A battery is a device consisting of one or more electrochemical cells with external connections provided to power electrical devices such as flashlights and electric cars. When a battery is supplying electric power, its positive terminal is the cathode and its negative terminal is the anode; the terminal marked negative is the source of electrons that will flow through an external electric circuit to the positive terminal. When a battery is connected to an external electric load, a redox reaction converts high-energy reactants to lower-energy products, the free-energy difference is delivered to the external circuit as electrical energy; the term "battery" referred to a device composed of multiple cells, however the usage has evolved to include devices composed of a single cell. Primary batteries are discarded. Common examples are the alkaline battery used for flashlights and a multitude of portable electronic devices. Secondary batteries can be discharged and recharged multiple times using an applied electric current.
Examples include the lead-acid batteries used in vehicles and lithium-ion batteries used for portable electronics such as laptops and smartphones. Batteries come in many shapes and sizes, from miniature cells used to power hearing aids and wristwatches to small, thin cells used in smartphones, to large lead acid batteries or lithium-ion batteries in vehicles, at the largest extreme, huge battery banks the size of rooms that provide standby or emergency power for telephone exchanges and computer data centers. According to a 2005 estimate, the worldwide battery industry generates US$48 billion in sales each year, with 6% annual growth. Batteries have much lower specific energy than common fuels such as gasoline. In automobiles, this is somewhat offset by the higher efficiency of electric motors in converting chemical energy to mechanical work, compared to combustion engines; the usage of "battery" to describe a group of electrical devices dates to Benjamin Franklin, who in 1748 described multiple Leyden jars by analogy to a battery of cannon.
Italian physicist Alessandro Volta built and described the first electrochemical battery, the voltaic pile, in 1800. This was a stack of copper and zinc plates, separated by brine-soaked paper disks, that could produce a steady current for a considerable length of time. Volta did not understand, he thought that his cells were an inexhaustible source of energy, that the associated corrosion effects at the electrodes were a mere nuisance, rather than an unavoidable consequence of their operation, as Michael Faraday showed in 1834. Although early batteries were of great value for experimental purposes, in practice their voltages fluctuated and they could not provide a large current for a sustained period; the Daniell cell, invented in 1836 by British chemist John Frederic Daniell, was the first practical source of electricity, becoming an industry standard and seeing widespread adoption as a power source for electrical telegraph networks. It consisted of a copper pot filled with a copper sulfate solution, in, immersed an unglazed earthenware container filled with sulfuric acid and a zinc electrode.
These wet cells used liquid electrolytes, which were prone to leakage and spillage if not handled correctly. Many used glass jars to hold their components, which made them fragile and dangerous; these characteristics made. Near the end of the nineteenth century, the invention of dry cell batteries, which replaced the liquid electrolyte with a paste, made portable electrical devices practical. Batteries convert chemical energy directly to electrical energy. In many cases, the electrical energy released is the difference in the cohesive or bond energies of the metals, oxides, or molecules undergoing the electrochemical reaction. For instance, energy can be stored in Zn or Li, which are high-energy metals because they are not stabilized by d-electron bonding, unlike transition metals. Batteries are designed such that the energetically favorable redox reaction can occur only if electrons move through the external part of the circuit. A battery consists of some number of voltaic cells; each cell consists of two half-cells connected in series by a conductive electrolyte containing metal cations.
One half-cell includes electrolyte and the negative electrode, the electrode to which anions migrate. Cations are reduced at the cathode; some cells use different electrolytes for each half-cell. Each half-cell has an electromotive force relative to a standard; the net emf of the cell is the difference between the emfs of its half-cells. Thus, if the electrodes have emfs E 1 and E 2 the net emf is E 2 − E 1.
Electronics comprises the physics, engineering and applications that deal with the emission and control of electrons in vacuum and matter. The identification of the electron in 1897, along with the invention of the vacuum tube, which could amplify and rectify small electrical signals, inaugurated the field of electronics and the electron age. Electronics deals with electrical circuits that involve active electrical components such as vacuum tubes, diodes, integrated circuits and sensors, associated passive electrical components, interconnection technologies. Electronic devices contain circuitry consisting or of active semiconductors supplemented with passive elements; the nonlinear behaviour of active components and their ability to control electron flows makes amplification of weak signals possible. Electronics is used in information processing, telecommunication, signal processing; the ability of electronic devices to act as switches makes digital information-processing possible. Interconnection technologies such as circuit boards, electronics packaging technology, other varied forms of communication infrastructure complete circuit functionality and transform the mixed electronic components into a regular working system, called an electronic system.
