, the previous form for E.

Energy in the electric field.

The total energy per unit volume stored by the electromagnetic field is  where "ε" is the permittivity of the medium in which the field exists, formula_65 its magnetic permeability, and E and B are the electric and magnetic field vectors.  As E and B fields are coupled, it would be misleading to split this expression into "electric" and "magnetic" contributions. However, in the steady-state case, the fields are no longer coupled (see Maxwell's equations). It makes sense in that case to compute the electrostatic energy per unit volume:  The total energy "U" stored in the electric field in a given volume "V" is therefore

Further extensions.

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:  where P is the electric polarization – the volume density of electric dipole moments, and D is the electric displacement field. Since E and P are defined separately, this equation can be used to define D. The physical interpretation of D is not as clear as E (effectively the field applied to the material) or P (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 E and D fields are related by the permittivity of the material, "ε".  For linear, homogeneous, isotropic materials E and D are proportional and constant throughout the region, there is no position dependence: For inhomogeneous materials, there is a position dependence throughout the material:  For anisotropic materials the E and D fields are not parallel, and so E and D are related by the permittivity tensor (a 2nd order tensor field), in component form:  For non-linear media, E and D are not proportional. Materials can have varying extents of linearity, homogeneity and isotropy.

An electric field (sometimes abbreviated as E-field) surrounds an electric charge, and exerts force on other charges in the field, attracting or repelling them. Electric fields are created by electric charges, or by time-varying magnetic fields. Electric fields and magnetic fields are both manifestations of the electromagnetic force, one of the four fundamental forces (or interactions) of nature.

电场是存在于电荷周围能传递电荷与电荷之间相互作用的物理场。在电荷周围总有电场存在;同时电场对场中其他电荷发生力的作用。观察者相对于电荷静止时所观察到的场称为静电场。如果电荷相对于观察者运动,则除静电场外,还有磁场出现。除了电荷以外,随着时间流逝而变化的磁场也可以生成电场,这种电场叫做涡旋电场或感应电场。迈克尔·法拉第最先提出电场的概念。


Terminology.

The term "loudspeaker" may refer to individual transducers (also known as "drivers") or to complete speaker systems consisting of an enclosure including one or more drivers. To adequately reproduce a wide range of frequencies with even coverage, most loudspeaker systems employ more than one driver, particularly for higher sound pressure level or maximum accuracy. Individual drivers are used to reproduce different frequency ranges. The drivers are named subwoofers (for very low frequencies); woofers (low frequencies); mid-range speakers (middle frequencies); tweeters (high frequencies); and sometimes supertweeters, optimized for the highest audible frequencies. The terms for different speaker drivers differ,

History.

Johann Philipp Reis installed an electric loudspeaker in his "telephone" in 1861; it was capable of reproducing clear tones, but also could reproduce muffled speech after a few revisions. Alexander Graham Bell patented his first electric loudspeaker (capable of reproducing intelligible speech) as part of his telephone in 1876, which was followed in 1877 by an improved version from Ernst Siemens. During this time, Thomas Edison was issued a British patent for a system using compressed air as an amplifying mechanism for his early cylinder phonographs, but he ultimately settled for the familiar metal horn driven by a membrane attached to the stylus. In 1898, Horace Short patented a design for a loudspeaker driven by compressed air; he then sold the rights to Charles Parsons, who was issued several additional British patents before 1910. A few companies, including the Victor Talking Machine Company and Pathé, produced record players using compressed-air loudspeakers. However, these designs were significantly limited by their poor sound quality and their inability to reproduce sound at low volume. Variants of the system were used for public address applications, and more recently, other variations have been used to test space-equipment resistance to the very loud sound and vibration levels that the launching of rockets produces.

Moving-coil.

The first experimental moving-coil (also called "dynamic") loudspeaker was invented by Oliver Lodge in 1898. The first practical moving-coil loudspeakers were manufactured by Danish engineer Peter L. Jensen and Edwin Pridham in 1915, in Napa, California. Like previous loudspeakers these used horns to amplify the sound produced by a small diaphragm. Jensen was denied patents. Being unsuccessful in selling their product to telephone companies, in 1915 they changed their target market to radios and public address systems, and named their product Magnavox. Jensen was, for years after the invention of the loudspeaker, a part owner of The Magnavox Company.  The moving-coil principle commonly used today in speakers was patented in 1925 by Chester W. Rice and Edward W. Kellogg. The key difference between previous attempts and

First loudspeaker systems.

In the 1930s, loudspeaker manufacturers began to combine two and three bandpasses' worth of drivers in order to increase frequency response and sound pressure level. In 1937, the first film industry-standard loudspeaker system, "The Shearer Horn System for Theatres" (a two-way system), was introduced by Metro-Goldwyn-Mayer. It used four 15′′ low-frequency drivers, a crossover network set for 375 Hz, and a single multi-cellular horn with two compression drivers providing the high frequencies. John Kenneth Hilliard, James Bullough Lansing, and Douglas Shearer all played roles in creating the system. At the 1939 New York World's Fair, a very large two-way public address system was mounted on a tower at Flushing Meadows. The eight 27′′ low-frequency drivers were designed by Rudy Bozak in his role as chief engineer for Cinaudagraph. High-frequency drivers were likely made by Western Electric.  Altec Lansing introduced the "604", which became their most famous coaxial

Driver design: dynamic loudspeakers.

The most common type of driver, commonly called a dynamic loudspeaker, uses a lightweight diaphragm, or "cone", connected to a rigid "basket", or "frame", via a flexible suspension, commonly called a "spider", that constrains a voice coil to move axially through a cylindrical magnetic gap. A protective cap glued in the cone's center prevents dust, especially iron filings, from entering the gap. When an electrical signal is applied to the voice coil, a magnetic field is created by the electric current in the voice coil, making it a variable electromagnet. The coil and the driver's magnetic system interact, generating a mechanical force that causes the coil (and thus, the attached cone) to move back and forth, accelerating and reproducing sound under the control of the applied electrical signal coming from the amplifier. The following is a description of the individual components of this type of loudspeaker.

Diaphragm.

The diaphragm is usually manufactured with a cone- or dome-shaped profile. A variety of different materials may be used, but the most common are paper, plastic, and metal. The ideal material would 1) be rigid, to prevent uncontrolled cone motions; 2) have low mass, to minimize starting force requirements and energy storage issues; 3) be well damped, to reduce vibrations continuing after the signal has stopped with little or no audible ringing due to its resonance frequency as determined by its

Basket.

The chassis, frame, or basket, is designed to be rigid, preventing deformation that could change critical alignments with the magnet gap, perhaps allowing the voice coil to rub against the magnet around the gap. Chassis are typically cast from aluminum alloy, in heavier magnet-structure speakers; or stamped from thin sheet steel in lighter-structure drivers. Other materials such as molded plastic and damped plastic compound baskets are becoming common, especially for inexpensive, low-mass drivers. Metallic chassis can play an important role in conducting heat away from the voice coil; heating during operation changes resistance, causes physical dimensional changes, and if extreme, broils the varnish on the voice

Cone materials.

The cone surround can be rubber or polyester foam, treated paper or a ring of corrugated, resin coated fabric; it is attached to both the outer cone circumference and to the upper frame. These diverse surround materials, their shape and treatment can dramatically affect the acoustic output of a driver; each implementation has advantages and disadvantages. Polyester foam, for example, is lightweight and economical, though usually leaks air to some degree, but is degraded by time, exposure to ozone, UV light, humidity and elevated temperatures, limiting useful life before failure. Treated paper surrounds will eventually fail.  The wire in a voice coil is usually made of copper, though aluminum—and, rarely, silver—may be used. The advantage of aluminum is its light weight, which reduces the moving mass compared to copper. This raises the resonant frequency of the speaker and increases its efficiency. A disadvantage

Driver types.

Individual electrodynamic drivers provide their best performance within a limited frequency range. Multiple drivers (e.g., subwoofers, woofers, mid-range drivers, and tweeters) are generally combined into a complete loudspeaker system to provide performance beyond that constraint. The three most commonly used sound radiation systems are the cone, dome and horn type drivers.

Full-range drivers.

