Single-mode optical fiber
In fiber-optic communication, a single-mode optical fiber is an optical fiber designed to carry light only directly down the fiber - the transverse mode. Modes are the possible solutions of the Helmholtz equation for waves, obtained by combining Maxwell's equations and the boundary conditions; these modes define the way the wave travels through space, i.e. how the wave is distributed in space. Waves have different frequencies; this is the case in single-mode fibers, where we can have waves with different frequencies, but of the same mode, which means that they are distributed in space in the same way, that gives us a single ray of light. Although the ray travels parallel to the length of the fiber, it is called transverse mode since its electromagnetic oscillations occur perpendicular to the length of the fiber; the 2009 Nobel Prize in Physics was awarded to Charles K. Kao for his theoretical work on the single-mode optical fiber. In 1961, Elias Snitzer while working at American Optical published a comprehensive theoretical description of single mode fibers in the Journal of the Optical Society of America.
At the Corning Glass Works, Robert Maurer, Donald Keck and Peter Schultz started with fused silica, a material that can be made pure, but has a high melting point and a low refractive index. They made cylindrical preforms by depositing purified materials from the vapor phase, adding controlled levels of dopants to make the refractive index of the core higher than that of the cladding, without raising attenuation dramatically. In September 1970, they announced they had made single-mode fibers with attenuation at the 633-nanometer helium-neon line below 20 dB/km. Like multi-mode optical fibers, single-mode fibers do exhibit modal dispersion resulting from multiple spatial modes but with narrower modal dispersion. Single-mode fibers are therefore better at retaining the fidelity of each light pulse over longer distances than multi-mode fibers. For these reasons, single-mode fibers can have a higher bandwidth than multi-mode fibers. Equipment for single-mode fiber is more expensive than equipment for multi-mode optical fiber, but the single-mode fiber itself is cheaper in bulk.
A typical single-mode optical fiber has a core diameter between 8 and 10.5 µm and a cladding diameter of 125 µm. There are a number of special types of single-mode optical fiber which have been chemically or physically altered to give special properties, such as dispersion-shifted fiber and nonzero dispersion-shifted fiber. Data rates are limited by chromatic dispersion; as of 2005, data rates of up to 10 gigabits per second were possible at distances of over 80 km with commercially available transceivers. By using optical amplifiers and dispersion-compensating devices, state-of-the-art DWDM optical systems can span thousands of kilometers at 10 Gbit/s, several hundred kilometers at 40 Gbit/s; the lowest-order bounds mode is ascertained for the wavelength of interest by solving Maxwell's equations for the boundary conditions imposed by the fiber, which are determined by the core diameter and the refractive indices of the core and cladding. The solution of Maxwell's equations for the lowest order bound mode will permit a pair of orthogonally polarized fields in the fiber, this is the usual case in a communication fiber.
In step-index guides, single-mode operation occurs when the normalized frequency, V, is less than or equal to 2.405. For power-law profiles, single-mode operation occurs for a normalized frequency, V, less than 2.405 g + 2 g,where g is the profile parameter. In practice, the orthogonal polarizations may not be associated with degenerate modes. OS1 and OS2 are standard single-mode optical fiber used with wavelengths 1310 nm and 1550 nm with a maximum attenuation of 1 dB/km and 0.4 dB/km. OS1 is defined in ISO/IEC 11801, OS2 is defined in ISO/IEC 24702. Optical fiber connectors are used to join optical fibers where a connect/disconnect capability is required; the basic connector unit is a connector assembly. A connector assembly consists of two connector plugs. Due to the sophisticated polishing and tuning procedures that may be incorporated into optical connector manufacturing, connectors are assembled onto optical fiber in a supplier’s manufacturing facility. However, the assembly and polishing operations involved can be performed in the field, for example to make cross-connect jumpers to size.
Optical fiber connectors are used in telephone company central offices, at installations on customer premises, in outside plant applications. Their uses include: Making the connection between equipment and the telephone plant in the central office Connecting fibers to remote and outside plant electronics such as Optical Network Units and Digital Loop Carrier systems Optical cross connects in the central office Patching panels in the outside plant to provide architectural flexibility and to interconnect fibers belonging to different service providers Connecting couplers and Wavelength Division Multiplexers to optical fibers Connecting optical test equipment to fibers for testing and maintenance. Outside plant applications may involve locating connectors underground in subsurface enclosures that may be subject to flooding, on outdoor walls, or on utility poles; the closures that enclose them may be hermetic, or may be “free-breathing.” Hermetic closures will prevent the connectors within being subjected to temperature swings unless they are breached.
