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 ...
, the Larmor formula is used to calculate the total
power
Power most often refers to:
* Power (physics), meaning "rate of doing work"
** Engine power, the power put out by an engine
** Electric power
* Power (social and political), the ability to influence people or events
** Abusive power
Power may a ...
radiated by a nonrelativistic point charge as it accelerates. It was first derived by
J. J. Larmor in 1897, in the context of the
wave theory of light
In physics, physical optics, or wave optics, is the branch of optics that studies Interference (wave propagation), interference, diffraction, Polarization (waves), polarization, and other phenomena for which the ray approximation of geometric opti ...
.
When any charged particle (such as an
electron
The electron ( or ) is a subatomic particle with a negative one elementary electric charge. Electrons belong to the first generation of the lepton particle family,
and are generally thought to be elementary particles because they have no kn ...
, a
proton
A proton is a stable subatomic particle, symbol , H+, or 1H+ with a positive electric charge of +1 ''e'' elementary charge. Its mass is slightly less than that of a neutron and 1,836 times the mass of an electron (the proton–electron mass ...
, or an
ion
An ion () is an atom or molecule with a net electrical charge.
The charge of an electron is considered to be negative by convention and this charge is equal and opposite to the charge of a proton, which is considered to be positive by conve ...
) accelerates, energy is radiated in the form of
electromagnetic wave
In physics, electromagnetic radiation (EMR) consists of waves of the electromagnetic (EM) field, which propagate through space and carry momentum and electromagnetic radiant energy. It includes radio waves, microwaves, infrared, (visib ...
s. For a particle whose velocity is small relative to the
speed of light
The speed of light in vacuum, commonly denoted , is a universal physical constant that is important in many areas of physics. The speed of light is exactly equal to ). According to the special theory of relativity, is the upper limit ...
(i.e., nonrelativistic), the total power that the particle radiates (when considered as a point charge) can be calculated by the Larmor formula:
where
or
— is the proper acceleration,
— is the charge, and
— is the speed of light. A relativistic generalization is given by the
Liénard–Wiechert potential
The Liénard–Wiechert potentials describe the classical electromagnetic effect of a moving electric point charge in terms of a vector potential and a scalar potential in the Lorenz gauge. Stemming directly from Maxwell's equations, these descr ...
s.
In either unit system, the power radiated by a single electron can be expressed in terms of the
classical electron radius
The classical electron radius is a combination of fundamental physical quantities that define a length scale for problems involving an electron interacting with electromagnetic radiation. It links the classical electrostatic self-interaction energ ...
and
electron mass
The electron mass (symbol: ''m''e) is the mass of a stationary electron, also known as the invariant mass of the electron. It is one of the fundamental constants of physics. It has a value of about or about , which has an energy-equivalent of ...
as:
One implication is that an electron orbiting around a nucleus, as in the
Bohr model
In atomic physics, the Bohr model or Rutherford–Bohr model, presented by Niels Bohr and Ernest Rutherford in 1913, is a system consisting of a small, dense nucleus surrounded by orbiting electrons—similar to the structure of the Solar Syst ...
, should lose energy, fall to the nucleus and the atom should collapse. This puzzle was not solved until
quantum theory
Quantum theory may refer to:
Science
*Quantum mechanics, a major field of physics
*Old quantum theory, predating modern quantum mechanics
* Quantum field theory, an area of quantum mechanics that includes:
** Quantum electrodynamics
** Quantum ...
was introduced.
Derivation
Derivation 1: Mathematical approach (using CGS units)
We first need to find the form of the electric and magnetic fields. The fields can be written (for a fuller derivation see
Liénard–Wiechert potential
The Liénard–Wiechert potentials describe the classical electromagnetic effect of a moving electric point charge in terms of a vector potential and a scalar potential in the Lorenz gauge. Stemming directly from Maxwell's equations, these descr ...
)
and
where
is the charge's velocity divided by
,
is the charge's acceleration divided by ,
is a unit vector in the
direction,
is the magnitude of
,
is the charge's location, and
. The terms on the right are evaluated at the
retarded time
In electromagnetism, electromagnetic waves in vacuum travel at the speed of light ''c'', according to Maxwell's Equations. The retarded time is the time when the field began to propagate from the point where it was emitted to an observer. The term ...
.
The right-hand side is the sum of the electric fields associated with the velocity and the acceleration of the charged particle. The velocity field depends only upon
while the acceleration field depends on both
and
and the angular relationship between the two. Since the velocity field is proportional to
, it falls off very quickly with distance. On the other hand, the acceleration field is proportional to
, which means that it falls off more slowly with distance. Because of this, the acceleration field is representative of the radiation field and is responsible for carrying most of the energy away from the charge.
We can find the
energy
In physics, energy (from Ancient Greek: ἐνέργεια, ''enérgeia'', “activity”) is the quantitative property that is transferred to a body or to a physical system, recognizable in the performance of work and in the form of heat a ...
flux
Flux describes any effect that appears to pass or travel (whether it actually moves or not) through a surface or substance. Flux is a concept in applied mathematics and vector calculus which has many applications to physics. For transport ph ...
density of the radiation field by computing its
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 ...
:
where the 'a' subscripts emphasize that we are taking only the acceleration field. Substituting in the relation between the magnetic and electric fields while assuming that the particle instantaneously at rest at time
and simplifying gives
[The case where is more complicated and is treated, for example, in Griffiths's ''Introduction to Electrodynamics''.]
