In
physicsPhysics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...
,
power is the rate at which energy is transferred, used, or transformed. For example, the rate at which a light bulb transforms electrical energy into heat and light is measured in watts—the more wattage, the more power, or equivalently the more electrical energy is used per unit time.
Energy transfer can be used to do work, so power is also the rate at which this work is performed. The output power of an electric motor is the product of the torque the motor generates and the angular velocity of its output shaft. The power expended to move a vehicle is the product of the traction force of the wheels and the velocity of the vehicle.
The integral of power over time defines the work done. Because this integral depends on the trajectory of the point of application of the force and torque, this calculation of work is said to be "path dependent."
Units
The dimension of power is energy divided by time. The
SIThe International System of Units is the modern form of the metric system and is generally a system of units of measurement devised around seven base units and the convenience of the number ten. The older metric system included several groups of units...
unit of power is the
wattThe watt is a derived unit of power in the International System of Units , named after the Scottish engineer James Watt . The unit, defined as one joule per second, measures the rate of energy conversion.-Definition:...
(W), which is equal to one
jouleThe joule ; symbol J) is a derived unit of energy or work in the International System of Units. It is equal to the energy expended in applying a force of one newton through a distance of one metre , or in passing an electric current of one ampere through a resistance of one ohm for one second...
per second. Other units of power include
ergAn erg is the unit of energy and mechanical work in the centimetre-gram-second system of units, symbol "erg". Its name is derived from the Greek ergon, meaning "work"....
s per second (erg/s),
horsepowerHorsepower is the name of several units of measurement of power. The most common definitions equal between 735.5 and 750 watts.Horsepower was originally defined to compare the output of steam engines with the power of draft horses in continuous operation. The unit was widely adopted to measure the...
(hp), metric horsepower (Pferdestärke (PS) or cheval vapeur, CV), and
foot-poundsThe foot-pound force, or simply foot-pound is a unit of work or energy in the Engineering and Gravitational Systems in United States customary and Imperial units of measure. It is the energy transferred on applying a force of 1 pound-force through a displacement of 1 foot...
per minute. One horsepower is equivalent to 33,000 foot-pounds per minute, or the power required to lift 550
poundsThe pound or pound-mass is a unit of mass used in the Imperial, United States customary and other systems of measurement...
by one foot in one second, and is equivalent to about 746 watts. Other units include
dBmdBm is an abbreviation for the power ratio in decibels of the measured power referenced to one milliwatt . It is used in radio, microwave and fiber optic networks as a convenient measure of absolute power because of its capability to express both very large and very small values in a short form...
, a relative logarithmic measure with 1 milliwatt as reference; (food)
calorieThe calorie is a pre-SI metric unit of energy. It was first defined by Nicolas Clément in 1824 as a unit of heat, entering French and English dictionaries between 1841 and 1867. In most fields its use is archaic, having been replaced by the SI unit of energy, the joule...
s per hour (often referred to as kilocalories per hour); Btu per hour (Btu/h); and tons of refrigeration (12,000 Btu/h).
Average power
As a simple example, burning a kilogram of
coalCoal is a combustible black or brownish-black sedimentary rock usually occurring in rock strata in layers or veins called coal beds or coal seams. The harder forms, such as anthracite coal, can be regarded as metamorphic rock because of later exposure to elevated temperature and pressure...
releases much more energy than does detonating a kilogram of
TNT, but because the TNT reaction releases energy much more quickly, it delivers far more power than the coal.
If Δ
W is the amount of
workIn physics, work is a scalar quantity that can be described as the product of a force times the distance through which it acts, and it is called the work of the force. Only the component of a force in the direction of the movement of its point of application does work...
performed during a period of
timeTime is a part of the measuring system used to sequence events, to compare the durations of events and the intervals between them, and to quantify rates of change such as the motions of objects....
of duration Δ
t, 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 Δ
t 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
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
mechanicsMechanics is the branch of physics concerned with the behavior of physical bodies when subjected to forces or displacements, and the subsequent effects of the bodies on their environment....