Free-breathing enclosures will subject them to temperature
General Services Administration
The General Services Administration, an independent agency of the United States government, was established in 1949 to help manage and support the basic functioning of federal agencies. GSA supplies products and communications for U. S. government offices, provides transportation and office space to federal employees, develops government-wide cost-minimizing policies and other management tasks. GSA employs about 12,000 federal workers and has an annual operating budget of $20.9 billion. GSA oversees $66 billion of procurement annually, it contributes to the management of about $500 billion in U. S. federal property, divided chiefly among 8,700 owned and leased buildings and a 215,000 vehicle motor pool. Among the real estate assets managed by GSA are the Ronald Reagan Building and International Trade Center in Washington, D. C. – the largest U. S. federal building after the Pentagon – and the Hart-Dole-Inouye Federal Center. GSA's business lines include the Federal Acquisition Service and the Public Buildings Service, as well as several Staff Offices including the Office of Government-wide Policy, the Office of Small Business Utilization, the Office of Mission Assurance.
As part of FAS, GSA's Technology Transformation Services helps federal agencies improve delivery of information and services to the public. Key initiatives include FedRAMP, Cloud.gov, the USAGov platform, Data.gov, Performance.gov, Challenge.gov. GSA is a member of the Procurement G6, an informal group leading the use of framework agreements and e-procurement instruments in public procurement. In 1947 President Harry Truman asked former President Herbert Hoover to lead what became known as the Hoover Commission to make recommendations to reorganize the operations of the federal government. One of the recommendations of the commission was the establishment of an "Office of the General Services." This proposed office would combine the responsibilities of the following organizations: U. S. Treasury Department's Bureau of Federal Supply U. S. Treasury Department's Office of Contract Settlement National Archives Establishment All functions of the Federal Works Agency, including the Public Buildings Administration and the Public Roads Administration War Assets AdministrationGSA became an independent agency on July 1, 1949, after the passage of the Federal Property and Administrative Services Act.
General Jess Larson, Administrator of the War Assets Administration, was named GSA's first Administrator. The first job awaiting Administrator Larson and the newly formed GSA was a complete renovation of the White House; the structure had fallen into such a state of disrepair by 1949 that one inspector of the time said the historic structure was standing "purely from habit." Larson explained the nature of the total renovation in depth by saying, "In order to make the White House structurally sound, it was necessary to dismantle, I mean dismantle, everything from the White House except the four walls, which were constructed of stone. Everything, except the four walls without a roof, was stripped down, that's where the work started." GSA worked with President Truman and First Lady Bess Truman to ensure that the new agency's first major project would be a success. GSA completed the renovation in 1952. In 1986 GSA headquarters, U. S. General Services Administration Building, located at Eighteenth and F Streets, NW, was listed on the National Register of Historic Places, at the time serving as Interior Department offices.
In 1960 GSA created the Federal Telecommunications System, a government-wide intercity telephone system. In 1962 the Ad Hoc Committee on Federal Office Space created a new building program to address obsolete office buildings in Washington, D. C. resulting in the construction of many of the offices that now line Independence Avenue. In 1970 the Nixon administration created the Consumer Product Information Coordinating Center, now part of USAGov. In 1974 the Federal Buildings Fund was initiated, allowing GSA to issue rent bills to federal agencies. In 1972 GSA established the Automated Data and Telecommunications Service, which became the Office of Information Resources Management. In 1973 GSA created the Office of Federal Management Policy. GSA's Office of Acquisition Policy centralized procurement policy in 1978. GSA was responsible for emergency preparedness and stockpiling strategic materials to be used in wartime until these functions were transferred to the newly-created Federal Emergency Management Agency in 1979.