A full-range driver is a speaker designed to be used alone to reproduce an audio channel without the help of other drivers, and therefore must cover the entire audio frequency range. These drivers are small, typically in diameter to permit reasonable high frequency response, and carefully designed to give low-distortion output at low frequencies, though with reduced maximum output level. Full-range (or more accurately, wide-range) drivers are most commonly heard in public address systems, in televisions (although some models are suitable for hi-fi listening), small radios, intercoms, some computer speakers, etc. In hi-fi speaker systems, the use of wide-range drive units can avoid undesirable interactions between multiple drivers caused by non-coincident driver location or crossover network issues. Fans of wide-range driver hi-fi speaker systems claim a coherence of sound due to the single source and a resulting lack of interference, and likely also to the lack of crossover components. Detractors typically cite wide-range drivers' limited frequency response and modest output abilities (most especially at low frequencies), together with

Subwoofer.

A subwoofer is a woofer driver used only for the lowest-pitched part of the audio spectrum: typically below 200 Hz for consumer systems, below 100 Hz for professional live sound, and below 80 Hz in THX-approved systems. Because the intended range of frequencies is limited, subwoofer system design is usually simpler in many respects than for conventional loudspeakers, often consisting of a single driver enclosed in a suitable box or enclosure. Since sound in this frequency range can easily bend around corners by diffraction, the speaker aperture does not have to face the audience, and subwoofers can be mounted in the bottom of the enclosure, facing the floor. This is eased by the limitations of human hearing at low frequencies; such sounds cannot be located in space, due to their large wavelengths compared to higher frequencies which produce differential effects in the ears due to shadowing by the head, and diffraction around it, both of which we rely upon for localization clues.  To accurately reproduce

Woofer.

A woofer is a driver that reproduces low frequencies. The driver works with the characteristics of the enclosure to produce suitable low frequencies (see speaker enclosure for some of the design choices available). Indeed, both are so closely connected that they must be considered together in use. Only at design time

Mid-range driver.

A mid-range speaker is a loudspeaker driver that reproduces a band of frequencies generally between 1–6 kHz, otherwise known as the'mid' frequencies (between the woofer and tweeter). Mid-range driver diaphragms can be made of paper or composite materials, and

Tweeter.

A tweeter is a high-frequency driver that reproduces the highest frequencies in a speaker system. A major problem in tweeter design is achieving wide angular sound coverage (off-axis response), since high frequency sound tends to leave the speaker in narrow beams. Soft-dome

Coaxial drivers.

A coaxial driver is a loudspeaker driver with two or several combined concentric drivers. Coaxial drivers have been produced by many companies, such as Altec, Tannoy, Pioneer, KEF, SEAS, B&C Speakers, BMS, Cabasse and Genelec.

System design.

Crossover.

Used in multi-driver speaker systems, the crossover is an assembly of filters that separate the input signal into different frequency ranges (i.e. "bands"), according to the requirements of each driver. Hence the drivers receive power only at their operating frequency (the sound frequency range they were designed for), thereby reducing distortion in the drivers and interference between them. The ideal characteristics of a crossover may include perfect out-of-band attenuation at the output of each filter, no amplitude variation ("ripple") within each passband, no phase delay between overlapping frequency

Enclosures.

Most loudspeaker systems consist of drivers mounted in an enclosure, or cabinet. The role of the enclosure is to prevent sound waves emanating from the back of a driver from interfering destructively with those from the front. The sound waves emitted from the back are 180° out of phase with those emitted forward, so without an enclosure they typically cause cancellations which significantly degrade the level and quality of sound at low frequencies.  The simplest driver mount is a flat panel (i.e., baffle) with the drivers mounted in holes in it. However, in this approach, sound frequencies with a wavelength longer than the baffle dimensions are canceled out, because the antiphase radiation from the rear of the cone interferes with the radiation from the front. With an infinitely large panel, this interference could be entirely prevented. A sufficiently large sealed box can approach this behavior.  Since panels of infinite dimensions are impossible, most enclosures function by containing the rear radiation from the moving diaphragm. A sealed enclosure prevents transmission of the sound emitted from the rear of the loudspeaker by confining the sound in a rigid and airtight box. Techniques used to reduce transmission of sound through the walls of the cabinet include thicker cabinet walls, lossy wall

Wiring connections.

Most home hi-fi loudspeakers use two wiring points to connect to the source of the signal (for example, to the audio amplifier or receiver). To accept the wire connection, the loudspeaker enclosure may have binding posts, spring clips, or a panel-mount jack. If the wires for a pair of speakers are not connected with respect to the proper electrical polarity (the + and − connections on the speaker and amplifier should be connected + to + and − to −; speaker cable is almost always marked so that one conductor of a pair can be distinguished from the other, even if it has run under or behind things in its run from amplifier to speaker location), the loudspeakers

Wireless speakers.

Wireless speakers are very similar to traditional (wired) loudspeakers, but they receive audio signals using radio frequency (RF) waves rather than over audio cables. There is normally an amplifier integrated in the speaker's cabinet because the RF waves alone are not

Specifications.

Speaker specifications generally include:  and optionally:

Electrical characteristics of dynamic loudspeakers.

The load that a driver presents to an amplifier consists of a complex electrical impedance—a combination of resistance and both capacitive and inductive reactance, which combines properties of the driver, its mechanical motion, the effects of crossover components (if any are in the signal path between amplifier and driver), and the effects of air loading on the driver as modified by the enclosure and its environment. Most amplifiers' output specifications are given at a specific power into an ideal resistive load; however, a loudspeaker does not have a constant impedance across its frequency range. Instead, the voice coil is inductive,

Electromechanical measurements.

Examples of typical measurements are: amplitude and phase characteristics vs. frequency; impulse response under one or more conditions (e.g., square waves, sine wave bursts, etc.); directivity vs. frequency (e.g., horizontally, vertically, spherically, etc.); harmonic and intermodulation distortion vs. sound pressure level (SPL) output, using any of several test

Efficiency vs. sensitivity.

Loudspeaker efficiency is defined as the sound power output divided by the electrical power input. Most loudspeakers are inefficient transducers; only about 1% of the electrical energy sent by an amplifier to a typical home loudspeaker is converted to acoustic energy. The remainder is converted to heat, mostly in the voice coil and magnet assembly. The main reason for this is the difficulty of achieving proper impedance matching between the acoustic impedance of the drive unit and the air it radiates into. (At low frequencies, improving this match is the main purpose of speaker enclosure designs). The efficiency of loudspeaker drivers varies with

Listening environment.

The interaction of a loudspeaker system with its environment is complex and is largely out of the loudspeaker designer's control. Most listening rooms present a more or less reflective environment, depending on size, shape, volume, and furnishings. This means the sound reaching a listener's ears consists not only of sound directly from the speaker system, but also the same sound delayed by traveling to and from (and being modified by) one or more surfaces. These reflected sound waves, when added to the direct sound, cause cancellation and addition at assorted frequencies (e.g., from resonant room modes), thus changing the timbre and character of the sound at the listener's ears. The human brain is very sensitive to small variations, including some of these, and this is part of the reason why a loudspeaker system sounds different at different listening positions or in different rooms.  A significant factor in the sound of a loudspeaker system is the amount of absorption and diffusion present in the environment. Clapping one's hands in a typical empty room, without draperies or carpet, produces a zippy, fluttery echo due both to a lack of absorption and to reverberation (that is, repeated echoes) from flat reflective walls, floor, and ceiling. The addition of hard surfaced furniture, wall hangings, shelving and even baroque plaster ceiling decoration changes the echoes, primarily because of diffusion caused by reflective objects with shapes and surfaces having sizes on the order of the sound wavelengths. This somewhat breaks up the simple reflections otherwise caused by bare flat surfaces, and spreads the reflected energy of an incident wave over a larger angle on reflection.

Placement.

In a typical rectangular listening room, the hard, parallel surfaces of the walls, floor and ceiling cause primary acoustic resonance nodes in each of the three dimensions: left-right, up-down and forward-backward. Furthermore, there are more complex resonance modes involving three, four, five and even all six boundary surfaces combining to create standing waves. Low frequencies excite these modes the most, since long wavelengths are not much affected by furniture

Directivity.

Acousticians, in studying the radiation of sound sources have developed some concepts important to understanding how loudspeakers are perceived. The simplest possible radiating source is a point source, sometimes called a simple source. An ideal point source is an infinitesimally small point radiating sound. It may be easier to imagine a tiny pulsating sphere, uniformly increasing and decreasing in diameter, sending out sound waves in all directions equally, independent of frequency.  Any object radiating sound, including a loudspeaker system, can be thought of as being composed of combinations of such simple point sources. The radiation pattern of a combination of point sources is not the same as for a single source, but depends on the distance and orientation between the sources, the position relative to them from which the listener hears the combination, and the frequency of

Other speaker designs.

