Power (physics)

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 (J/s), 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 (comparing to the power of a horse). Being the rate of work, the equation for power can be written:

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, which is 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.[1][2]

Common symbols
SI unitwatt
In SI base unitskgm2s−3
Derivations from
other quantities


The dimension of power is energy divided by time. The SI unit of power is the watt (W), which is equal to one joule per second. Other units of power include ergs per second (erg/s), horsepower (hp), metric horsepower (Pferdestärke (PS) or cheval vapeur (CV)), and 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, and is equivalent to about 746 watts. Other units include dBm, a logarithmic measure relative to a reference of 1 milliwatt; food calories per hour (often referred to as kilocalories per hour); BTU per hour (BTU/h); and tons of refrigeration (12,000 BTU/h).

Equations for power

Power, as a function of time, is the rate (i.e. derivative) at which work is done, so can be expressed by this equation:

where P is power, W is work, and t is time. Because work is a force F applied over a distance x,

for a constant force, power can be rewritten as:

In fact, this is valid for any force, as a consequence of applying the fundamental theorem of calculus.

Average power

As a simple example, burning one kilogram of coal releases much more energy than does detonating a kilogram of TNT,[3] but because the TNT reaction releases energy much more quickly, 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

It is the average amount of work done or energy converted per unit of time. The average power is often simply called "power" when the context makes it clear.

The instantaneous power is then the limiting value of the average power as the time interval Δt approaches zero.

In the case of constant power P, the amount of work performed during a period of duration t is given by:

In the context of energy conversion, it is more customary to use the symbol E rather than W.

Mechanical power

Horsepower plain
One metric horsepower is needed to lift 75 kilograms by 1 meter in 1 second.

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 also 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:

where x defines the path C and v is the velocity along this path.

If the force F is derivable from a potential (conservative), then applying the gradient theorem (and remembering that force is the negative of the gradient of the potential energy) yields:

where A and B are the beginning and end of the path along which the work was done.

The power at any point along the curve C is the time derivative

In one dimension, this can be simplified to:

In rotational systems, power is the product of the torque τ and angular velocity ω,

where ω measured in radians per second. The represents scalar product.

In fluid power systems such as hydraulic actuators, power is given by

where p is pressure in pascals, or N/m2 and Q is volumetric flow rate in m3/s in SI units.

Mechanical advantage

If a mechanical system has no losses, then the input power must equal the output power. This provides a simple formula for the mechanical advantage of the system.

Let the input power to a device be a force FA acting on a point that moves with velocity vA and the output power be a force FB acts on a point that moves with velocity vB. If there are no losses in the system, then

and the mechanical advantage of the system (output force per input force) is given by

The similar relationship is obtained for rotating systems, where TA and ωA are the torque and angular velocity of the input and TB and ωB are the torque and angular velocity of the output. If there are no losses in the system, then

which yields the mechanical advantage

These relations are important because they define the maximum performance of a device in terms of velocity ratios determined by its physical dimensions. See for example gear ratios.

Electrical power

Ansel Adams - National Archives 79-AAB-02
Ansel Adams photograph of electrical wires of the Boulder Dam Power Units, 1941–1942

The instantaneous electrical power P delivered to a component is given by


P(t) is the instantaneous power, measured in watts (joules per second)
V(t) is the potential difference (or voltage drop) across the component, measured in volts
I(t) is the current through it, measured in amperes

If the component is a resistor with time-invariant voltage to current ratio, then:


is the resistance, measured in ohms.

Peak power and duty cycle

In a train of identical pulses, the instantaneous power is a periodic function of time. The ratio of the pulse duration to the period is equal to the ratio of the average power to the peak power. It is also called the duty cycle (see text for definitions).

In the case of a periodic signal of period , like a train of identical pulses, the instantaneous power is also a periodic function of period . The peak power is simply defined by:


The peak power is not always readily measurable, however, and the measurement of the average power is more commonly performed by an instrument. If one defines the energy per pulse as:

then the average power is:


One may define the pulse length such that so that the ratios

are equal. These ratios are called the duty cycle of the pulse train.

