In
electrodynamics
In physics, electromagnetism is an interaction that occurs between particles with electric charge. It is the second-strongest of the four fundamental interactions, after the strong force, and it is the dominant force in the interactions of a ...
, Poynting's theorem is a statement of
conservation of energy
In physics and chemistry, the law of conservation of energy states that the total energy of an isolated system remains constant; it is said to be ''conserved'' over time. This law, first proposed and tested by Émilie du Châtelet, means th ...
for
electromagnetic fields
An electromagnetic field (also EM field or EMF) is a classical (i.e. non-quantum) field produced by (stationary or moving) electric charges. It is the field described by classical electrodynamics (a classical field theory) and is the classical co ...
developed by British
physicist
A physicist is a scientist who specializes in the field of physics, which encompasses the interactions of matter and energy at all length and time scales in the physical universe.
Physicists generally are interested in the root or ultimate caus ...
John Henry Poynting
John Henry Poynting FRS (9 September 185230 March 1914) was an English physicist. He was the first professor of physics at Mason Science College from 1880 to 1900, and then the successor institution, the University of Birmingham until his deat ...
.
It states that in a given volume, the stored energy changes at a rate given by the
work
Work may refer to:
* Work (human activity), intentional activity people perform to support themselves, others, or the community
** Manual labour, physical work done by humans
** House work, housework, or homemaking
** Working animal, an animal tr ...
done on the
charges
Charge or charged may refer to:
Arts, entertainment, and media Films
* ''Charge, Zero Emissions/Maximum Speed'', a 2011 documentary
Music
* ''Charge'' (David Ford album)
* ''Charge'' (Machel Montano album)
* '' Charge!!'', an album by The Aqu ...
within the volume, minus the rate at which energy leaves the volume. It is only strictly true in media which is not
dispersive, but can be extended for the dispersive case.
The theorem is analogous to the
work-energy theorem
In physics, work is the energy transferred to or from an object via the application of force along a displacement. In its simplest form, for a constant force aligned with the direction of motion, the work equals the product of the force stren ...
in
classical mechanics
Classical mechanics is a physical theory describing the motion of macroscopic objects, from projectiles to parts of machinery, and astronomical objects, such as spacecraft, planets, stars, and galaxies. For objects governed by classical ...
, and mathematically similar to the
continuity equation
A continuity equation or transport equation is an equation that describes the transport of some quantity. It is particularly simple and powerful when applied to a conserved quantity, but it can be generalized to apply to any extensive quantity. S ...
.
Definition
Poynting's theorem states that the rate of energy transfer per unit volume from a region of space equals the rate of
work
Work may refer to:
* Work (human activity), intentional activity people perform to support themselves, others, or the community
** Manual labour, physical work done by humans
** House work, housework, or homemaking
** Working animal, an animal tr ...
done on the charge distribution in the region, plus the
energy flux Energy flux is the rate of transfer of energy through a surface. The quantity is defined in two different ways, depending on the context:
# Total rate of energy transfer (not per unit area); SI units: W = J⋅s−1.
# Specific rate of energy transfe ...
leaving that region.
Mathematically:
where:
*
is the rate of change of the energy density in the volume.
* ∇•S is the energy flow out of the volume, given by the
divergence
In vector calculus, divergence is a vector operator that operates on a vector field, producing a scalar field giving the quantity of the vector field's source at each point. More technically, the divergence represents the volume density of the ...
of the
Poynting vector
In physics, the Poynting vector (or Umov–Poynting vector) represents the directional energy flux (the energy transfer per unit area per unit time) or '' power flow'' of an electromagnetic field. The SI unit of the Poynting vector is the watt ...
S.
* J•E is the rate at which the fields do work on charges in the volume (J is the
current density
In electromagnetism, current density is the amount of charge per unit time that flows through a unit area of a chosen cross section. The current density vector is defined as a vector whose magnitude is the electric current per cross-sectional ar ...
corresponding to the motion of charge, E is the
electric field
An electric field (sometimes E-field) is the physical field that surrounds electrically charged particles and exerts force on all other charged particles in the field, either attracting or repelling them. It also refers to the physical field fo ...
, and • is the
dot product
In mathematics, the dot product or scalar productThe term ''scalar product'' means literally "product with a scalar as a result". It is also used sometimes for other symmetric bilinear forms, for example in a pseudo-Euclidean space. is an algebra ...
).
Integral Form
Using the
divergence theorem
In vector calculus, the divergence theorem, also known as Gauss's theorem or Ostrogradsky's theorem, reprinted in is a theorem which relates the ''flux'' of a vector field through a closed surface to the ''divergence'' of the field in the vol ...
, Poynting's theorem can also be written in integral form:
where
*S is the energy flow, given by the Poynting Vector.
*
is the energy density in the volume.
*
is the boundary of the volume. The shape of the volume is arbitrary but fixed for the calculation.
Continuity Equation Analog
In an
electrical engineering
Electrical engineering is an engineering discipline concerned with the study, design, and application of equipment, devices, and systems which use electricity, electronics, and electromagnetism. It emerged as an identifiable occupation in the l ...
context the theorem is sometimes written with the energy density term ''u'' expanded as shown. This form resembles the
continuity equation
A continuity equation or transport equation is an equation that describes the transport of some quantity. It is particularly simple and powerful when applied to a conserved quantity, but it can be generalized to apply to any extensive quantity. S ...
:
:
where
*''ε''
0 is the
vacuum permittivity
Vacuum permittivity, commonly denoted (pronounced "epsilon nought" or "epsilon zero"), is the value of the absolute dielectric permittivity of classical vacuum. It may also be referred to as the permittivity of free space, the electric consta ...
and ''μ''
0 is the
vacuum permeability
The vacuum magnetic permeability (variously ''vacuum permeability'', ''permeability of free space'', ''permeability of vacuum''), also known as the magnetic constant, is the magnetic permeability in a classical vacuum. It is a physical constant, ...