In 1984 GSA introduced the federal government to the use of charge cards, known as the GMA SmartPay system. The National Archives and Records Administration was spun off into an independent agency in 1985; the same year, GSA began to provide governmentwide policy oversight and guidance for federal real property management as a result of an Executive Order signed by President Ronald Reagan. In 2003 the Federal Protective Service was moved to the Department of Homeland Security. In 2005 GSA reorganized to merge the Federal Supply Service and Federal Technology Service business lines into the Federal Acquisition Service. On April 3, 2009, President Barack Obama nominated Martha N. Johnson to serve as GSA Administrator. After a nine-month delay, the United States Senate confirmed her nomination on February 4, 2010. On April 2, 2012, Johnson resigned in the wake of a management-deficiency report that detailed improper payments for a 2010 "Western Regions" training conference put on by the Public Buildings Service in Las Vegas.
In July 1991 GSA contractors began the excavation of what is now the Ted Weiss Federal Building in New York City. The planning for that buildin
In physics, power is the rate of doing work or of transferring heat, i.e. the amount of energy transferred or converted per unit time. Having no direction, it is a scalar quantity. In the International System of Units, the unit of power is the joule per second, known as the watt in honour of James Watt, the eighteenth-century developer of the condenser steam engine. Another common and traditional measure is horsepower. Being the rate of work, the equation for power can be written: power = work time As a physical concept, power requires both a change in the physical system and a specified time in which the change occurs; this is distinct from the concept of work, only measured in terms of a net change in the state of the physical system. The same amount of work is done when carrying a load up a flight of stairs whether the person carrying it walks or runs, but more power is needed for running because the work is done in a shorter amount of time; the output power of an electric motor is the product of the torque that the motor generates and the angular velocity of its output shaft.
The power involved in moving a vehicle is the product of the traction force of the wheels and the velocity of the vehicle. The rate at which a light bulb converts electrical energy into light and heat is measured in watts—the higher the wattage, the more power, or equivalently the more electrical energy is used per unit time; the dimension of power is energy divided by time. The SI unit of power is the watt, equal to one joule per second. Other units of power include ergs per second, metric horsepower, foot-pounds per minute. One horsepower is equivalent to 33,000 foot-pounds per minute, or the power required to lift 550 pounds by one foot in one second, is equivalent to about 746 watts. Other units include a logarithmic measure relative to a reference of 1 milliwatt. Power, as a function of time, is the rate at which work is done, so can be expressed by this equation: P = d W d t where P is power, W is work, t is time; because work is a force F applied over a distance x, W = F ⋅ x for a constant force, power can be rewritten as: P = d W d t = d d t = F ⋅ d x d t = F ⋅ v In fact, this is valid for any force, as a consequence of applying the fundamental theorem of calculus.
As a simple example, burning one kilogram of coal releases much more energy than does detonating a kilogram of TNT, but because the TNT reaction releases energy much more it delivers far more power than the coal. If ΔW is the amount of work performed during a period of time of duration Δt, the average power Pavg over that period is given by the formula P a v g = Δ W Δ t, it is the average amount of energy converted per unit of time. The average power is simply called "power" when the context makes it clear; the instantaneous power is the limiting value of the average power as the time interval Δt approaches zero. P = lim Δ t → 0 P a v g = lim Δ t → 0 Δ W Δ t = d W d t. In the case of constant power P, the amount of work performed during a period of duration t is given by: W = P t. In the context of energy conversion, it is more customary to use the symbol E rather than W. Power in mechanical systems is the combination of forces and movement. In particular, power is the product of a force on an object and the object's velocity, or the product of a torque on a shaft and the shaft's angular velocity.
Mechanical power is described as the time derivative of work. In mechanics, the work done by a force F on an object that travels along a curve C is given by the line integral: W C = ∫ C F ⋅ v d t = ∫ C F ⋅ d x, where x defines the path C and v is the velocity along this path. If the force F is derivable from a potential applying the gradi
In optics, a Gaussian beam is a beam of monochromatic electromagnetic radiation whose transverse magnetic and electric field amplitude profiles are given by the Gaussian function. This fundamental transverse gaussian mode describes the intended output of most lasers, as such a beam can be focused into the most concentrated spot; when such a beam is refocused by a lens, the transverse phase dependence is altered. The electric and magnetic field amplitude profiles along any such circular Gaussian beam are determined by a single parameter: the so-called waist w0. At any position z relative to the waist along a beam having a specified w0, the field amplitudes and phases are thereby determined as detailed below; the equations below assume a beam with a circular cross-section at all values of z. Beams with elliptical cross-sections, or with waists at different positions in z for the two transverse dimensions can be described as Gaussian beams, but with distinct values of w0 and of the z = 0 location for the two transverse dimensions x and y.