While dynamic cone speakers remain the most popular choice, many other speaker technologies exist.

With a diaphragm.

Moving-iron loudspeakers.

The moving iron speaker was the first type of speaker that was invented. Unlike the newer dynamic (moving coil) design, a moving-iron speaker uses a stationary coil to vibrate a magnetized piece of metal (called the iron, reed, or armature). The metal is either attached to the diaphragm or is the diaphragm itself. This design was the original

Piezoelectric speakers.

Piezoelectric speakers are frequently used as beepers in watches and other electronic devices, and are sometimes used as tweeters in less-expensive speaker systems, such as computer speakers and portable radios. Piezoelectric speakers have several advantages over conventional loudspeakers: they are resistant to overloads that would normally destroy most high frequency drivers, and they can be used without a crossover due to their electrical properties. There are also disadvantages: some amplifiers can oscillate when driving capacitive loads like most piezoelectrics, which results in distortion or damage to the amplifier. Additionally, their frequency response, in most cases, is inferior to that of other technologies. This is why they

Magnetostatic loudspeakers.

Instead of a voice coil driving a speaker cone, a magnetostatic speaker uses an array of metal strips bonded to a large film membrane. The magnetic field produced

Magnetostrictive speakers.

Magnetostrictive transducers, based on magnetostriction, have been predominantly used as sonar ultrasonic sound wave radiators, but their use has spread also to audio speaker systems. Magnetostrictive speaker drivers have some special advantages: they can provide greater force (with

Electrostatic loudspeakers.

Electrostatic loudspeakers use a high voltage electric field (rather than a magnetic field) to drive a thin statically charged membrane. Because they are driven over the entire membrane surface rather than from a small voice coil, they ordinarily provide a more linear and lower-distortion motion than dynamic drivers. They also have a relatively narrow dispersion pattern that can make for precise sound-field positioning. However, their optimum listening area is small and they are not very efficient speakers. They have the disadvantage that the diaphragm excursion is severely limited because of practical construction limitations—the further apart the stators are positioned, the higher the voltage must be to achieve acceptable efficiency. This increases the tendency for electrical arcs as well as increasing the speaker's attraction of dust particles.

Ribbon and planar magnetic loudspeakers.

A ribbon speaker consists of a thin metal-film ribbon suspended in a magnetic field. The electrical signal is applied to the ribbon, which moves with it to create the sound. The advantage of a ribbon driver is that the ribbon has very little mass; thus, it can accelerate very quickly, yielding very good high-frequency response. Ribbon loudspeakers are often very fragile—some can be torn by a strong gust of air. Most ribbon tweeters emit sound in a dipole pattern. A few have backings that limit the dipole radiation pattern. Above and below the ends of the more or less rectangular ribbon, there is less audible output due to phase cancellation, but the precise amount of directivity depends on ribbon length. Ribbon designs generally require exceptionally powerful magnets, which makes them

Bending wave loudspeakers.

Bending wave transducers use a diaphragm that is intentionally flexible. The rigidity of the material increases from the center to the outside. Short wavelengths radiate primarily from the inner area, while longer waves reach the edge of the speaker. To prevent reflections from the outside back into the center, long waves are absorbed by a surrounding damper. Such transducers can cover a wide frequency range (80 Hz to 35,000 Hz) and have been promoted as being close to an ideal point sound source. This uncommon approach is being taken by only a very few manufacturers, in very different arrangements.  The Ohm Walsh loudspeakers use a unique driver designed by Lincoln Walsh, who had

Flat panel loudspeakers.

There have been many attempts to reduce the size of speaker systems, or alternatively to make them less obvious. One such attempt was the development of "exciter" transducer coils mounted to flat panels to act as sound sources, most accurately called exciter/panel drivers. These can then be made in a neutral color and hung on walls where they are less noticeable

Heil air motion transducers.

Oskar Heil invented the air motion transducer in the 1960s. In this approach, a pleated diaphragm is mounted in a magnetic field and forced to close and open under control of a music signal. Air is forced from between the pleats in accordance with the imposed signal, generating sound. The drivers are less fragile than ribbons and considerably more efficient (and able to produce higher absolute output levels) than ribbon,

Transparent ionic conduction speaker.

In 2013, a research team introduced Transparent ionic conduction speaker which a 2 layers transparent conductive gel and a layer of transparent rubber in between to make high voltage and high actuation work to reproduce good sound quality. The speaker is suitable for robotics, mobile computing and adaptive optics fields.

Without a diaphragm.

Plasma arc speakers.

Plasma arc loudspeakers use electrical plasma as a radiating element. Since plasma has minimal mass, but is charged and therefore can be manipulated by an electric field, the result is a very linear output at frequencies far higher than the audible range. Problems of maintenance and reliability for this approach tend to make it unsuitable for mass market use. In 1978 Alan E. Hill of the Air Force Weapons Laboratory in

Thermoacoustic speakers.

In 2008, researchers of Tsinghua University demonstrated a thermoacoustic loudspeaker of carbon nanotube thin film, whose working mechanism is a thermoacoustic effect. Sound frequency electric currents are used to periodically heat the CNT and

Rotary woofers.

A rotary woofer is essentially a fan with blades that constantly change their pitch, allowing them to easily push the air back and forth. Rotary woofers are able to efficiently reproduce infrasound frequencies, which are difficult to impossible to achieve on a traditional speaker with a diaphragm. They are often employed in movie theaters to recreate rumbling bass effects, such as explosions.

New technologies.

Digital speakers.

Digital speakers have been the subject of experiments performed by Bell Labs as far back as the 1920s. The design is simple; each bit controls a driver, which is either fully 'on' or 'off'. Problems with this design have led manufacturers to abandon it as impractical for the present. First, for a reasonable number of bits (required for adequate sound reproduction quality), the physical size of a speaker system becomes very large. Secondly, due to inherent analog-to-digital conversion problems, the effect of aliasing is unavoidable, so that the audio output is "reflected" at equal amplitude in the frequency domain, on the other side of the Nyquist limit (half the sampling frequency), causing an unacceptably high level of ultrasonics to accompany the desired output. No workable scheme has been found to adequately deal with this.  The term "digital" or "digital-ready" is often used for marketing purposes on speakers or headphones, but these systems are not digital in the sense described above. Rather, they are conventional speakers that can be used with digital sound sources (e.g., optical media, MP3 players, etc.), as can any conventional speaker.

A loudspeaker is an electroacoustic transducer; a device which converts an electrical audio signal into a corresponding sound. The most widely used type of speaker in the 2010s is the dynamic speaker, invented in 1924 by Edward W. Kellogg and Chester W. Rice. The dynamic speaker operates on the same basic principle as a dynamic microphone, but in reverse, to produce sound from an electrical signal. When an alternating current electrical audio signal is applied to its voice coil, a coil of wire suspended in a circular gap between the poles of a permanent magnet, the coil is forced to move rapidly back and forth due to Faraday's law of induction, which causes a diaphragm (usually conically shaped) attached to the coil to move back and forth, pushing on the air to create sound waves. Besides this most common method, there are several alternative technologies that can be used to convert an electrical signal into sound. The sound source (e.g., a sound recording or a microphone) must be amplified or strengthened with an audio power amplifier before the signal is sent to the speaker.

扬声器(;英语:Loudspeaker;),又称喇叭、音响。是一种转换电子信号成为声音的换能器、电子元件,可以由一个或多个组成音响组。


Biography.

Wilhelm was born on 27 January 1859 at the Crown Prince's Palace, Berlin, to Victoria, Princess Royal, the wife of Prince Frederick William of Prussia (the future Frederick III). His mother was the eldest daughter of Britain's Queen Victoria. At the time of his birth, his great-uncle Frederick William IV was king of Prussia, and his grandfather and namesake Wilhelm was acting as regent. He was the first grandchild of Queen Victoria and Prince Albert and one of the two grandchildren born in Albert's lifetime, but more importantly, the first son of the crown prince of Prussia. From 1861, Wilhelm was second in the line of succession to Prussia, and also, after 1871, to the newly created German Empire, which, according to the constitution of the German Empire, was ruled by the Prussian king. At the time of his birth, he was also sixth in the line of succession to the British throne, after his maternal uncles and his mother.  A traumatic breech birth resulted in Erb's palsy, which left him with a withered left arm about six inches (15 centimetres) shorter than his right. He tried with some success to conceal this; many photographs show him holding a pair of white gloves in his left hand to make the arm seem longer. In others, he holds his left hand with his right, has his crippled arm on the hilt of a sword, or holds a cane to give the illusion of a useful limb posed at a dignified angle. Historians have suggested that this disability affected his emotional development.