.
*
is the density of
reactive power
Reactive may refer to:
*Generally, capable of having a reaction (disambiguation)
*An adjective abbreviation denoting a bowling ball coverstock made of reactive resin
*Reactivity (chemistry)
*Reactive mind
*Reactive programming
See also
*Reactanc ...
driving the build-up of electric field,
*
is the density of
reactive power
Reactive may refer to:
*Generally, capable of having a reaction (disambiguation)
*An adjective abbreviation denoting a bowling ball coverstock made of reactive resin
*Reactivity (chemistry)
*Reactive mind
*Reactive programming
See also
*Reactanc ...
driving the build-up of magnetic field, and
*
is the density of
electric power
Electric power is the rate at which electrical energy is transferred by an electric circuit. The SI unit of power is the watt, one joule per second. Standard prefixes apply to watts as with other SI units: thousands, millions and billions o ...
dissipated by the
Lorentz force
In physics (specifically in electromagnetism) the Lorentz force (or electromagnetic force) is the combination of electric and magnetic force on a point charge due to electromagnetic fields. A particle of charge moving with a velocity in an elect ...
acting on charge carriers.
Derivation
For an individual charge in an electromagnetic field, the rate of work done by the field on the charge is given by the
Lorentz Force Law
Lorentz is a name derived from the Roman surname, Laurentius, which means "from Laurentum". It is the German form of Laurence. Notable people with the name include:
Given name
* Lorentz Aspen (born 1978), Norwegian heavy metal pianist and keyboar ...
as:
Extending this to a continuous distribution of charges, moving with current density J, gives:
By
Ampère's circuital law:
(Note that the H and D forms of the magnetic and electric fields are used here. The B and E forms could also be used in an equivalent derivation.)
Substituting this into the expression for rate of work gives:
Using the
vector identity
The following are important identities involving derivatives and integrals in vector calculus.
Operator notation
Gradient
For a function f(x, y, z) in three-dimensional Cartesian coordinate variables, the gradient is the vector field:
\o ...
:
By
Faraday's Law:
giving:
Continuing the derivation requires the following assumptions:
* the charges are moving in a medium which is not
dispersive.
* the total electromagnetic energy density, even for time-varying fields, is given by
It can be shown that:
and
and so:
Returning to the equation for rate of work,
Since the volume is arbitrary, this can be cast in differential form as:
where
is the Poynting vector.
Poynting vector in macroscopic media
In a macroscopic medium, electromagnetic effects are described by spatially averaged (macroscopic) fields. The Poynting vector in a macroscopic medium can be defined self-consistently with microscopic theory, in such a way that the spatially averaged microscopic Poynting vector is exactly predicted by a macroscopic formalism. This result is strictly valid in the limit of low-loss and allows for the unambiguous identification of the Poynting vector form in macroscopic electrodynamics.
Alternative forms
It is possible to derive alternative versions of Poynting's theorem.
[
] Instead of the flux vector as above, it is possible to follow the same style of derivation, but instead choose , the
Minkowski form , or perhaps . Each choice represents the response of the propagation medium in its own way: the form above has the property that the response happens only due to electric currents, while the form uses only (fictitious)
magnetic monopole
In particle physics, a magnetic monopole is a hypothetical elementary particle that is an isolated magnet with only one magnetic pole (a north pole without a south pole or vice versa). A magnetic monopole would have a net north or south "magneti ...
currents. The other two forms (Abraham and Minkowski) use complementary combinations of electric and magnetic currents to represent the polarization and magnetization responses of the medium.
[
]
Modification
The derivation of the statement is dependent on the assumption that the materials the equation models can be described by a set of susceptibility properties that are linear
Linearity is the property of a mathematical relationship (''function'') that can be graphically represented as a straight line. Linearity is closely related to '' proportionality''. Examples in physics include rectilinear motion, the linear r ...
, isotropic
Isotropy is uniformity in all orientations; it is derived . Precise definitions depend on the subject area. Exceptions, or inequalities, are frequently indicated by the prefix ' or ', hence ''anisotropy''. ''Anisotropy'' is also used to describe ...
, homogenous and independent of frequency
Frequency is the number of occurrences of a repeating event per unit of time. It is also occasionally referred to as ''temporal frequency'' for clarity, and is distinct from ''angular frequency''. Frequency is measured in hertz (Hz) which is eq ...
. The assumption that the materials have no absorption must also be made. A modification to Poynting's theorem to account for variations includes a term for the rate of non-Ohmic absorption
Absorption may refer to:
Chemistry and biology
* Absorption (biology), digestion
**Absorption (small intestine)
*Absorption (chemistry), diffusion of particles of gas or liquid into liquid or solid materials
*Absorption (skin), a route by which ...
in a material, which can be calculated by a simplified approximation based on the Drude model
The Drude model of electrical conduction was proposed in 1900 by Paul Drude to explain the transport properties of electrons in materials (especially metals). Basically, Ohm's law was well established and stated that the current ''J'' and voltage ...
.[
]
Complex Poynting vector theorem
This form of the theorem is useful in Antenna theory, where one has often to consider harmonic fields propagating in the space.
In this case, using phasor notation, and .
Then the following mathematical identity holds:
:
where is the current density.
Note that in free space, and are real, thus,
taking the real part of the above formula, it expresses the fact that the averaged radiated power flowing through is equal to the work on the charges.
References
External links
Eric W. Weisstein "Poynting Theorem" From ScienceWorld – A Wolfram Web Resource.
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Electrodynamics
Physics theorems
Circuit theorems