Arbitrary solutions of the paraxial Helmholtz equation can be expressed as combinations of Hermite–Gaussian modes or as combinations of Laguerre–Gaussian modes. At any point along the beam z these modes include the same Gaussian factor as the fundamental Gaussian mode multiplying the additional geometrical factors for the specified mode; however different modes propagate with a different Gouy phase, why the net transverse profile due to a superposition of modes evolves in z, whereas the propagation of any single Hermite–Gaussian mode retains the same form along a beam. Although there are other possible modal decompositions, these families of solutions are the most useful for problems involving compact beams, that is, where the optical power is rather confined along an axis; when a laser is not operating in the fundamental Gaussian mode, its power will be found among the lowest-order modes using these decompositions, as the spatial extent of higher order modes will tend to exceed the bounds of a laser's resonator.
"Gaussian beam" implies radiation confined to the fundamental Gaussian mode. The Gaussian beam is a transverse electromagnetic mode; the mathematical expression for the electric field amplitude is a solution to the paraxial Helmholtz equation. Assuming polarization in the x direction and propagation in the +z direction, the electric field in phasor notation is given by: E = E 0 x ^ w 0 w exp exp, where r is the radial distance from the center axis of the beam, z is the axial distance from the beam's focus, i is the imaginary unit, k = 2 π / λ is the wave number for a wavelength λ, E 0 = E, the electric field amplitude at the origin at time 0, w is the radius at which the field amplitudes fall to 1/e of their axial values, at the plane z along the beam, w 0 = w is the waist radius, R is the radius of curvature of the beam's wavefronts at z, ψ is the Gouy phase at z, an extra phase term beyond that attributable to the phase velocity of light. There is an understood time dependence e i ω t multiplying such phasor quantities.
Since this solution relies on the paraxial approximation, it is not accurate for strongly diverging beams. In most practical cases the above form is valid; the wave's associated magnetic field is everywhere directly proportional to the electric field and perpendicular to it. Since the electric field is
An optical fiber is a flexible, transparent fiber made by drawing glass or plastic to a diameter thicker than that of a human hair. Optical fibers are used most as a means to transmit light between the two ends of the fiber and find wide usage in fiber-optic communications, where they permit transmission over longer distances and at higher bandwidths than electrical cables. Fibers are used instead of metal wires. Fibers are used for illumination and imaging, are wrapped in bundles so they may be used to carry light into, or images out of confined spaces, as in the case of a fiberscope. Specially designed fibers are used for a variety of other applications, some of them being fiber optic sensors and fiber lasers. Optical fibers include a core surrounded by a transparent cladding material with a lower index of refraction. Light is kept in the core by the phenomenon of total internal reflection which causes the fiber to act as a waveguide. Fibers that support many propagation paths or transverse modes are called multi-mode fibers, while those that support a single mode are called single-mode fibers.
Multi-mode fibers have a wider core diameter and are used for short-distance communication links and for applications where high power must be transmitted. Single-mode fibers are used for most communication links longer than 1,000 meters. Being able to join optical fibers with low loss is important in fiber optic communication; this is more complex than joining electrical wire or cable and involves careful cleaving of the fibers, precise alignment of the fiber cores, the coupling of these aligned cores. For applications that demand a permanent connection a fusion splice is common. In this technique, an electric arc is used to melt the ends of the fibers together. Another common technique is a mechanical splice, where the ends of the fibers are held in contact by mechanical force. Temporary or semi-permanent connections are made by means of specialized optical fiber connectors; the field of applied science and engineering concerned with the design and application of optical fibers is known as fiber optics.