Early years.

In 1863, Wilhelm was taken to England to be present at the wedding of his Uncle Bertie (later King Edward VII), and Princess Alexandra of Denmark. Wilhelm attended the ceremony in a Highland costume, complete with a small toy dirk. During the ceremony, the four-year-old became restless. His eighteen-year-old uncle Prince Alfred, charged with keeping an eye on him, told him to be quiet, but Wilhelm drew his dirk and threatened Alfred. When Alfred attempted to subdue him by force, Wilhelm bit him on the leg. His grandmother, Queen Victoria, missed seeing the fracas; to her Wilhelm remained "a clever, dear, good little child, the great favourite of my beloved Vicky".  His mother, Vicky, was obsessed with his damaged arm, blaming herself for the child's handicap and insisted that he become a good rider. The thought that he, as heir to the throne, should not be able to ride was intolerable to her. Riding lessons began when Wilhelm was eight and were a matter of endurance for

Accession.

Kaiser Wilhelm I died in Berlin on 9 March 1888, and Prince Wilhelm's father ascended the throne as Frederick III. He was already suffering from an incurable throat cancer and spent all 99 days of his reign fighting the disease before dying. On 15 June of that same year, his 29-year-old son succeeded him as German Emperor and King of Prussia. Although in his youth he had been a great admirer of Otto von Bismarck, Wilhelm's characteristic impatience soon brought him into conflict with the "Iron Chancellor", the

Break with Bismarck.

The impetuous young Kaiser rejected Bismarck's "peaceful foreign policy" and instead plotted with senior generals to work "in favour of a war of aggression". Bismarck told an aide, "That young man wants war with Russia, and would like to draw his sword straight away if he could. I shall not be a party to it." Bismarck, after gaining an absolute majority in favour of his policies in the Reichstag, decided to make the anti-Socialist laws permanent. His "Kartell", the majority of the amalgamated Conservative Party and the National Liberal Party, favoured making the laws permanent, with one exception: the police power to expel Socialist agitators from their homes. The "Kartell" split over this issue and nothing was passed.  As the debate continued, Wilhelm became more and more interested in social problems, especially the treatment of mine workers who went on strike in 1889. He routinely interrupted Bismarck in Council to make clear where he stood on social policy; Bismarck, in turn, sharply disagreed with Wilhelm's policy and worked to circumvent it. Bismarck, feeling pressured and unappreciated by the young Emperor and undermined by his ambitious advisors, refused to sign a proclamation regarding the protection of workers along with

Wilhelm in control.

Dismissal of Bismarck.

Bismarck resigned at Wilhelm II's insistence in 1890, at the age of 75, to be succeeded as Chancellor of Germany and Minister-President of Prussia by Leo von Caprivi, who in turn was replaced by Chlodwig, Prince of Hohenlohe-Schillingsfürst, in 1894. Following the dismissal of Hohenlohe in 1900, Wilhelm appointed the man whom he regarded as "his own Bismarck", Bernhard von Bülow.  In foreign policy Bismarck had achieved a fragile balance of interests between Germany, France and Russia—peace was at hand and Bismarck tried to keep it that way despite growing popular sentiment against Britain (regarding colonies) and especially against Russia. With Bismarck's dismissal the Russians now expected a reversal of policy in Berlin, so they quickly came to terms with France, beginning the process that by 1914 largely isolated Germany. In appointing Caprivi and then Hohenlohe, Wilhelm was embarking upon what is known to history as "the New Course", in which he hoped to exert decisive influence in the government of the empire. There is debate amongst historians as to the precise degree to which Wilhelm succeeded in implementing "personal rule" in this era, but what is clear is the

Promoter of arts and sciences.

Wilhelm enthusiastically promoted the arts and sciences, as well as public education and social welfare. He sponsored the Kaiser Wilhelm Society for the promotion of scientific research; it was funded by wealthy private donors and by the state and comprised a number of research institutes in both pure and applied sciences. The Prussian Academy of Sciences was unable to avoid the Kaiser's pressure and lost some of its autonomy when it was forced to incorporate new programs in engineering, and

Personality.

Historians have frequently stressed the role of Wilhelm's personality in shaping his reign. Thus, Thomas Nipperdey concludes he was: Historian David Fromkin states that Wilhelm had a love–hate relationship with Britain. According to Fromkin "From the outset, the half-German side of him was at war with the half-English side. He was wildly jealous of the British, wanting to be British, wanting to be better at being British than the British were, while at the same time hating them and resenting them because he never could be fully accepted by them".  Langer et al. (1968) emphasise the negative international consequences of Wilhelm's erratic personality: "He believed in force, and the'survival of the fittest' in domestic as well as foreign politics... William was not lacking in intelligence, but he did lack stability, disguising his deep insecurities by swagger and tough talk. He frequently fell into depressions and hysterics... William's personal instability was reflected in vacillations of policy. His actions, at home as well as abroad, lacked guidance, and therefore often bewildered or infuriated public opinion. He was not so much concerned with gaining specific objectives, as had been the case with Bismarck, as with asserting his will. This trait in the ruler of the leading Continental power was one of the main causes of the uneasiness prevailing in Europe at the turn-of-the-century".

Relationships with foreign relatives.

As a grandchild of Queen Victoria, Wilhelm was a first cousin of the future King George V, as well as of Queens Marie of Romania, Maud of Norway, Victoria Eugenie of Spain, and the Empress Alexandra of Russia. In 1889, Wilhelm's younger sister, Sophia, married the future King Constantine I of Greece. Wilhelm was infuriated by his sister's conversion to Greek Orthodoxy; upon her marriage, he attempted to ban her from entering Germany.  Wilhelm's most contentious relationships were with his British relations. He craved the acceptance of his grandmother, Queen Victoria, and of the rest of her family. Despite the fact that his grandmother treated him with courtesy and tact, his other relatives found him arrogant and obnoxious, and they largely denied him acceptance. He had an especially bad relationship with his Uncle Bertie, the Prince of Wales (later King Edward

Foreign affairs.

German foreign policy under Wilhelm II was faced with a number of significant problems. Perhaps the most apparent was that Wilhelm was an impatient man, subjective in his reactions and affected strongly by sentiment and impulse. He was personally ill-equipped to steer German foreign policy along a rational course. It is now widely recognised that the various spectacular acts which Wilhelm undertook in the international sphere were often partially encouraged by the German foreign policy elite. There were a number of notorious examples, such as the Kruger telegram of 1896 in which Wilhelm congratulated President Paul Kruger of the Transvaal Republic on the suppression of the British Jameson Raid, thus alienating British public opinion.  British public opinion had been quite favourable toward the Kaiser in his first twelve years on the throne, but it turned sour in the late 1890s. During the First World War, he became the central target of British anti-German propaganda and the personification of a hated enemy.  Wilhelm invented and spread fears of a yellow peril trying to interest other European rulers in the perils they faced by invading China; few other leaders paid attention. Wilhelm used the Japanese victory in the Russo-Japanese War to try to incite fear in the west of the yellow peril that they faced by a resurgent Japan, which Wilhelm claimed would ally with China to overrun the west. Under Wilhelm, Germany invested in strengthening its colonies in Africa and the Pacific, but few became profitable and all were lost during the First World War. In South West Africa (now Namibia), a native revolt against German rule led to the Herero and Namaqua Genocide, although Wilhelm eventually ordered it to be stopped.  One of the few times when Wilhelm succeeded in personal diplomacy was when in 1900 he supported the marriage of Archduke Franz Ferdinand of Austria to Countess Sophie Chotek, against the wishes of Emperor Franz Joseph I of Austria.  A domestic triumph for Wilhelm was when his daughter Victoria Louise married the Duke of Brunswick in 1913; this helped heal the rift between the House of Hanover and the House of Hohenzollern that had followed the annexation of Hanover by Prussia in 1866.

Political visits to the Ottoman Empire.