, the
workIn physics, work is a scalar quantity that can be described as the product of a force times the distance through which it acts, and it is called the work of the force. Only the component of a force in the direction of the movement of its point of application does work...
done by a force
F on an object that travels along a curve
C is given by the
line integralIn mathematics, a line integral is an integral where the function to be integrated is evaluated along a curve.The function to be integrated may be a scalar field or a vector field...
:
-
where
x defines the path
C and
v is the velocity along this path. The time derivative of the equation for work yields the instantaneous power,
In rotational systems, power is the product of the
torqueTorque, moment or moment of force , is the tendency of a force to rotate an object about an axis, fulcrum, or pivot. Just as a force is a push or a pull, a torque can be thought of as a twist....
τ and
angular velocityIn physics, the angular velocity is a vector quantity which specifies the angular speed of an object and the axis about which the object is rotating. The SI unit of angular velocity is radians per second, although it may be measured in other units such as degrees per second, revolutions per...
ω,
where
ω measured in radians per second.
In fluid power systems such as hydraulic actuators, power is given by
where
p is
pressurePressure is the force per unit area applied in a direction perpendicular to the surface of an object. Gauge pressure is the pressure relative to the local atmospheric or ambient pressure.- Definition :...
in
pascalsThe pascal is the SI derived unit of pressure, internal pressure, stress, Young's modulus and tensile strength, named after the French mathematician, physicist, inventor, writer, and philosopher Blaise Pascal. It is a measure of force per unit area, defined as one newton per square metre...
, or N/m
2 and
Q is
volumetric flow rateThe volumetric flow rate in fluid dynamics and hydrometry, is the volume of fluid which passes through a given surface per unit time...
in m
3/s in SI units.
Mechanical advantage
If a mechanical system has no losses then the fact that the input power must equal the output power provides a simple formula for the
mechanical advantageMechanical advantage is a measure of the force amplification achieved by using a tool, mechanical device or machine system. Ideally, the device preserves the input power and simply trades off forces against movement to obtain a desired amplification in the output force...
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 advantageMechanical advantage is a measure of the force amplification achieved by using a tool, mechanical device or machine system. Ideally, the device preserves the input power and simply trades off forces against movement to obtain a desired amplification in the output force...
of the system is given by
-
A similar relationship is obtained for rotating systems, where
TA and
ωB 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 advantageMechanical advantage is a measure of the force amplification achieved by using a tool, mechanical device or machine system. Ideally, the device preserves the input power and simply trades off forces against movement to obtain a desired amplification in the output force...
Electrical power
The instantaneous electrical power
P delivered to a component is given by
where
- P(t) is the instantaneous power, measured in watt
The watt is a derived unit of power in the International System of Units , named after the Scottish engineer James Watt . The unit, defined as one joule per second, measures the rate of energy conversion.-Definition:...
s (jouleThe joule ; symbol J) is a derived unit of energy or work in the International System of Units. It is equal to the energy expended in applying a force of one newton through a distance of one metre , or in passing an electric current of one ampere through a resistance of one ohm for one second...
s per secondThe second is a unit of measurement of time, and is the International System of Units base unit of time. It may be measured using a clock....
)
- V(t) is the potential difference (or voltage drop) across the component, measured in volt
The volt is the SI derived unit for electric potential, electric potential difference, and electromotive force. The volt is named in honor of the Italian physicist Alessandro Volta , who invented the voltaic pile, possibly the first chemical battery.- Definition :A single volt is defined as the...
s
- I(t) is the current
Electric current is a flow of electric charge through a medium.This charge is typically carried by moving electrons in a conductor such as wire...
through it, measured in ampereThe ampere , often shortened to amp, is the SI unit of electric current and is one of the seven SI base units. It is named after André-Marie Ampère , French mathematician and physicist, considered the father of electrodynamics...