The term was coined by Indian physicist Narinder Singh Kapany, acknowledged as the father of fiber optics. Guiding of light by refraction, the principle that makes fiber optics possible, was first demonstrated by Daniel Colladon and Jacques Babinet in Paris in the early 1840s. John Tyndall included a demonstration of it in his public lectures in London, 12 years later. Tyndall wrote about the property of total internal reflection in an introductory book about the nature of light in 1870:When the light passes from air into water, the refracted ray is bent towards the perpendicular... When the ray passes from water to air it is bent from the perpendicular... If the angle which the ray in water encloses with the perpendicular to the surface be greater than 48 degrees, the ray will not quit the water at all: it will be reflected at the surface.... The angle which marks the limit where total reflection begins is called the limiting angle of the medium. For water this angle is 48°27′, for flint glass it is 38°41′, while for diamond it is 23°42′.
In the late 19th and early 20th centuries, light was guided through bent glass rods to illuminate body cavities. Practical applications such as close internal illumination during dentistry appeared early in the twentieth century. Image transmission through tubes was demonstrated independently by the radio experimenter Clarence Hansell and the television pioneer John Logie Baird in the 1920s. In the 1930s, Heinrich Lamm showed that one could transmit images through a bundle of unclad optical fibers and used it for internal medical examinations, but his work was forgotten. In 1953, Dutch scientist Bram van Heel first demonstrated image transmission through bundles of optical fibers with a transparent cladding; that same year, Harold Hopkins and Narinder Singh Kapany at Imperial College in London succeeded in making image-transmitting bundles with over 10,000 fibers, subsequently achieved image transmission through a 75 cm long bundle which combined several thousand fibers. Their article titled "A flexible fibrescope, using static scanning" was published in the journal Nature in 1954.
The first practical fiber optic semi-flexible gastroscope was patented by Basil Hirschowitz, C. Wilbur Peters, Lawrence E. Curtiss, researchers at the University of Michigan, in 1956. In the process of developing the gastroscope, Curtiss produced the first glass-clad fibers. A variety of other image transmission applications soon followed. Kapany coined the term fiber optics, wrote a 1960 article in Scientific American that introduced the topic to a wide audience, wrote the first book about the new field; the first working fiber-optical data transmission system was demonstrated by German physicist Manfred Börner at Telefunken Research Labs in Ulm in 1965, followed by the first patent application for this technology in 1966. NASA used fiber optics in the television cameras. At the time, the use in the cameras was classified confidential, employees handling the cameras had to be supervised by someone with an appropriate security clearance. Charles K. Kao and George A. Hockham of the British company Standard Telephones and Cables were the first, in 1965, to promote the idea that the attenuation in optical fibers could be reduced below 20 decibels per kilometer, making fibers a practical communication medium.
They proposed th
A magnetic field is a vector field that describes the magnetic influence of electric charges in relative motion and magnetized materials. Magnetic fields are observed from subatomic particles to galaxies. In everyday life, the effects of magnetic fields are seen in permanent magnets, which pull on magnetic materials and attract or repel other magnets. Magnetic fields surround and are created by magnetized material and by moving electric charges such as those used in electromagnets. Magnetic fields exert forces on nearby moving electrical torques on nearby magnets. In addition, a magnetic field that varies with location exerts a force on magnetic materials. Both the strength and direction of a magnetic field vary with location; as such, it is an example of a vector field. The term'magnetic field' is used for two distinct but related fields denoted by the symbols B and H. In the International System of Units, H, magnetic field strength, is measured in the SI base units of ampere per meter. B, magnetic flux density, is measured in tesla, equivalent to newton per meter per ampere.
H and B differ in. In a vacuum, B and H are the same aside from units. Magnetic fields are produced by moving electric charges and the intrinsic magnetic moments of elementary particles associated with a fundamental quantum property, their spin. Magnetic fields and electric fields are interrelated, are both components of the electromagnetic force, one of the four fundamental forces of nature. Magnetic fields are used throughout modern technology in electrical engineering and electromechanics. Rotating magnetic fields are used in both electric generators; the interaction of magnetic fields in electric devices such as transformers is studied in the discipline of magnetic circuits. Magnetic forces give information about the charge carriers in a material through the Hall effect; the Earth produces its own magnetic field, which shields the Earth's ozone layer from the solar wind and is important in navigation using a compass. Although magnets and magnetism were studied much earlier, the research of magnetic fields began in 1269 when French scholar Petrus Peregrinus de Maricourt mapped out the magnetic field on the surface of a spherical magnet using iron needles.