In his first visit to Istanbul in 1889, Wilhelm secured the sale of German-made rifles to the Ottoman Army. Later on, he had his second political visit to the Ottoman Empire as a guest of Sultan Abdülhamid II. The Kaiser started his journey to the Ottoman Eyalets with Istanbul on 16 October 1898; then he went by yacht to Haifa on 25 October. After visiting Jerusalem and Bethlehem, the

Hun speech of 1900.

The Boxer rebellion, an anti-western uprising in China, was put down in 1900 by an international force of British, French, Russian, Austrian, Italian, American, Japanese, and German troops. The Germans, however, forfeited any prestige that they might have gained for their participation by arriving only after the British and Japanese forces had taken Peking, the site of the fiercest fighting. Moreover, the poor impression left by the German troops' late arrival was made worse by the Kaiser's ill-conceived farewell address, in which

Eulenberg Scandal.

In the years 1906–09, a succession of homosexual revelations, trials, and suicides involving ministers, courtiers, and Wilhelm's closest friend and advisor, Prince Philipp zu Eulenberg, evolved into the most tumultuous "cause célèbre" of its era. Fuelled by the journalist Maximilian Harden, who, like some in the upper echelons of the military and Foreign Office, resented Eulenberg's

Moroccan Crisis.

One of Wilhelm's diplomatic blunders sparked the Moroccan Crisis of 1905, when he made a spectacular visit to Tangier, in Morocco on 31 March 1905. He conferred with representatives of Sultan Abdelaziz of Morocco. The Kaiser proceeded to tour the city on the back of a white horse. The Kaiser declared he had come to support the sovereignty of the Sultan—a statement which amounted

"Daily Telegraph" affair.

Wilhelm's most damaging personal blunder cost him much of his prestige and power and had a far greater impact in Germany than overseas. The "Daily Telegraph" Affair of 1908 involved the publication in Germany of an interview with a British daily newspaper that included wild statements and diplomatically damaging remarks. Wilhelm had seen the interview as an opportunity to promote his views and ideas on Anglo-German friendship, but due to his emotional outbursts during the course of the interview, he ended up further alienating not only the British, but also the French, Russians, and Japanese. He implied, among other things, that

Naval expansion.

Nothing Wilhelm did in the international arena was of more influence than his decision to pursue a policy of massive naval construction. A powerful navy was Wilhelm's pet project. He had inherited from his mother a love of the British Royal Navy, which was at that time the world's largest. He once confided to his uncle, the Prince of Wales, that his dream was to have a "fleet of my own some day". Wilhelm's frustration over his fleet's poor showing

World War I.

Historians typically argue that Wilhelm was largely confined to ceremonial duties during the war—there were innumerable parades to review and honours to award. "The man who in peace had believed himself omnipotent became in war a'shadow Kaiser', out of sight, neglected, and relegated to the sidelines."

The Sarajevo crisis.

Wilhelm was a friend of Archduke Franz Ferdinand of Austria, and he was deeply shocked by his assassination on 28 June 1914. Wilhelm offered to support Austria-Hungary in crushing the Black Hand, the secret organisation that had plotted the killing, and even sanctioned the use of force by Austria against the perceived source of the movement—Serbia (this is often called "the blank cheque"). He wanted to remain in Berlin until the crisis

July 1914.

On the night of 30 July, when handed a document stating that Russia would not cancel its mobilisation, Wilhelm wrote a lengthy commentary containing these observations:  More recent British authors state that Wilhelm II really declared, "Ruthlessness and weakness will start the most terrifying war of the world, whose purpose is to destroy Germany. Because there can no longer be any doubts, England, France and Russia have conspired themselves together to fight an annihilation war against us".  When it became clear that Germany would experience a war on two fronts and that Britain would enter the war if Germany attacked France through neutral Belgium, the panic-stricken Wilhelm

Shadow-Kaiser.

Wilhelm's role in wartime was one of ever-decreasing power as he increasingly handled awards ceremonies and honorific duties. The high command continued with its strategy even when it was clear that the Schlieffen plan had failed. By 1916 the Empire had effectively become a military dictatorship under the control of Field Marshal Paul von Hindenburg and General Erich Ludendorff. Increasingly cut off from reality and the political decision-making process, Wilhelm vacillated between defeatism and dreams of victory, depending upon the fortunes of his armies. Nevertheless, Wilhelm still retained the ultimate authority in matters of political appointment, and it was only after his consent had been gained that major changes to

Abdication and flight.

Wilhelm was at the Imperial Army headquarters in Spa, Belgium, when the uprisings in Berlin and other centres took him by surprise in late 1918. Mutiny among the ranks of his beloved Kaiserliche Marine, the imperial navy, profoundly shocked him. After the outbreak of the German Revolution, Wilhelm could not make up his mind whether or not to abdicate. Up to that point, he accepted that he would likely have to give up the imperial crown, but still hoped to retain the Prussian kingship. However, this was impossible under the imperial constitution. While Wilhelm thought he ruled as emperor in a personal union with Prussia, the constitution actually tied the imperial crown to the Prussian crown, meaning that Wilhelm could not renounce one crown without renouncing the other.  Wilhelm's hope of retaining at least one of his crowns was revealed as unrealistic when, in the hope of preserving the monarchy in the face of growing revolutionary unrest, Chancellor Prince Max of Baden announced Wilhelm's abdication of both titles on 9 November 1918. Prince Max himself was forced to resign later the same day, when it became clear that only Friedrich Ebert, leader of the SPD, could effectively exert control. Later that day, one of Ebert's secretaries of state (ministers), Social Democrat Philipp Scheidemann, proclaimed Germany a republic.  Wilhelm consented to the abdication only after Ludendorff's replacement, General Wilhelm Groener, had informed him that the officers and men of the army would march back in good order under Paul von Hindenburg's command, but would certainly not fight for Wilhelm's throne on the home front. The monarchy's last and strongest support had been broken, and finally even Hindenburg, himself a lifelong royalist, was obliged, with some embarrassment, to advise the Emperor to give up the crown.  The fact that the High Command might one day abandon the Kaiser had been foreseen in December 1897, when Wilhelm had visited Otto von Bismarck for the last time. Bismarck had again warned the Kaiser about the increasing influence of militarists, especially of the admirals who were pushing for the construction of a battle fleet. Bismarck's last warning had been:  Subsequently, Bismarck had predicted accurately:  On 10 November, Wilhelm crossed the border by train and went into exile in the Netherlands, which had remained neutral throughout the war. Upon the conclusion of the Treaty of Versailles in early 1919, Article 227 expressly provided for the prosecution of Wilhelm "for a supreme offence against international morality and the sanctity of treaties", but the Dutch government refused to extradite him, despite appeals from the Allies. King George V wrote that he looked on his cousin as "the greatest criminal in history", but opposed Prime Minister David Lloyd George's proposal to "hang the Kaiser".  It was reported, however, that there was little zeal in Britain to prosecute. On 1 January 1920, it was stated in official circles in London that Great Britain would “welcome refusal by Holland to deliver the former kaiser for trial,” and it was hinted that this had been conveyed to the Dutch government through diplomatic channels.  President Woodrow Wilson of the United States opposed extradition, arguing that prosecuting Wilhelm would destabilise international order and lose the peace.  Wilhelm first settled in Amerongen, where on 28 November he issued a belated statement of abdication from both the Prussian and imperial thrones, thus formally ending the Hohenzollerns' 400-year rule over Prussia. Accepting the reality that he had lost both of his crowns for good, he gave up his rights to "the throne of Prussia and to the German Imperial throne connected therewith." He also released his soldiers and officials in both Prussia and the empire from their oath of loyalty to him. He purchased a country house in the municipality of Doorn, known as Huis Doorn, and moved in on 15 May 1920. This was to be his home for the remainder of his life. The Weimar Republic allowed Wilhelm to remove twenty-three railway wagons of furniture, twenty-seven containing packages of all sorts, one bearing a car and another a boat, from the New Palace at Potsdam.

Historiography.

Three trends have characterised the writing about Wilhelm. First, the court-inspired writers considered him a martyr and a hero, often uncritically accepting the justifications provided in the Kaiser's own memoirs. Second, there came those who judged Wilhelm to be completely unable to handle the great responsibilities of his position, a ruler too reckless to deal with power. Third, after 1950, later scholars have sought to transcend the passions of the early 20th century and attempted an objective portrayal of Wilhelm and his rule.  On 8 June 1913, a year before the Great War began, "The New York Times" published a special supplement devoted to the 25th anniversary of the Kaiser's accession. The banner headline read: ""Kaiser,

First marriage and issue.