An electronic system may be a component of a standalone device. Electrical and electromechanical science and technology deals with the generation, switching and conversion of electrical energy to and from other energy forms; this distinction started around 1906 with the invention by Lee De Forest of the triode, which made electrical amplification of weak radio signals and audio signals possible with a non-mechanical device. Until 1950 this field was called "radio technology" because its principal application was the design and theory of radio transmitters and vacuum tubes; as of 2018 most electronic devices use semiconductor components to perform electron control. The study of semiconductor devices and related technology is considered a branch of solid-state physics, whereas the design and construction of electronic circuits to solve practical problems come under electronics engineering; this article focuses on engineering aspects of electronics. Digital electronics Analogue electronics Microelectronics Circuit design Integrated circuits Power electronics Optoelectronics Semiconductor devices Embedded systems An electronic component is any physical entity in an electronic system used to affect the electrons or their associated fields in a manner consistent with the intended function of the electronic system.
Components are intended to be connected together by being soldered to a printed circuit board, to create an electronic circuit with a particular function. Components may be packaged singly, or in more complex groups as integrated circuits; some common electronic components are capacitors, resistors, transistors, etc. Components are categorized as active or passive. Vacuum tubes were among the earliest electronic components, they were solely responsible for the electronics revolution of the first half of the twentieth century. They allowed for vastly more complicated systems and gave us radio, phonographs, long-distance telephony and much more, they played a leading role in the field of microwave and high power transmission as well as television receivers until the middle of the 1980s. Since that time, solid-state devices have all but taken over. Vacuum tubes are still used in some specialist applications such as high power RF amplifiers, cathode ray tubes, specialist audio equipment, guitar amplifiers and some microwave devices.
In April 1955, the IBM 608 was the first IBM product to use transistor circuits without any vacuum tubes and is believed to be the first all-transistorized calculator to be manufactured for the commercial market. The 608 contained more than 3,000 germanium transistors. Thomas J. Watson Jr. ordered all future IBM products to use transistors in their design. From that time on transistors were exclusively used for computer logic and peripherals. Circuits and components can be divided into two groups: digital. A particular device may consist of circuitry that has a mix of the two types. Most analog electronic appliances, such as radio receivers, are constructed from combinations of a few types of basic circuits. Analog circuits use a continuous range of voltage or current as opposed to discrete levels as in digital circuits; the number of different analog circuits so far devised is huge because a'circuit' can be defined as anything from a single component, to systems containing thousands of components.
Analog circuits are sometimes called linear circuits although many non-linear effects are used in analog circuits such as mixers, etc. Good examples of analog circuits include vacuum tube and transistor amplifiers, operational amplifiers and oscillators. One finds modern circuits that are analog; these days analog circuitry may use digital or microprocessor techniques to improve performance. This type of circuit is called "mixed signal" rather than analog or digital. Sometimes it may be difficult to differentiate between analog and digital circuits as they have elements of both linear and non-linear
A compressor is a mechanical device that increases the pressure of a gas by reducing its volume. An air compressor is a specific type of gas compressor. Compressors are similar to pumps: both increase the pressure on a fluid and both can transport the fluid through a pipe; as gases are compressible, the compressor reduces the volume of a gas. Liquids are incompressible; the main and important types of gas compressors are illustrated and discussed below: A positive displacement compressor is a system which compresses the air by the displacement of a mechanical linkage reducing the volume. Reciprocating compressors use pistons driven by a crankshaft, they can be either stationary or portable, can be single or multi-staged, can be driven by electric motors or internal combustion engines. Small reciprocating compressors from 5 to 30 horsepower are seen in automotive applications and are for intermittent duty. Larger reciprocating compressors well over 1,000 hp are found in large industrial and petroleum applications.
Discharge pressures can range from low pressure to high pressure. In certain applications, such as air compression, multi-stage double-acting compressors are said to be the most efficient compressors available, are larger, more costly than comparable rotary units. Another type of reciprocating compressor employed in automotive cabin air conditioning systems, is the swash plate or wobble plate compressor, which uses pistons moved by a swash plate mounted on a shaft. Household, home workshop, smaller job site compressors are reciprocating compressors 1½ hp or less with an attached receiver tank. A linear compressor is a reciprocating compressor with the piston being the rotor of a linear motor. An ionic liquid piston compressor, ionic compressor or ionic liquid piston pump is a hydrogen compressor based on an ionic liquid piston instead of a metal piston as in a piston-metal diaphragm compressor. Rotary screw compressors use two meshed rotating positive-displacement helical screws to force the gas into a smaller space.