s
If the component is a
resistorA linear resistor is a linear, passive two-terminal electrical component that implements electrical resistance as a circuit element.The current through a resistor is in direct proportion to the voltage across the resistor's terminals. Thus, the ratio of the voltage applied across a resistor's...
with time-invariant
voltageVoltage, otherwise known as electrical potential difference or electric tension is the difference in electric potential between two points — or the difference in electric potential energy per unit charge between two points...
to
currentElectric current is a flow of electric charge through a medium.This charge is typically carried by moving electrons in a conductor such as wire...
ratio, then:
where
is the
resistanceThe electrical resistance of an electrical element is the opposition to the passage of an electric current through that element; the inverse quantity is electrical conductance, the ease at which an electric current passes. Electrical resistance shares some conceptual parallels with the mechanical...
, measured in ohms.
Peak power and duty cycle
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.
Power in optics
In
opticsOptics is the branch of physics which involves the behavior and properties of light, including its interactions with matter and the construction of instruments that use or detect it. Optics usually describes the behavior of visible, ultraviolet, and infrared light...
, or
radiometryIn optics, radiometry is a set of techniques for measuring electromagnetic radiation, including visible light. Radiometric techniques characterize the distribution of the radiation's power in space, as opposed to photometric techniques, which characterize the light's interaction with the human eye...
, the term
power sometimes refers to
radiant fluxIn radiometry, radiant flux or radiant power is the measure of the total power of electromagnetic radiation...
, the average rate of energy transport by electromagnetic radiation, measured in
wattThe watt is a derived unit of power in the International System of Units , named after the Scottish engineer James Watt . The unit, defined as one joule per second, measures the rate of energy conversion.-Definition:...
s. The term "power" is also, however, used to express the ability of a
lensA lens is an optical device with perfect or approximate axial symmetry which transmits and refracts light, converging or diverging the beam. A simple lens consists of a single optical element...
or other optical device to
focusIn geometrical optics, a focus, also called an image point, is the point where light rays originating from a point on the object converge. Although the focus is conceptually a point, physically the focus has a spatial extent, called the blur circle. This non-ideal focusing may be caused by...
light. It is measured in
dioptreA dioptre, or diopter, is a unit of measurement of the optical power of a lens or curved mirror, which is equal to the reciprocal of the focal length measured in metres . It is thus a unit of reciprocal length. For example, a 3-dioptre lens brings parallel rays of light to focus at metre...
s (inverse
metreThe metre , symbol m, is the base unit of length in the International System of Units . Originally intended to be one ten-millionth of the distance from the Earth's equator to the North Pole , its definition has been periodically refined to reflect growing knowledge of metrology...
s), and equals the inverse of the
focal lengthThe focal length of an optical system is a measure of how strongly the system converges or diverges light. For an optical system in air, it is the distance over which initially collimated rays are brought to a focus...
of the optical device.
See also
- Motive power
In thermodynamics, motive power is an agency, as water or steam, used to impart motion. Generally, motive power is defined as a natural agent, as water, steam, wind, electricity, etc., used to impart motion to machinery; a motor; a mover. The term may also define something, as a locomotive or a...
- 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, and each section covers three orders of magnitude, or a factor of one thousand.-zeptowatt :...
- Pulsed power
Pulsed power is the term used to describe the science and technology of accumulating energy over a relatively long period of time and releasing it very quickly thus increasing the instantaneous power.-Overview:...
- Intensity
In physics, intensity is a measure of the energy flux, averaged over the period of the wave. The word "intensity" here is not synonymous with "strength", "amplitude", or "level", as it sometimes is in colloquial speech...
— in the radiative sense, power per area
- Power gain
The power gain of an electrical network is the ratio of an output power to an input power. Unlike other signal gains, such as voltage and current gain, "power gain" may be ambiguous as the meaning of terms "input power" and "output power" is not always clear. Three important power gains are...
— for linear, two-port networks.