Wilhelm and his first wife, Princess Augusta Victoria of Schleswig-Holstein, were married on 27 February 1881. They had seven children:  Empress Augusta, known affectionately as "Dona", was a constant companion to Wilhelm, and her death on 11 April 1921 was a devastating blow. It also came less than a year after their son Joachim committed suicide.

Remarriage.

The following January, Wilhelm received a birthday greeting from a son of the late Prince Johann George Ludwig Ferdinand August Wilhelm of Schönaich-Carolath. The 63-year-old Wilhelm invited the boy and his mother, Princess Hermine Reuss of Greiz, to Doorn. Wilhelm found

Religion.

Own views.

Emperor Wilhelm II was a Lutheran member of the Evangelical State

Attitude toward Islam.

Wilhelm II was on friendly terms with the Muslim world. He described himself as a "friend" to "300 million Mohammedans". Following his trip to Constantinople (which he visited three times – an unbeaten record for any European monarch)

Antisemitism.

Wilhelm's biographer Lamar Cecil identified Wilhelm's "curious but well-developed anti-Semitism", noting that in 1888 a friend of Wilhelm "declared that the young Kaiser's dislike of his Hebrew subjects, one rooted in a perception that they possessed an overweening influence in Germany, was so strong that it could not be overcome". Cecil concludes:  On 2 December 1919, Wilhelm wrote to Field Marshal August

Wilhelm II (; 27 January 18594 June 1941) was the last German Emperor ("Kaiser") and King of Prussia. He reigned from 15 June 1888 until his abdication on 9 November 1918 shortly before Germany's defeat in World War I.

弗里德里希·威廉·维克托·阿尔贝特·冯·普鲁士(,1859年-1月27日-1941年-6月3日),史称威廉二世(),末代德意志皇帝和普鲁士国王,1888年-1918年在位。


In classical mechanics.

Definition.

Orbital angular momentum in two dimensions.

Angular momentum is a vector quantity (more precisely, a pseudovector) that represents the product of a body's rotational inertia and rotational velocity (in radians/sec) about a particular axis. However, if the particle's trajectory lies in a single plane, it is sufficient to discard the vector nature of angular momentum, and treat it as a scalar (more precisely, a pseudoscalar). Angular momentum can be considered a rotational analog of linear momentum. Thus, where linear momentum formula_1 is proportional to mass formula_2 and linear speed  angular momentum formula_4 is proportional to moment of inertia formula_5 and angular speed formula_6 measured in radians per second.  Unlike mass, which depends only on amount of matter, moment of inertia is also dependent on the position of the axis of rotation and the shape of the matter. Unlike linear speed, which does not depend upon the choice of origin, angular velocity is always measured with respect to a fixed origin. Therefore, strictly speaking, formula_4 should be referred to as the angular momentum "relative to that center".  Because formula_9 for a single particle and formula_10 for circular motion, angular momentum can be expanded, formula_11 and reduced to,  the product of the radius of rotation formula_13 and the linear momentum of the particle formula_14, where formula_15 in this case is the equivalent linear (tangential) speed at the radius (formula_16).  This simple analysis can also apply to non-circular motion if only the component of the motion which is perpendicular to the radius vector is considered. In that case,  where formula_18 is the perpendicular component of the motion. Expanding, formula_19 rearranging, formula_20 and reducing, angular momentum can also be expressed,  where formula_22 is the length of the "moment arm", a line dropped perpendicularly from the origin onto the path of the particle. It is this definition, to which the term "moment of momentum" refers.

Scalar—angular momentum from Lagrangian mechanics.

Another approach is to define angular momentum as the conjugate momentum (also called canonical momentum) of the angular coordinate formula_23 expressed in the Lagrangian of the mechanical system. Consider a mechanical system with a mass formula_2 constrained to move in a circle of radius formula_25 in the absence of any external force field. The kinetic energy of the system is  And the potential energy is  Then the Lagrangian is  The "generalized momentum" "canonically conjugate to" the coordinate formula_23 is defined by

Orbital angular momentum in three dimensions.

To completely define orbital angular momentum in three dimensions, it is required to know the rate at which the position vector sweeps out angle, the direction perpendicular to the instantaneous plane of angular displacement, and the mass involved, as well as how this mass is distributed in space. By retaining this vector nature of angular momentum, the general nature of the equations is also retained, and can describe any sort of three-dimensional motion about the center of rotation – circular, linear, or otherwise. In vector notation, the orbital angular momentum of a point particle in motion about the origin can be expressed as:  where  This can be expanded, reduced, and by the rules of vector algebra, rearranged:  which is the cross product of the position vector formula_34 and the linear momentum formula_40 of the particle. By the definition of the cross product, the formula_41 vector is perpendicular to both formula_34 and formula_43. It is directed perpendicular to the plane of angular displacement, as indicated by the right-hand rule – so that the angular velocity is seen as counter-clockwise from the head of the vector. Conversely, the formula_41 vector defines the plane in which formula_34 and formula_43 lie.  By defining a unit vector formula_47 perpendicular to the plane of angular displacement, a scalar angular speed formula_6 results, where  The two-dimensional scalar equations of the previous section can thus be given direction: and formula_53 for circular motion, where all of the motion is perpendicular to the radius formula_13.  In the spherical coordinate system the angular momentum vector expresses as

Discussion.

Angular momentum can be described as the rotational analog of linear momentum. Like linear momentum it involves elements of mass and displacement. Unlike linear momentum it also involves elements of position and shape.  Many problems in physics involve matter in motion about some certain point in space, be it in actual rotation about it, or simply moving past it, where it is desired to know what effect the moving matter has on the point—can it exert energy upon it or perform work about it? Energy, the ability to do work, can be stored in matter by setting it in motion—a combination of its inertia and its displacement. Inertia is measured by its mass, and displacement by its velocity. Their product,  is the matter's momentum. Referring this momentum to a central point introduces a complication: the momentum is not applied to the point directly. For instance, a particle of matter at the outer edge of a wheel is, in effect, at the end of a lever of the same length as the wheel's radius, its momentum turning the lever about the center point. This imaginary lever is known as the "moment arm". It has the effect of multiplying the momentum's effort in proportion to its length, an effect known as a "moment". Hence, the particle's momentum referred to a particular point,  is the "angular momentum", sometimes called, as here, the "moment of momentum" of the particle versus that particular center point. The equation formula_58 combines a moment (a mass formula_2 turning moment arm formula_13) with a linear (straight-line equivalent) speed formula_15. Linear speed referred to the central point is simply the product of the distance formula_13 and the angular speed formula_6 versus the point: formula_64 another moment. Hence, angular momentum contains a double moment: formula_65 Simplifying slightly, formula_66 the quantity formula_67 is the particle's moment of inertia, sometimes called the second moment of mass. It is a measure of rotational inertia.  Because moment of inertia is a crucial part of the spin angular momentum, the latter necessarily includes all of the complications of the former, which is calculated by multiplying elementary bits of the mass by the squares of their distances from the center of rotation. Therefore, the total moment of inertia, and the angular momentum, is a complex function of the configuration of the matter about the center of rotation and the orientation of the rotation for the various bits.  For a rigid body, for instance a wheel or an asteroid, the orientation of rotation is simply the position of the rotation axis versus the matter of the body. It may or may not pass through the center of mass, or it may lie completely outside of the body. For the same body, angular momentum may take a different value for every possible axis about which rotation may take place. It reaches a minimum when the axis passes through the center of mass.  For a collection of objects revolving about a center, for instance all of the bodies of the Solar System, the orientations may be somewhat organized, as is the Solar System, with most of the bodies' axes lying close to the system's axis. Their orientations may also be completely random.  In brief, the more mass and the farther it is from the center of rotation (the longer the moment arm), the greater the moment of inertia, and therefore the greater the angular momentum for a given angular velocity. In many cases the moment of inertia, and hence the angular momentum, can be simplified by, Similarly, for a point mass formula_2 the moment of inertia is defined as, and for any collection of particles formula_74 as the sum,  Angular momentum's dependence on position and shape is reflected in its units versus linear momentum: kg⋅m/s, N⋅m⋅s, or J⋅s for angular momentum versus kg⋅m/s or N⋅s for linear momentum. When calculating angular momentum as the product of the moment of inertia times the angular velocity, the angular velocity must be expressed in radians per second, where the radian assumes the dimensionless value of unity. (When performing dimensional analysis, it may be productive to use which treats radians as a base unit, but this is outside the scope of the International system of units). Angular momentum's units can be interpreted as torque⋅time or as energy⋅time per angle. An object with angular momentum of can be reduced to zero rotation (all of the rotational energy can be transferred out of it) by an angular impulse of or equivalently, by torque or work of for one second, or energy of for one second.  The plane perpendicular to the axis of angular momentum and passing through the center of mass is sometimes called the "invariable plane", because the direction of the axis remains fixed if only the interactions of the bodies within the system, free from outside influences, are considered. One such plane is the invariable plane of the Solar System.