These are used for continuous operation in commercial and industrial applications and may be either stationary or portable. Their application can be from 3 horsepower to over 1,200 horsepower and from low pressure to moderately high pressure; the classifications of rotary screw compressors vary based on stages, cooling methods, drive types among others. Rotary screw compressors are commercially produced in Water Flooded and Dry type; the efficiency of rotary compressors depends on the air drier, the selection of air drier is always 1.5 times volumetric delivery of the compressor. Designs with a single screw or three screws instead of two exist. Rotary vane compressors consist of a rotor with a number of blades inserted in radial slots in the rotor; the rotor is mounted offset in a larger housing, either circular or a more complex shape. As the rotor turns, blades slide in and out of the slots keeping contact with the outer wall of the housing. Thus, a series of increasing and decreasing volumes is created by the rotating blades.
Rotary Vane compressors are, with piston compressors one of the oldest of compressor technologies. With suitable port connections, the devices may be either a vacuum pump, they can be either stationary or portable, can be single or multi-staged, can be driven by electric motors or internal combustion engines. Dry vane machines are used at low pressures for bulk material movement while oil-injected machines have the necessary volumetric efficiency to achieve pressures up to about 13 bar in a single stage. A rotary vane compressor is well suited to electric motor drive and is quieter in operation than the equivalent piston compressor. Rotary vane compressors can have mechanical efficiencies of about 90%; the Rolling piston in a rolling piston style compressor plays the part of a partition between the vane and the rotor. Rolling piston forces gas against a stationary vane. 2 of these compressors can be mounted on the same shaft to increase capacity and reduce vibration and noise. A design without a spring is known as a swing compressor.
In refrigeration and air conditioning, this type of compressor is known as a rotary compressor, with rotary screw compressors being known as screw compressors. A scroll compressor known as scroll pump and scroll vacuum pump, uses two interleaved spiral-like vanes to pump or compress fluids such as liquids and gases; the vane geometry may be archimedean spiral, or hybrid curves. They operate more smoothly and reliably than other types of compressors in the lower volume range. One of the scrolls is fixed, while the other orbits eccentrically without rotating, thereby trapping and pumping or compressing pockets of fluid between the scrolls. Due to minimum clearance volume between the fixed scroll and the orbiting scroll, these compressors have a high volumetric efficiency; these compressors are extensively used in air conditioning and refrigeration because they are lighter and have fewer moving parts than reciprocating compressors and they are more reliable. They are more expensive though, so peltier coolers or rotary and reciprocating compressors may be used in applications where cost is the most important or one of the most important factors to consider when designing a refrigeration or air conditioining
A check valve, clack valve, non-return valve, reflux valve, retention valve or one-way valve is a valve that allows fluid to flow through it in only one direction. Check valves are two-port valves, meaning they have two openings in the body, one for fluid to enter and the other for fluid to leave. There are various types of check valves used in a wide variety of applications. Check valves are part of common household items. Although they are available in a wide range of sizes and costs, check valves are small, simple, or inexpensive. Check valves work automatically and most are not controlled by any external control; the bodies of most check valves are made of metal. An important concept in check valves is the cracking pressure, the minimum differential upstream pressure between inlet and outlet at which the valve will operate; the check valve is designed for and can therefore be specified for a specific cracking pressure. Heart valves are inlet and outlet check valves for the heart ventricles, since the ventricles act as pumps.
Cracking pressure — Refers to the minimum pressure differential needed between the inlet and outlet of the valve at which the first indication of flow occurs. Cracking pressure is known as unseating head or opening pressure. Reseal pressure — Refers to the pressure differential between the inlet and outlet of the valve during the closing process of the CV, at which there shall be no visible leak rate. Reseal pressure is known as sealing pressure, seating head or closing pressure. Back pressure — a pressure higher at the outlet of a fitting than that at the inlet or a point upstream A ball check valve is a check valve in which the closing member, the movable part to block the flow, is a ball. In some ball check valves, the ball is spring-loaded to help keep. For those designs without a spring, reverse flow is required to move the ball toward the seat and create a seal; the interior surface of the main seats of ball check valves are more or less conically-tapered to guide the ball into the seat and form a positive seal when stopping reverse flow.
