power, in mathematics: see exponent.
power, in physics, time rate of doing work or of producing or expending energy. The unit of power based on the English units of measurement is the horsepower, devised for describing mechanical power by James Watt, who estimated that a horse can do 550 ft-lb of work per sec; a foot-pound is the work done when a weight (force) of 1 lb is moved through a distance of 1 ft. The unit of power in the metric system is the watt, named in honor of James Watt and equal to 1 joule per sec; the watt is used for measuring electric power in most countries, even those still using English units for other quantities. In common usage, the terms power and energy have become synonymous; for example, electrical energy is usually referred to as electric power (see power, electric). See also energy, sources of.
power, electric, energy dissipated in an electrical or electronic circuit or device per unit of time. The electrical energy supplied by a current to an appliance enables it to do work or provide some other form of energy such as light or heat. Electric power is usually measured in Watts, kilowatts (1,000 watts), and megawatts (1,000,000 watts). The amount of electrical energy used by an appliance is found by multiplying its consumed power by the length of time of operation. The units of electrical energy are usually watt-seconds (joules), watt-hours, or kilowatt-hours. For commercial purposes the kilowatt-hour is the unit of choice.

Sources of Electrical Energy

Electrical energy occurs naturally, but seldom in forms that can be used. For example, although the energy dissipated as lightning exceeds the world's demand for electricity by a large factor, lightning has not been put to practical use because of its unpredictability and other problems. Generally, practical electric-power-generating systems convert the mechanical energy of moving parts into electrical energy (see generator). While systems that operate without a mechanical step do exist, they are at present either excessively inefficient or expensive because of a dependence on elaborate technology. While some electric plants derive mechanical energy from moving water (hydroelectric power), the vast majority derive it from heat engines in which the working substance is steam. Roughly 89% of power in the United States is generated this way. The steam is generated with heat from combustion of fossil fuels or from nuclear fission (see nuclear energy; nuclear reactor).

Steam as an Energy Source

The conversion of mechanical energy to electrical energy can be accomplished with an efficiency of about 80%. In a hydroelectric plant, the losses occur in the turbines, bearings, penstocks, and generators. The basic limitations of thermodynamics fix the maximum efficiency obtainable in converting heat to electrical energy. The necessity of limiting the temperature to safe levels also helps to keep the efficiency down to about 41% for a fossil-fuel plant. Most nuclear plants use low-pressure, low-temperature steam operation, and have an even lower efficiency of about 30%. Nuclear plants have been able to achieve efficiency up to 40% with liquid-metal cooling. It is thought that by using magnetohydrodynamic "topping" generators in conjunction with normal steam turbines, the efficiency of conventional plants can be raised to close to 50%. These devices remove the restrictions imposed by the blade structure of turbines by using the steam or gasses produced by combustion as the working fluid.

Environmental Concerns

The heat generated by an electric-power plant that is not ultimately converted into electrical energy is called waste heat. The environmental impact of this waste is potentially catastrophic, especially when, as is often the case, the heat is absorbed by streams or other bodies of water. Cooling towers help to dispose waste heat into the atmosphere. Associated with nuclear plants, in addition to the problem of waste heat, are difficulties attending the disposal and confinement of reaction products that remain dangerously radioactive for many thousands of years and the adjustment of such plants to variable demands for power. Public concern about such issues—fueled in part by the accidents at the Three Mile Island nuclear plant in Harrisburg Pennsylvania in 1979, and the nuclear plant explosion in the Soviet Union at Chernobyl in 1986—forced the U.S. government to introduce extensive safety regulations for nuclear plants. Partly because of those regulations, nuclear plants are proving to be uneconomical. Several are being shut down and replaced by conventionally fueled plants.

Alternative Energy Sources

Fuel cells develop electricity by direct conversion of hydrogen, hydrocarbons, alcohol, or other fuels, with an efficiency of 50% to 60%. Although they have been used to produce electric power in space vehicles and some terrestrial locations, several problems have kept them from being widely used. Most important, the catalyst, which is an important component of a fuel cell, especially one that is operating at around room temperature, is very expensive. Controlled nuclear fusion could provide a virtually unlimited source of heat energy to produce steam in generating plants; however, many problems surround its development, and no appreciable contribution is expected from this source in the near future.

Solar energy has been recognized as a feasible alternative. It has been suggested that efficient collection of the solar energy incident on 14% of the western desert areas of the United States would provide enough electricity to satisfy current demands. Two main solar processes could be used. Photovoltaic cells (see solar cell) convert sunlight directly into electrical energy. Another method would use special coatings that absorb sunlight readily and emit infrared radiation slowly, making it possible to heat fluids to 1,000°F; (540°C;) by solar radiation. The heat in turn can be converted to electricity. Some of this heat would be stored to allow operation at night and during periods of heavy cloud cover. The projected efficiency of such a plant would be about 30%, but this fairly low efficiency must be balanced against the facts that energy from the sun costs nothing and that the waste heat from such a plant places virtually no additional burden on the environment. The principal problem with this and other exotic systems for generating electricity is that the time needed for their implementation may be considerable.