Angular momentum and torque.

Newton's second law of motion can be expressed mathematically, or force = mass × acceleration. The rotational equivalent for point particles may be derived as follows: which means that the torque (i.e. the time derivative of the angular momentum) is  Because the moment of inertia is formula_79, it follows that formula_80, and formula_81 which, reduces to  This is the rotational analog of Newton's Second Law. Note that the torque is not necessarily proportional or parallel to the angular acceleration (as one might expect). The reason for this is that the moment of inertia of a particle can change with time, something that cannot occur for ordinary mass.

Conservation of angular momentum.

A rotational analog of Newton's third law of motion might be written, "In a closed system, no torque can be exerted on any matter without the exertion on some other matter of an equal and opposite torque." Hence, "angular momentum can be exchanged between objects in a closed system, but total angular momentum before and after an exchange remains constant (is conserved)."  Seen another way, a rotational analogue of Newton's first law of motion might be written, "A rigid body continues in a state of uniform rotation unless acted by an external influence." Thus "with no external influence to act upon it, the original angular momentum of the system remains constant".  The conservation of angular momentum is used in analyzing "central force motion". If the net force on some body is directed always toward some point, the "center", then there is no torque on the body with respect to the center, as all of the force is directed along the radius vector, and none is perpendicular to the radius. Mathematically, torque formula_83 because in this case formula_34 and formula_85 are parallel vectors. Therefore, the angular momentum of the body about the center is constant. This is the case with gravitational attraction in the orbits of planets and satellites, where the gravitational force is always directed toward the primary body and orbiting bodies conserve angular momentum by exchanging distance and velocity as they move about the primary. Central force motion is also used in the analysis of the Bohr model of the atom.  For a planet, angular momentum is distributed between the spin of the planet and its revolution in its orbit, and these are often exchanged by various mechanisms. The conservation of angular momentum in the Earth–Moon system results in the transfer of angular momentum from Earth to Moon, due to tidal torque the Moon exerts on the Earth. This in turn results in the slowing down of the rotation rate of Earth, at about 65.7 nanoseconds per day, and in gradual increase of the radius of Moon's orbit, at about 3.82 centimeters per year.  The conservation of angular momentum explains the angular acceleration of an ice skater as she brings her arms and legs close to the vertical axis of rotation. By bringing part of the mass of her body closer to the axis, she decreases her body's moment of inertia. Because angular momentum is the product of moment of inertia and angular velocity, if the angular momentum remains constant (is conserved), then the angular velocity (rotational speed) of the skater must increase.  The same phenomenon results in extremely fast spin of compact stars (like white dwarfs, neutron stars and black holes) when they are formed out of much larger and slower rotating stars. Decrease in the size of an object "n" times results in increase of its angular velocity by the factor of "n".  Conservation is not always a full explanation for the dynamics of a system but is a key constraint. For example, a spinning top is subject to gravitational torque making it lean over and change the angular momentum about the nutation axis, but neglecting friction at the point of spinning contact, it has a conserved angular momentum about its spinning axis, and another about its precession axis. Also, in any planetary system, the planets, star(s), comets, and asteroids can all move in numerous complicated ways, but only so that the angular momentum of the system is conserved.  Noether's theorem states that every conservation law is associated with a symmetry (invariant) of the underlying physics. The symmetry associated with conservation of angular momentum is rotational invariance. The fact that the physics of a system is unchanged if it is rotated by any angle about an axis implies that angular momentum is conserved.

Angular momentum in orbital mechanics.

In astrodynamics and celestial mechanics, a "massless" (or "per unit mass") angular momentum is defined called "specific angular momentum". Note that formula_87 Mass is often unimportant in orbital mechanics calculations, because motion is defined by gravity. The primary body of the system is often so much larger than any bodies in motion about it that the smaller bodies have a negligible gravitational effect on it; it is, in effect, stationary. All bodies are apparently attracted by its gravity in the same way, regardless of mass, and therefore all move approximately the same way under the same conditions.

Solid bodies.

For a continuous mass distribution with density function "ρ"(r), a differential volume element "dV" with position vector r within the mass has a mass element "dm" = "ρ"(r)"dV". Therefore, the infinitesimal angular momentum of this element is:  and integrating this differential over the volume of the entire mass gives its total angular momentum:  In the derivation which follows, integrals similar to this can replace the sums for the case of continuous mass.

Collection of particles.

Center of mass.

For a collection of particles in motion about an arbitrary origin, it is informative to develop the equation of angular momentum by resolving their motion into components about their own center of mass and about the origin. Given,  The total mass of the particles is simply their sum,  The position vector of the center of mass is defined by,  By inspection,  The total angular momentum of the collection of particles is the sum of the angular momentum of each particle, Expanding formula_92,  Expanding formula_94,  It can be shown that (see sidebar), therefore the second and third terms vanish,  The first term can be rearranged,  and total angular momentum for the collection of particles is finally, The first term is the angular momentum of the center of mass relative to the origin. Similar to Single particle, below, it is the angular momentum of one particle of mass "M" at the center of mass moving with velocity V. The second term is the angular momentum of the particles moving relative to the center of mass, similar to Fixed center of mass, below. The result is general—the motion of the particles is not restricted to rotation or revolution about the origin or center of mass. The particles need not be individual masses, but can be elements of a continuous distribution, such as a solid body.  Rearranging equation () by vector identities, multiplying both terms by "one", and grouping appropriately,  gives the total angular momentum of the system of particles in terms of moment of inertia formula_5 and angular velocity formula_116,

Simplifications.

Single particle.

In the case of a single particle moving about the arbitrary origin,

Fixed center of mass.

For the case of the center of mass fixed in space with respect to the origin,

Angular momentum (modern definition).

In modern (20th century) theoretical physics, angular momentum (not including any intrinsic angular momentum – see below) is described using a different formalism, instead of a classical pseudovector. In this formalism, angular momentum is the 2-form Noether charge associated with rotational invariance. As a result, angular momentum is not conserved for general curved spacetimes, unless it happens to be asymptotically rotationally invariant.  In classical mechanics, the angular momentum of a particle can be reinterpreted as a plane element:  in which the exterior product ∧ replaces the cross product × (these products have similar characteristics but are nonequivalent). This has the advantage of a clearer geometric interpretation as a plane element, defined from the x and p vectors, and the expression is true in any number of dimensions (two or higher). In Cartesian coordinates:  or more compactly in index notation:  The angular velocity can also be defined as an antisymmetric second order tensor, with components "ω". The relation between the two antisymmetric tensors is given by the moment of inertia which must now be a fourth order tensor:  Again, this equation in L and ω as tensors is true in any number of dimensions. This equation also appears in the geometric algebra formalism, in which L and ω are bivectors, and the moment of inertia is a mapping between them.  In relativistic mechanics, the relativistic angular momentum of a particle is expressed as an antisymmetric tensor of second order:  in the language of four-vectors, namely the four position "X" and the four momentum "P", and absorbs the above L together with the motion of the centre of mass of the particle.  In each of the above cases, for a system of particles, the total angular momentum is just the sum of the individual particle angular momenta, and the centre of mass is for the system.

In quantum mechanics.

Angular momentum in quantum mechanics differs in many profound respects from angular momentum in classical mechanics. In relativistic quantum mechanics, it differs even more, in which the above relativistic definition becomes a tensorial operator.

Spin, orbital, and total angular momentum.