Ball check valves are very small and cheap. They are used in liquid or gel minipump dispenser spigots, spray devices, some rubber bulbs for pumping air, etc. manual air pumps and some other pumps, refillable dispensing syringes. Although the balls are most made of metal, they can be made of other materials. High pressure HPLC pumps and similar applications use small inlet and outlet ball check valves with balls of ruby and seats made of sapphire or both ball and seat of ruby, for both hardness and chemical resistance. After prolonged use, such check valves can wear out or the seat can develop a crack, requiring replacement. Therefore, such valves are made to be replaceable, sometimes placed in a small plastic body tightly-fitted inside a metal fitting which can withstand high pressure and, screwed into the pump head. There are similar check valves where the disc is not a ball, but some other shape, such as a poppet energized by a spring. Ball check valves should not be confused with ball valves, a different type of valve in which a ball acts as a controllable rotor to stop or direct flow.
A diaphragm check valve uses a flexing rubber diaphragm positioned to create a normally-closed valve. Pressure on the upstream side must be greater than the pressure on the downstream side by a certain amount, known as the pressure differential, for the check valve to open allowing flow. Once positive pressure stops, the diaphragm automatically flexes back to its original closed position. A swing check valve or tilting disc check valve is a check valve in which the disc, the movable part to block the flow, swings on a hinge or trunnion, either onto the seat to block reverse flow or off the seat to allow forward flow; the seat opening cross-section may be perpendicular to the centerline between the two ports or at an angle. Although swing check valves can come in various sizes, large check valves are swing check valves. A common issue caused by swing check valves is known as water hammer; this can occur when the swing check closes and the flow abruptly stops, causing a surge of pressure resulting in high velocity shock waves that act against the piping and valves, placing large stress on the metals and vibrations in the system.
Undetected, water hammer can rupture pumps and pipes within the system. The flapper valve in a flush-toilet mechanism is an example of this type of valve. Tank pressure holding it closed is overcome by manual lift of the flapper, it remains open until the tank drains and the flapper falls due to gravity. Another variation of this mechanism is the clapper valve, used in applications such firefighting and fire life safety systems. A hinged gate only remains open in the inflowing direction; the clapper valve also has a spring that keeps the gate shut when there is no forward pressure. Another example is the backwater valve that protects against flooding caused by return flow of sewage waters; such risk occurs most in sanitary drainage systems connected to combined sewerage systems and in rainwater drainage systems. It may be caused by intense rainfall, flood. A stop-check valve is a check valve with override control to stop flow regardless of flow direction or pressure. In addition to closing in response to backflow or insufficient forward pressure, it can be deliberately shut by an external mechanism, the
A current source is an electronic circuit that delivers or absorbs an electric current, independent of the voltage across it. A current source is the dual of a voltage source; the term current sink is sometimes used for sources fed from a negative voltage supply. Figure 1 shows the schematic symbol for an ideal current source driving a resistive load. There are two types. An independent current source delivers a constant current. A dependent current source delivers a current, proportional to some other voltage or current in the circuit. An ideal current source generates a current, independent of the voltage changes across it. An ideal current source is a mathematical model, which real devices can approach closely. If the current through an ideal current source can be specified independently of any other variable in a circuit, it is called an independent current source. Conversely, if the current through an ideal current source is determined by some other voltage or current in a circuit, it is called a dependent or controlled current source.
Symbols for these sources are shown in Figure 2. The internal resistance of an ideal current source is infinite. An independent current source with zero current is identical to an ideal open circuit; the voltage across an ideal current source is determined by the circuit it is connected to. When connected to a short circuit, there is zero voltage and thus zero power delivered; when connected to a load resistance, the voltage across the source approaches infinity as the load resistance approaches infinity. No physical current source is ideal. For example, no physical current source can operate. There are two characteristics. One is its internal resistance and the other is its compliance voltage; the compliance voltage is the maximum voltage. Over a given load range, it is possible for some types of real current sources to exhibit nearly infinite internal resistance. However, when the current source reaches its compliance voltage, it abruptly stops being a current source. In circuit analysis, a current source having finite internal resistance is modeled by placing the value of that resistance across an ideal current source.