Radiant power

Power is related to intensity at a radius ; the power emitted by a source can be written as:

See also


  1. ^ Halliday and Resnick (1974). "6. Power". Fundamentals of Physics.CS1 maint: Uses authors parameter (link)
  2. ^ Chapter 13, § 3, pp 13-2,3 The Feynman Lectures on Physics Volume I, 1963
  3. ^ Burning coal produces around 15-30 megajoules per kilogram, while detonating TNT produces about 4.7 megajoules per kilogram. For the coal value, see Fisher, Juliya (2003). "Energy Density of Coal". The Physics Factbook. Retrieved 30 May 2011. For the TNT value, see the article TNT equivalent. Neither value includes the weight of oxygen from the air used during combustion.

External links

Audio power

Audio power is the electrical power transferred from an audio amplifier to a loudspeaker, measured in watts. The electrical power delivered to the loudspeaker, together with its efficiency, determines the sound power generated (with the rest of the electrical power being converted to heat).

Amplifiers are limited in the electrical energy they can output, while loudspeakers are limited in the electrical energy they can convert to sound energy without being damaged or distorting the audio signal. These limits, or power ratings, are important to consumers finding compatible products and comparing competitors.

Brake-specific fuel consumption

Brake-specific fuel consumption (BSFC) is a measure of the fuel efficiency of any prime mover that burns fuel and produces rotational, or shaft power. It is typically used for comparing the efficiency of internal combustion engines with a shaft output.

It is the rate of fuel consumption divided by the power produced. It may also be thought of as power-specific fuel consumption, for this reason. BSFC allows the fuel efficiency of different engines to be directly compared.

Bridged and paralleled amplifiers

Multiple electronic amplifiers can be connected such that they drive a single floating load (bridge) or a single common load (parallel), to increase the amount of power (physics) available in different situations. This is commonly encountered in audio applications.

Effective radiated power

Effective radiated power (ERP), synonymous with equivalent radiated power, is an IEEE standardized definition of directional radio frequency (RF) power, such as that emitted by a radio transmitter. It is the total power in watts that would have to be radiated by a half-wave dipole antenna to give the same radiation intensity (signal strength in watts per square meter) as the actual source at a distant receiver located in the direction of the antenna's strongest beam (main lobe). ERP measures the combination of the power emitted by the transmitter and the ability of the antenna to direct that power in a given direction. It is equal to the input power to the antenna multiplied by the gain of the antenna. It is used in electronics and telecommunications, particularly in broadcasting to quantify the apparent power of a broadcasting station experienced by listeners in its reception area.

An alternate parameter that measures the same thing is effective (or equivalent) isotropic radiated power (EIRP). Effective isotropic radiated power is the total power that would have to be radiated by a hypothetical isotropic antenna to give the same signal strength as the actual source in the direction of the antenna's strongest beam. The difference between EIRP and ERP is that ERP compares the actual antenna to a half-wave dipole antenna, while EIRP compares it to a theoretical isotropic antenna. Since a half-wave dipole antenna has a gain of 1.64, or 2.15 decibels compared to an isotropic radiator, if ERP and EIRP are expressed in watts their relation is

If they are expressed in decibels

Engine power

Engine power or horsepower is the maximum power that an engine can put out. It can be expressed in kilowatts or horsepower. The power output depends on the size and design of the engine, but also on the speed at which it is running and the load or torque. Maximum power is achieved at relatively high speeds and at high load.

Linear transformer driver

A linear transformer driver (LTD) is an annular parallel connection of switches and capacitors designed to deliver rapid high power pulses. The LTD was designed at the Institute of High Current Electronics (IHCE) in Tomsk, Russia. The LTD is capable of producing high current pulses, up to 1 mega amps (106 ampere), with a risetime of less than 100 ns. This is an improvement over Marx generator based pulsed power devices which require pulse compression to achieve such fast risetimes. It is being considered as a driver for z-pinch based inertial confinement fusion.