Windmills, once widely used for pumping water, have become viable for electric-power generation because of advances in their design and the development of increasingly efficient generators. Windmill "farms," at which rows of windmills are joined together as the source of electrical energy, serve as a significant, though minor, source of electrical energy in coastal and plains areas. However, the vagaries of the wind make this a difficult solution to implement on a large scale.

See also energy, sources of.

Transmission of Electrical Energy

Electrical energy is of little use unless it can be made available at the place where it is to be used. To minimize energy losses from heating of conductors and to economize on the material needed for conductors, electricity is usually transmitted at the highest voltages possible. As modern transformers are virtually loss free, the necessary steps upward or downward in voltage are easily accomplished. Transmission lines for alternating current using voltages as high as 765,000 volts are not uncommon. For voltages higher than this it is advantageous to transmit direct current rather than alternating current. Recent advances in rectifiers, which turn alternating current into direct current, and inverters, which convert direct into alternating, have made possible transmission lines that operate at 800,000 volts and above. Such lines are still very expensive, however.

Electric utilities are tied together by transmission lines into large systems called power grids. They are thus able to exchange power so that a utility with a low demand can assist another with a high demand to help prevent a blackout, which involves the partial or total shutdown of a utility. Under such a system a utility experiencing too great a load, as when peak demand coincides with equipment failure, must remove itself from the grid or endanger other utilities. During periods in which demand exceeds supply a utility can reduce the power drawn from it by lowering its voltage. These voltage reductions, which are normally of 3%, 5%, or 8%, result in power reductions, or brownouts, of about 6%, 10%, or 15%, causing inefficient operation of some electrical devices. The power distribution system, because of its generation of low-frequency electromagnetic fields, has been suggested as a possible source of health problems.

Reactive Power

Reactive power is a concept used by engineers to describe the loss of power in a system arising from the production of electric and magnetic fields. Although reactive loads such as inductors and capacitors dissipate no power, they drop voltage and draw current, which creates the impression that they actually do. This "imaginary power" or "phantom power" is called reactive power. It is measured in a unit called Volt-Amps-Reactive (VAR). The actual amount of power being used, or dissipated, is called true power, and is measured in the unit of watts. The combination of reactive power and true power is called apparent power, and it is the product of a circuit's voltage and current. Apparent power is measured in the unit of Volt-Amps (VA). Devices which store energy by virtue of a magnetic field produced by a flow of current are said to absorb reactive power; those which store energy by virtue of electric fields are said to generate reactive power. Reactive power is significant because it must be provided and maintained to insure continuous, steady voltage on transmission networks. Reactive power thus is produced for maintenance of the system and not for end-use consumption. Power losses incurred in transmission from heat and electromagnetic emissions are included in the total reactive power requirement as are the needs of power hungry devices, such as electric motors, electromagnetic generators, and alternators. This power is supplied for many purposes by condensers, capacitors, and similar devices, which can react to changes in current flow by releasing energy to normalize the flow. If elements of the power grid cannot get the reactive power they need from nearby sources, they will pull it across transmission lines and destabilize the grid. In this way, poor management of reactive power can cause major blackouts.


See K. W. Li and A. P. Priddy, Power Plant System Design (1985); L. F. Drbal et al., Power Plant Engineering (1996).

Power produced by a stream of water as it turns a wheel or similar device. The waterwheel, probably invented in the 1st century BC, was widely used throughout the Middle Ages and into modern times for grinding grain, operating bellows for furnaces, and other purposes. The more compact water turbine, which passes water through a series of fixed and rotating blades, was introduced in 1827. Water turbines, used originally for direct mechanical drive for irrigation, now are used almost exclusively to generate hydroelectric power.

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Electricity produced by turbines operated by tide flow. Large amounts of power are potentially available from the tides in certain locations, such as Canada's Bay of Fundy, where the tidal range reaches more than 50 ft (15 m), but this potential power is not continuous and varies with the seasons. The first working modern tidal power plant was built in France in 1961–67 and has 24 power units of 10,000 kilowatts each.

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Radiation from the Sun that can produce heat, generate electricity, or cause chemical reactions. Solar collectors collect solar radiation and transfer it as heat to a carrier fluid. It can then be used for heating. Solar cells convert solar radiation directly into electricity by means of the photovoltaic effect. Solar energy is inexhaustible and nonpolluting, but converting solar radiation to electricity is not yet commercially competitive, because of the high cost of producing large-scale solar cell arrays and the inherent inefficiency in converting light to electricity.