However, this model is only useful. The simplest non-ideal current source consists of a voltage source in series with a resistor; the amount of current available from such a source is given by the ratio of the voltage across the voltage source to the resistance of the resistor. This value of current will only be delivered to a load with zero voltage drop across its terminals The current delivered to a load with nonzero voltage across its terminals will always be different, it is given by the ratio of the voltage drop across the resistor to its resistance. For a nearly ideal current source, the value of the resistor should be large but this implies that, for a specified current, the voltage source must be large. Thus, efficiency is low and it is impractical to construct a'good' current source this way. Nonetheless, it is the case that such a circuit will provide adequate performance when the specified current and load resistance are small. For example, a 5 V voltage source in series with a 4.7 kilohm resistor will provide an constant current of 1 mA ± 5% to a load resistance in the range of 50 to 450 ohm.
A Van de Graaff generator is an example of such a high voltage current source. It behaves as an constant current source because of its high output voltage coupled with its high output resistance and so it supplies the same few microamperes at any output voltage up to hundreds of thousands of volts for large laboratory versions. In these circuits the output current is not controlled by means of negative feedback, they are implemented by active electronic components having current-stable nonlinear output characteristic when driven by steady input quantity. These circuits behave as dynamic resistors changing their present resistance to compensate current variations. For example, if the load increases its resistance, the transistor decreases its present output resistance to keep up a constant total resistance in the circuit. Active current sources have many important applications in electronic circuits, they are used in place of ohmic resistors in analog integrated circuits to generate a current that depends on the voltage across the load.
The common emitter configuration driven by a constant input current or voltage and common source driven by a constant voltage behave as current sources because the output impedance of these devices is high. The output part of the simple current mirror is an example of such a current source used in integrated circuits; the common base, common gate and common grid configurations can serve as constant current sources as well. A JFET can be made to act as a current source by tying its gate to its source; the current flowing is the IDSS of the F
A diode is a two-terminal electronic component that conducts current in one direction. A diode vacuum tube or thermionic diode is a vacuum tube with two electrodes, a heated cathode and a plate, in which electrons can flow in only one direction, from cathode to plate. A semiconductor diode, the most common type today, is a crystalline piece of semiconductor material with a p–n junction connected to two electrical terminals. Semiconductor diodes were the first semiconductor electronic devices; the discovery of asymmetric electrical conduction across the contact between a crystalline mineral and a metal was made by German physicist Ferdinand Braun in 1874. Today, most diodes are made of silicon, but other materials such as gallium arsenide and germanium are used; the most common function of a diode is to allow an electric current to pass in one direction, while blocking it in the opposite direction. As such, the diode can be viewed as an electronic version of a check valve; this unidirectional behavior is called rectification, is used to convert alternating current to direct current.
Forms of rectifiers, diodes can be used for such tasks as extracting modulation from radio signals in radio receivers. However, diodes can have more complicated behavior than this simple on–off action, because of their nonlinear current-voltage characteristics. Semiconductor diodes begin conducting electricity only if a certain threshold voltage or cut-in voltage is present in the forward direction; the voltage drop across a forward-biased diode varies only a little with the current, is a function of temperature. Diodes' high resistance to current flowing in the reverse direction drops to a low resistance when the reverse voltage across the diode reaches a value called the breakdown voltage. A semiconductor diode's current–voltage characteristic can be tailored by selecting the semiconductor materials and the doping impurities introduced into the materials during manufacture; these techniques are used to create special-purpose diodes. For example, diodes are used to regulate voltage, to protect circuits from high voltage surges, to electronically tune radio and TV receivers, to generate radio-frequency oscillations, to produce light.
Tunnel, Gunn and IMPATT diodes exhibit negative resistance, useful in microwave and switching circuits. Diodes, both vacuum and semiconductor, can be used as shot-noise generators. Thermionic diodes and solid-state diodes were developed separately, at the same time, in the early 1900s, as radio receiver detectors; until the 1950s, vacuum diodes were used more in radios because the early point-contact semiconductor diodes were less stable. In addition, most receiving sets had vacuum tubes for amplification that could have the thermionic diodes included in the tube, vacuum-tube rectifiers and gas-filled rectifiers were capable of handling some high-voltage/high-current rectification tasks better than the semiconductor diodes that were available at that time. In 1873, Frederick Guthrie observed that a grounded, white hot metal ball brought in close proximity to an electroscope would discharge a positively charged electroscope, but not a negatively charged electroscope. In 1880, Thomas Edison observed unidirectional current between heated and unheated elements in a bulb called Edison effect, was granted a patent on application of the phenomenon for use in a dc voltmeter.