Low power

In electronics, low power may mean:

Radio transmitters that send out relatively little power.

QRP operation, using "the minimum power necessary to carry out the desired communications", in amateur radio.

Cognitive radio transceivers typically automatically reduce the transmitted power to much less than the power required for reliable one-way broadcasts.

Low-power broadcasting that the power of the broadcast is less, i.e. the radio waves are not intended to travel as far as from typical transmitters.

Low-power communication device, a radio transmitter used in low-power broadcasting.

Low-power electronics, the consumption of electric power is deliberately low, e.g. notebook processors.In science

Low power is due to small sample sizes or poorly designed experiments

Nominal power (radio broadcasting)

Nominal power is a measurement of a mediumwave radio station's output used in the United States. AM broadcasters are licensed by the Federal Communications Commission to operate at a specific nominal power, which may be (and usually is) different from the transmitter power output.

For non-directional stations, nominal power is normally equal to the RF power presented to the antenna, as determined from the base current and the antenna's nominal impedance at the carrier frequency.

For directional stations, nominal power is normally equal to the RF power at the common point (the point at which the transmitter output branches off into separate phasing networks for each tower).In both cases, nominal power excludes losses in transmission lines between the tower or phasor and the transmitter; however, it includes losses in a resistor network used to decrease the efficiency of the antenna system.

Nominal power is ultimately a historical artifact of the regulatory regime employed by the FCC prior to the 1980s. In the old system, rather than allowing licensees to choose any power level which would meet the efficiency and interference standards for their class, stations were restricted to a small set of power levels: 50, 100, 250, 500, 1000, 2500, 5000, 10000, 25000, and 50000 watts. A station whose maximum coverage would otherwise be available at 4500 watts (given a specific directional pattern and antenna system efficiency) had a choice of either living with 2500 watts, or reducing the antenna efficiency to a level which would allow for 5 kW. Newly-constructed stations could fairly easily design an antenna system to meet the requirements, but stations on or moving to a shared tower with higher efficiency had a problem. The resistor network exception was created to allow stations to reduce their antenna efficiency without having to modify the existing tower.

Rule changes in the 1980s did away with the fixed set of power choices, allowing stations to choose an appropriate power level for their antenna system ("dial-a-power"), so there should no longer be any need for the concept of nominal power. However, stations still take advantage of the resistor exception in some cases, simply because they perceive the marketing advantage of higher power (or at least "round" power) to be worth the cost of the wasted energy.

Orders of magnitude (power)

This page lists examples of the power in watts produced by various sources of energy. They are grouped by orders of magnitude.


Output may refer to:

The information produced by a computer, see Input/output

An output state of a system, see state (computer science)

Output (economics), the amount of goods and services produced

Gross output in economics, the value of net output or GDP plus intermediate consumption

Net output in economics, the gross revenue from production less the value of goods and services

Power (physics) or Work (physics) output of a machine

Dependent variable of a function, in mathematics

Output (album)

Power-to-weight ratio

Power-to-weight ratio (or specific power or power-to-mass ratio) is a calculation commonly applied to engines and mobile power sources to enable the comparison of one unit or design to another. Power-to-weight ratio is a measurement of actual performance of any engine or power source. It is also used as a measurement of performance of a vehicle as a whole, with the engine's power output being divided by the weight (or mass) of the vehicle, to give a metric that is independent of the vehicle's size. Power-to-weight is often quoted by manufacturers at the peak value, but the actual value may vary in use and variations will affect performance.

The inverse of power-to-weight, weight-to-power ratio (power loading) is a calculation commonly applied to aircraft, cars, and vehicles in general, to enable the comparison of one vehicle's performance to another. Power-to-weight ratio is equal to thrust per unit mass multiplied by the velocity of any vehicle.

Power density

Power density (or volume power density or volume specific power) is the amount of power (time rate of energy transfer) per unit volume.