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Means by which a nation extends its military power onto the seas. Measured in terms of a nation's capacity to use the seas in defiance of rivals, it consists of combat vessels and weapons, auxiliary craft, commercial shipping, bases, and trained personnel. It includes aircraft based on carriers or used in support of shipping. Its main purpose is to protect friendly shipping from enemy attack and to destroy or hinder the enemy's shipping. It may also be used to enforce a blockade. Finally, naval forces have been used to bombard land targets from the sea. The aircraft carrier added a new dimension to this capability, as did the missile-firing nuclear submarine. The classic exposition of the role of sea power as the basis of national greatness is Alfred Thayer Mahan's The Influence of Sea Power upon History (1890).

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In science and engineering, the time rate of doing work or delivering energy. Power (math.P) can be expressed as the amount of work done (math.W), or energy transferred, divided by the time interval (math.t): math.P = math.W/math.t. A given amount of work can be done by a low-powered motor in a long time or by a high-powered motor in a short time. Units of power are those of work (or energy) per unit time, such as foot-pounds per minute, joules per second (called watts), or ergs per second. Power can also be expressed as the product of the force (math.F) applied to move an object and the speed (math.v) of the object in the direction of the force: math.P = math.Fmath.v. Seealso horsepower.

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Power of a government to exercise reasonable control over people and property within its jurisdiction in the interest of general security, health, safety, morals, and welfare. It is generally regarded as one of the powers reserved to the states under the U.S. Constitution. In considering cases involving the exercise of police power, the courts have applied a doctrine called “balance of interests” to determine when the public's right to health and well-being outweighs private or individual concerns. Of equal concern is that due process of law be observed.

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A hydro station generates power by the controlled release of water from the reservoir of a dammed elipsis

Electricity produced from generators driven by water turbines that convert the energy in falling or fast-flowing water to mechanical energy. Water at a higher elevation flows downward through large pipes or tunnels (penstocks). The falling water rotates turbines, which drive the generators, which convert the turbines' mechanical energy into electricity. The advantages of hydroelectric power over such other sources as fossil fuels and nuclear fission are that it is continually renewable and produces no pollution. Norway, Sweden, Canada, and Switzerland rely heavily on hydroelectricity because they have industrialized areas close to mountainous regions with heavy rainfall. The U.S., Russia, China, India, and Brazil get a much smaller proportion of their electric power from hydroelectric generation. Seealso tidal power.

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In international relations, an equilibrium of power sufficient to discourage or prevent one nation or party from imposing its will on or interfering with the interests of another. The term came into use at the end of the Napoleonic Wars to denote the power relationships in the European state system. Until World War I, Britain played the role of balancer in a number of shifting alliances. After World War II, a Northern Hemisphere balance of power pitted the U.S. and its allies (see NATO) against the Soviet Union and its satellites (see Warsaw Pact) in a bipolar balance of power backed by the threat of nuclear war. China's defection from the Soviet camp to a nonaligned but covertly anti-Soviet stance produced a third node of power. With the Soviet Union's collapse (1991), the U.S. and its NATO allies were recognized universally as the world's paramount military power.

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Authorization to act as agent or attorney for another. Many of the general powers of attorney important in civil-law countries come under the powers of trust in common-law countries (see civil law, common law). Durable power of attorney becomes effective when the principal becomes unable to manage his or her affairs; general power of attorney authorizes the agent to carry on business for the principal; special power of attorney authorizes the agent to carry out a particular business transaction.

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or atomic energy

Energy released from atomic nuclei in significant amounts. In 1919 Ernest Rutherford discovered that alpha rays could split the nucleus of an atom. This led ultimately to the discovery of the neutron and the release of huge amounts of energy by the process of nuclear fission. Nuclear energy is also released as a result of nuclear fusion. The release of nuclear energy can be controlled or uncontrolled. Nuclear reactors carefully control the release of energy, whereas the energy release of a nuclear weapon or resulting from a core meltdown in a nuclear reactor is uncontrolled. Seealso chain reaction, nuclear power, radioactivity.

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For optical fibers, a power-law index profile is an index of refraction profile characterized by

n(r) =
begin{cases} n_1 sqrt{1-2Deltaleft({r over alpha}right)^g} & r le alpha n_1 sqrt{1-2Delta} & r ge alpha end{cases} where Delta = {n_1^2 - n_2^2 over 2 n_1^2},

and n(r) is the nominal refractive index as a function of distance from the fiber axis, n_1 is the nominal refractive index on axis, n_2 is the refractive index of the cladding, which is taken to be homogeneous (n(r)=n_2 mathrm{ for } r ge alpha), alpha is the core radius, and g is a parameter that defines the shape of the profile. alpha is often used in place of g. Hence, this is sometimes called an alpha profile.

For this class of profiles, multimode distortion is smallest when g takes a particular value depending on the material used. For most materials, this optimum value is approximately 2. In the limit of infinite g, the profile becomes a step-index profile.

See also


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