The classical definition of angular momentum as formula_133 can be carried over to quantum mechanics, by reinterpreting r as the quantum position operator and p as the quantum momentum operator. L is then an operator, specifically called the "orbital angular momentum operator". The components of the angular momentum operator satisfy the commutation relations of the Lie algebra so(3). Indeed, these operators are precisely the infinitesimal action of the rotation group on the quantum Hilbert space. (See also the discussion below of the angular momentum operators as the generators of rotations.)  However, in quantum physics, there is another type of angular momentum, called "spin angular momentum", represented by the spin operator S. Almost all elementary particles have nonvanishing spin. Spin is often depicted as a particle literally spinning around an axis, but this is a misleading and inaccurate picture: spin is an intrinsic property of a particle, unrelated to any sort of motion in space and fundamentally different from orbital angular momentum. All elementary particles have a characteristic spin (possibly zero), for example electrons have "spin 1/2" (this actually means "spin ħ/2"), photons have "spin 1" (this actually means "spin ħ"), and pi-mesons have spin 0.  Finally, there is total angular momentum J, which combines both the spin and orbital angular momentum of all particles and fields. (For one particle, J = L + S.) Conservation of angular momentum applies to J, but not to L or S; for example, the spin–orbit interaction allows angular momentum to transfer back and forth between L and S, with the total remaining constant. Electrons and photons need not have integer-based values for total angular momentum, but can also have fractional values.

Quantization.

In quantum mechanics, angular momentum is quantized – that is, it cannot vary continuously, but only in "quantum leaps" between certain allowed values. For any system, the following restrictions on measurement results apply, where formula_134 is the reduced Planck constant and formula_135 is any Euclidean vector such as x, y, or z:  The reduced Planck constant formula_134 is tiny by everyday standards, about 10 J s, and therefore this quantization does not noticeably affect the angular momentum of macroscopic objects. However, it is very important in the microscopic world. For example, the structure of electron shells and subshells in chemistry is significantly affected by the quantization of angular momentum.  Quantization of angular momentum was first postulated by Niels Bohr in his Bohr model of the atom and was later predicted by Erwin Schrödinger in his Schrödinger equation.

Uncertainty.

In the definition formula_137, six operators are involved: The position operators formula_138, formula_139, formula_140, and the momentum operators formula_141, formula_142, formula_143. However, the Heisenberg uncertainty principle tells us that it is not possible for all six of these quantities to be known simultaneously with arbitrary precision. Therefore, there are limits to what can be known or measured about a particle's angular momentum. It turns out that the best that one can do is to simultaneously measure both the angular momentum vector's magnitude and its component along one axis.  The uncertainty is closely related to the fact that different components of an angular momentum operator do not commute, for example formula_144. (For the precise commutation relations, see angular momentum operator.)

Total angular momentum as generator of rotations.

As mentioned above, orbital angular momentum L is defined as in classical mechanics: formula_137, but "total" angular momentum J is defined in a different, more basic way: J is defined as the "generator of rotations". More specifically, J is defined so that the operator is the rotation operator that takes any system and rotates it by angle formula_23 about the axis formula_148. (The "exp" in the formula refers to operator exponential) To put this the other way around, whatever our quantum Hilbert space is, we expect that the rotation group SO(3) will act on it. There is then an associated action of the Lie algebra so(3) of SO(3); the operators describing the action of so(3) on our Hilbert space are the (total) angular momentum operators.  The relationship between the angular momentum operator and the rotation operators is the same as the relationship between Lie algebras and Lie groups in mathematics. The close relationship between angular momentum and rotations is reflected in Noether's theorem that proves that angular momentum is conserved whenever the laws of physics are rotationally invariant.

In electrodynamics.

When describing the motion of a charged particle in an electromagnetic field, the canonical momentum P (derived from the Lagrangian for this system) is not gauge invariant. As a consequence, the canonical angular momentum L = r × P is not gauge invariant either. Instead, the momentum that is physical, the so-called "kinetic momentum" (used throughout this article), is (in SI units)  where "e" is the electric charge of the particle and A the magnetic vector potential of the electromagnetic field. The gauge-invariant angular momentum, that is "kinetic angular momentum", is given by  The interplay with quantum mechanics is discussed further in the article on canonical commutation relations.

In optics.

In "classical Maxwell electrodynamics" the Poynting vector is a linear momentum density of electromagnetic field.  The angular momentum density vector formula_152 is given by a vector product as in classical mechanics:  The above identities are valid "locally", i.e. in each space point formula_34 in a given moment formula_155.

History.

Newton, in the "Principia", hinted at angular momentum in his examples of the First Law of Motion,  He did not further investigate angular momentum directly in the "Principia", However, his geometric proof of the law of areas is an outstanding example of Newton's genius, and indirectly proves angular momentum conservation in the case of a central force.

The Law of Areas.

Newton's derivation.

As a planet orbits the Sun, the line between the Sun and the planet sweeps out equal areas in equal intervals of time. This had been known since Kepler expounded his second law of planetary motion. Newton derived a unique geometric proof, and went on to show that the attractive force of the Sun's gravity was the cause of all of Kepler's laws.  During the first interval of time, an object is in motion from point A to point B. Undisturbed, it would continue to point c during the second interval. When the object arrives at B, it receives an impulse directed toward point S. The impulse gives it a small added velocity toward S, such that if this were its only velocity, it would move from B to V during the second interval. By the rules of velocity composition, these two velocities add, and point C is found by construction of parallelogram BcCV. Thus the object's path is deflected by the impulse so that it arrives at point C at the end of the second interval. Because the triangles SBc and SBC have the same base SB and the same height Bc or VC, they have the same area. By symmetry, triangle SBc also has the same area as triangle SAB, therefore the object has swept out equal areas SAB and SBC in equal times.  At point C, the object receives another impulse toward S, again deflecting its path during the third interval from d to D. Thus it continues to E and beyond, the triangles SAB, SBc, SBC, SCd, SCD, SDe, SDE all having the same area. Allowing the time intervals to become ever smaller, the path ABCDE approaches indefinitely close to a continuous curve.  Note that because this derivation is geometric, and no specific force is applied, it proves a more general law than Kepler's second law of planetary motion. It shows that the Law of Areas applies to any central force, attractive or repulsive, continuous or non-continuous, or zero.

Conservation of angular momentum in the Law of Areas.

The proportionality of angular momentum to the area swept out by a moving object can be understood by realizing that the bases of the triangles, that is, the lines from S to the object, are equivalent to the radius, and that the heights of the triangles are proportional to the perpendicular component of velocity. Hence, if the area swept per unit time is constant, then by the triangular area formula, the product and therefore the product are constant: if and the base length are decreased, and height must increase proportionally. Mass is constant, therefore angular momentum is conserved by this exchange of distance and velocity.  In the case of triangle SBC, area is equal to (SB)(VC). Wherever C is eventually located due to the impulse applied at B, the product (SB)(VC), and therefore remain constant. Similarly so for each of the triangles.

After Newton.

Leonhard Euler, Daniel Bernoulli, and Patrick d'Arcy all understood angular momentum in terms of conservation of areal velocity, a result of their analysis of Kepler's second law of planetary motion. It is unlikely that they realized the implications for ordinary rotating matter.  In 1736 Euler, like Newton, touched on some of the equations of angular momentum in his "Mechanica" without further developing them.  Bernoulli wrote in a 1744 letter of a "moment of rotational motion", possibly the first conception of angular momentum as we now understand it.  In 1799, Pierre-Simon Laplace first realized that a fixed plane was associated with rotation—his "invariable plane".  Louis Poinsot in 1803 began representing rotations as a line segment perpendicular to the rotation, and elaborated on the "conservation of moments".  In 1852 Léon Foucault used a gyroscope in an experiment to display the Earth's rotation.  William J. M. Rankine's 1858 "Manual of Applied Mechanics" defined angular momentum in the modern sense for the first time:  In an 1872 edition of the same book, Rankine stated that "The term "angular momentum" was introduced by Mr. Hayward," probably referring to R.B. Hayward's article "On a Direct Method of estimating Velocities, Accelerations, and all similar Quantities with respect to Axes moveable in any manner in Space with Applications," which was introduced in 1856, and published in 1864. Rankine was mistaken, as numerous publications feature the term starting in the late 18th to early 19th centuries. However, Hayward's article apparently was the first use of the term and the concept seen by much of the English-speaking world. Before this, angular momentum was typically referred to as "momentum of rotation" in English.

In physics, angular momentum (rarely, moment of momentum or rotational momentum) is the rotational equivalent of linear momentum. It is an important quantity in physics because it is a conserved quantity—the total angular momentum of a closed system remains constant.

在物理学中,角动量是与物体的位置向量和动量相关的物理量。对于某惯性参考系的原点formula_1,物体的角动量是物体的位置向量和动量的叉积,通常写做formula_2。角动量是矢量。