In energy transformers including batteries, fuel cells, motors, etc., and also power supply units or similar, power density refers to a volume. It is then also called volume power density, which is expressed as W/m3.

Volume power density is sometimes an important consideration where space is constrained.

In reciprocating internal combustion engines, power density—power per swept volume or brake horsepower per cubic centimeter —is an important metric. This is based on the internal capacity of the engine, not its external size.

Pulsed power

Pulsed power is the science and technology of accumulating energy over a relatively long period of time and releasing it very quickly, thus increasing the instantaneous power.

Radiant flux

In radiometry, radiant flux or radiant power is the radiant energy emitted, reflected, transmitted or received, per unit time, and spectral flux or spectral power is the radiant flux per unit frequency or wavelength, depending on whether the spectrum is taken as a function of frequency or of wavelength. The SI unit of radiant flux is the watt (W), that is the joule per second (J/s) in SI base units, while that of spectral flux in frequency is the watt per hertz (W/Hz) and that of spectral flux in wavelength is the watt per metre (W/m)—commonly the watt per nanometre (W/nm).

Sound power

Sound power or acoustic power is the rate at which sound energy is emitted, reflected, transmitted or received, per unit time. It is defined as "through a surface, the product of the sound pressure, and the component of the particle velocity, at a point on the surface in the direction normal to the surface, integrated over that surface." The SI unit of sound power is the watt (W). It relates to the power of the sound force on a surface enclosing a sound source, in air. For a sound source, unlike sound pressure, sound power is neither room-dependent nor distance-dependent. Sound pressure is a property of the field at a point in space, while sound power is a property of a sound source, equal to the total power emitted by that source in all directions. Sound power passing through an area is sometimes called sound flux or acoustic flux through that area.

Thrust-specific fuel consumption

Thrust-specific fuel consumption (TSFC) is the fuel efficiency of an engine design with respect to thrust output.

TSFC may also be thought of as fuel consumption (grams/second) per unit of thrust (kilonewtons, or kN). It is thus thrust-specific, meaning that the fuel consumption is divided by the thrust.

TSFC or SFC for thrust engines (e.g. turbojets, turbofans, ramjets, rocket engines, etc.) is the mass of fuel needed to provide the net thrust for a given period e.g. lb/(h·lbf) (pounds of fuel per hour-pound of thrust) or g/(s·kN) (grams of fuel per second-kilonewton). Mass of fuel is used, rather than volume (gallons or litres) for the fuel measure, since it is independent of temperature.Specific fuel consumption of air-breathing jet engines at their maximum efficiency is more or less proportional to speed. The fuel consumption per mile or per kilometre is a more appropriate comparison for aircraft that travel at very different speeds. There also exists power specific fuel consumption, which equals speed times the thrust specific fuel consumption. It can have units of pounds per hour per horsepower.

This figure is inversely proportional to specific impulse.

Transmitter power output

In radio transmission, transmitter power output (TPO) is the actual amount of power (in watts) of radio frequency (RF) energy that a transmitter produces at its output.

This is not the amount of power that a radio station reports as its power, as in "we're 100,000 watts of rock 'n' roll", which is usually the effective radiated power (ERP). The TPO for VHF-/UHF-transmitters is normally more than the ERP, for LF-/MF-transmitters it has nearly the same value, while for VLF-transmitters it may be less.

The radio antenna's design "focuses" the signal toward the horizon, creating gain and increasing the ERP. There is also some loss (negative gain) from the feedline, which reduces some of the TPO to the antenna by both resistance and by radiating a small part of the signal.

The basic equation relating transmitter to effective power is:

Note that in this formula the Antenna Gain is expressed with reference to a tuned dipole (dBd)

Zero Power Physics Reactor

The Zero Power Physics Reactor or ZPPR (originally named Zero Power Plutonium Reactor) was a nuclear reactor located at the Idaho National Laboratory, Idaho, USA.ZPPR ran only at extremely low power, for testing nuclear reactor designs. ZPPR was operated as a critical facility from April 18, 1969 until 1990.

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