About 20 years John Ambrose Fleming realized that the Edison effect could be used as a radio detector. Fleming patented the first true thermionic diode, the Fleming valve, in Britain on November 16, 1904. Throughout the vacuum tube era, valve diodes were used in all electronics such as radios, sound systems and instrumentation, they lost market share beginning in the late 1940s due to selenium rectifier technology and to semiconductor diodes during the 1960s. Today they are still used in a few high power applications where their ability to withstand transient voltages and their robustness gives them an advantage over semiconductor devices, in musical instrument and audiophile applications. In 1874, German scientist Karl Ferdinand Braun discovered the "unilateral conduction" across a contact between a metal and a mineral. Indian scientist Jagadish Chandra Bose was the first to use a crystal for detecting radio waves in 1894; the crystal detector was developed into a practical device for wireless telegraphy by Greenleaf Whittier Pickard, who invented a silicon crystal detector in 1903 and received a patent for it on November 20, 1906.
Other experimenters tried a variety of other minerals as detectors. Semiconductor principles were unknown to the developers of these early rectifiers. During the 1930s understanding of physics advanced and in the mid 1930s researchers at Bell Telephone Laboratories recognized the potential of the crystal detector for application in microwave technology. Researchers at Bell Labs, Western Electric, MIT, Purdue and in the UK intensively developed point-contact diodes during World War II for application in ra
In classical mechanics, the gravitational potential at a location is equal to the work per unit mass that would be needed to move the object from a fixed reference location to the location of the object. It is analogous to the electric potential with mass playing the role of charge; the reference location, where the potential is zero, is by convention infinitely far away from any mass, resulting in a negative potential at any finite distance. In mathematics, the gravitational potential is known as the Newtonian potential and is fundamental in the study of potential theory, it may be used for solving the electrostatic and magnetostatic fields generated by uniformly charged or polarized ellipsoidal bodies. The gravitational potential at a location is the gravitational potential energy at that location per unit mass: V = U m, where m is the mass of the object. Potential energy is equal to the work done by the gravitational field moving a body to its given position in space from infinity. If the body has a mass of 1 unit the potential energy to be assigned to that body is equal to the gravitational potential.
So the potential can be interpreted as the negative of the work done by the gravitational field moving a unit mass in from infinity. In some situations, the equations can be simplified by assuming a field, nearly independent of position. For instance, in a region close to the surface of the Earth, the gravitational acceleration, g, can be considered constant. In that case, the difference in potential energy from one height to another is, to a good approximation, linearly related to the difference in height: Δ U ≈ m g Δ h; the potential V of a unit mass m at a distance x from a point mass of mass M can be defined as the work W that needs to be done by an external agent to bring the unit mass in from infinity to that point: V = W m = 1 m ∫ ∞ x F ⋅ d x = 1 m ∫ ∞ x G m M x 2 d x = − G M x, where G is the gravitational constant, F is the gravitational force. The potential has units of energy per e.g. J/kg in the MKS system. By convention, it is always negative where it is defined, as x tends to infinity, it approaches zero.
The gravitational field, thus the acceleration of a small body in the space around the massive object, is the negative gradient of the gravitational potential. Thus the negative of a negative gradient yields positive acceleration toward a massive object; because the potential has no angular components, its gradient is a = − G M x 3 x = − G M x 2 x ^, where x is a vector of length x pointing from the point mass toward the small body and x ^ is a unit vector pointing from the point mass toward the small body. The magnitude of the acceleration therefore follows an inverse square law: | a | = G M x 2; the potential associated with a mass distribution is the superposition of the potentials of point masses. If the mass distribution is a finite collection of point masses, if the point masses are located at the points x1... xn and have masses m1... mn the potential of the distribution at the point x is V = ∑ i = 1 n − G m i | x − x i |. If the mass distribution is given as a mass measure dm on three-dimensional Euclidean space R3 the potential is the convolution of −G/|r| with dm.
In good cases this equals the integral V = − ∫ R 3 G | x − r | d m, where |x − r| is the distance between the points x and r. If there is a function ρ representing the density of the distribution at r, so that dm= ρdv, where dv is the Euclidean volume element the gravitational potential is the volume integral V = − ∫ R 3 G | x −