Noting that the resulting field lines crossed at two points he named those points'poles' in analogy to Earth's poles. He clearly articulated the principle that magnets always have both a north and south pole, no matter how finely one slices them. Three centuries William Gilbert of Colchester replicated Petrus Peregrinus' work and was the first to state explicitly that Earth is a magnet. Published in 1600, Gilbert's work, De Magnete, helped to establish magnetism as a science. In 1750, John Michell stated that magnetic poles attract and repel in accordance with an inverse square law. Charles-Augustin de Coulomb experimentally verified this in 1785 and stated explicitly that the north and south poles cannot be separated. Building on this force between poles, Siméon Denis Poisson created the first successful model of the magnetic field, which he presented in 1824. In this model, a magnetic H-field is produced by'magnetic poles' and magnetism is due to small pairs of north/south magnetic poles. Three discoveries in 1820 challenged this foundation of magnetism, though.
Hans Christian Ørsted demonstrated that a current-carrying wire is surrounded by a circular magnetic field. André-Marie Ampère showed that parallel wires with currents attract one another if the currents are in the same direction and repel if they are in opposite directions. Jean-Baptiste Biot and Félix Savart announced empirical results about the forces that a current-carrying long, straight wire exerted on a small magnet, determining that the forces were inversely proportional to the perpendicular distance from the wire to the magnet. Laplace deduced, but did not publish, a law of force based on the differential action of a differential section of the wire, which became known as the Biot–Savart law. Extending these experiments, Ampère published his own successful model of magnetism in 1825. In it, he showed the equivalence of electrical currents to magnets and proposed that magnetism is due to perpetually flowing loops of current instead of the dipoles of magnetic charge in Poisson's model.
This has the additional benefit of explaining. Further, Ampère derived both Ampère's force law describing the force between two currents and Ampère's law, like the Biot–Savart law described the magnetic field generated by a steady current. In this work, Ampère introduced the term electrodynamics to describe the relationship between electricity and magnetism. In 1831, Michael Faraday discovered electromagnetic induction when he found that a changing magnetic field generates an encircling electric field, he described this phenomenon in. Franz Ernst Neumann proved that, for a moving conductor in a magnetic field, induction is a consequence of Ampère's force law. In the process, he introduced the magnetic vector potential, shown to be equivalent to the underlying mechanism proposed by Faraday. In 1850, Lord Kelvin known as William Thomson, distinguished between two magnetic fields now denoted H and B; the former applied to the latter to Ampère's model and induction. Further, he derived how H and B relate to each other
An electric field surrounds an electric charge, exerts force on other charges in the field, attracting or repelling them. Electric field is sometimes abbreviated as E-field. Mathematically the electric field is a vector field that associates to each point in space the force per unit of charge exerted on an infinitesimal positive test charge at rest at that point; the SI unit for electric field strength is volt per meter. Newtons per coulomb is used as a unit of electric field strengh. Electric fields are created by time-varying magnetic fields. Electric fields are important in many areas of physics, are exploited electrical technology. On an atomic scale, the electric field is responsible for the attractive force between the atomic nucleus and electrons that holds atoms together, the forces between atoms that cause chemical bonding. Electric fields and magnetic fields are both manifestations of the electromagnetic force, one of the four fundamental forces of nature. From Coulomb's law a particle with electric charge q 1 at position x 1 exerts a force on a particle with charge q 0 at position x 0 of F = 1 4 π ε 0 q 1 q 0 2 r ^ 1, 0 where r 1, 0 is the unit vector in the direction from point x 1 to point x 0, ε0 is the electric constant in C2 m−2 N−1When the charges q 0 and q 1 have the same sign this force is positive, directed away from the other charge, indicating the particles repel each other.
When the charges have unlike signs the force is negative, indicating the particles attract. To make it easy to calculate the Coulomb force on any charge at position x 0 this expression can be divided by q 0, leaving an expression that only depends on the other charge E = F q 0 = 1 4 π ε 0 q 1 2 r ^ 1, 0 This is the electric field at point x 0 due to the point charge q 1. Since this formula gives the electric field magnitude and direction at any point x 0 in space 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, toward the charge if it is negative, 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.
E = E 1 + E 2 + E 3 + ⋯ = 1 4 π ε 0 q 1 2 r ^ 1 + 1 4 π ε 0 q 2 ( x 2 −