If we let the angle between the acceleration and the observation vector be equal to
, and we introduce the acceleration
, then the power radiated per unit
solid angle
In geometry, a solid angle (symbol: ) is a measure of the amount of the field of view from some particular point that a given object covers. That is, it is a measure of how large the object appears to an observer looking from that point.
The poi ...
is
The total power radiated is found by integrating this quantity over all solid angles (that is, over
and
). This gives
which is the Larmor result for a nonrelativistic accelerated charge. It relates the power radiated by the particle to its acceleration. It clearly shows that the faster the charge accelerates the greater the radiation will be. We would expect this since the radiation field is dependent upon acceleration.
Derivation 2: Edward M. Purcell approach
The full derivation can be found here.
Here is an explanation which can help understanding the above page.
This approach is based on the finite speed of light. A charge moving with constant velocity has a radial electric field
(at distance
from the charge), always emerging from the future position of the charge, and there is no tangential component of the electric field
. This future position is completely deterministic as long as the velocity is constant. When the velocity of the charge changes, (say it bounces back during a short time) the future position "jumps", so from this moment and on, the radial electric field
emerges from a new position. Given the fact that the electric field must be continuous, a non-zero tangential component of the electric field
appears, which decreases like
(unlike the radial component which decreases like
).
Hence, at large distances from the charge, the radial component is negligible relative to the tangential component, and in addition to that, fields which behave like
cannot radiate, because the Poynting vector associated with them will behave like
.
The tangential component comes out (SI units):
And to obtain the Larmour formula, one has to integrate over all angles, at large distance
from the charge, 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 ...
associated with
, which is:
giving (SI units)
This is mathematically equivalent to:
Since
, we recover the result quoted at the top of the article, namely
Relativistic generalization
Covariant form
Written in terms of momentum, , the nonrelativistic Larmor formula is (in CGS units)
The power can be shown to be
Lorentz invariant
In a relativistic theory of physics, a Lorentz scalar is an expression, formed from items of the theory, which evaluates to a scalar, invariant under any Lorentz transformation. A Lorentz scalar may be generated from e.g., the scalar product of v ...
.
Any relativistic generalization of the Larmor formula must therefore relate to some other Lorentz invariant quantity. The quantity
appearing in the nonrelativistic formula suggests that the relativistically correct formula should include the Lorentz scalar found by taking the inner product of the
four-acceleration In the theory of relativity, four-acceleration is a four-vector (vector in four-dimensional spacetime) that is analogous to classical acceleration (a three-dimensional vector, see three-acceleration in special relativity). Four-acceleration has ap ...
with itself
ere__is_the_four-momentum.html" ;"title="four-momentum.html" ;"title="ere is the four-momentum">ere is the four-momentum">four-momentum.html" ;"title="ere is the four-momentum">ere is the four-momentum The correct relativistic generalization of the Larmor formula is (in CGS units)
It can be shown that this inner product is given by
and so in the limit , it reduces to
, thus reproducing the nonrelativistic case.
Non-covariant form
The above inner product can also be written in terms of and its time derivative. Then the relativistic generalization of the Larmor formula is (in CGS units)
This is the Liénard result, which was first obtained in 1898. The
means that when the Lorentz factor
is very close to one (i.e.
) the radiation emitted by the particle is likely to be negligible. However, as
the radiation grows like
as the particle tries to lose its energy in the form of EM waves. Also, when the acceleration and velocity are orthogonal the power is reduced by a factor of
, i.e. the factor
becomes
. The faster the motion becomes the greater this reduction gets.
We can use Liénard's result to predict what sort of radiation losses to expect in different kinds of motion.
Angular distribution
The angular distribution of radiated power is given by a general formula, applicable whether or not the particle is relativistic. In CGS units, this formula is
where
is a unit vector pointing from the particle towards the observer. In the case of linear motion (velocity parallel to acceleration), this simplifies to
where
is the angle between the observer and the particle's motion.
Issues and implications
Radiation reaction
The radiation from a charged particle carries energy and momentum. In order to satisfy energy and momentum conservation, the charged particle must experience a recoil at the time of emission. The radiation must exert an additional force on the charged particle. This force is known as
Abraham-Lorentz force while its non-relativistic limit is known as the Lorentz self-force and relativistic forms are known as Lorentz-Dirac force or
Abraham–Lorentz–Dirac force.
Atomic physics
A classical electron in the Bohr model orbiting a nucleus experiences acceleration and should radiate. Consequently, the electron loses energy and the electron should eventually spiral into the nucleus. Atoms, according to classical mechanics, are consequently unstable. This classical prediction is violated by the observation of stable electron orbits. The problem is resolved with a
quantum mechanical
Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, qua ...
description of
atomic physics
Atomic physics is the field of physics that studies atoms as an isolated system of electrons and an atomic nucleus. Atomic physics typically refers to the study of atomic structure and the interaction between atoms. It is primarily concerned wit ...
, initially provided by the Bohr model. Classical solutions to the stability of electron orbitals can be demonstrated using
non-radiation conditions and in accordance with known physical laws.
See also
*
Atomic theory
Atomic theory is the scientific theory that matter is composed of particles called atoms. Atomic theory traces its origins to an ancient philosophical tradition known as atomism. According to this idea, if one were to take a lump of matter a ...
*
Cyclotron radiation Cyclotron radiation is electromagnetic radiation emitted by non-relativistic accelerating charged particles deflected by a magnetic field. The Lorentz force on the particles acts perpendicular to both the magnetic field lines and the particles' mot ...
*
Electromagnetic wave equation
The electromagnetic wave equation is a second-order partial differential equation that describes the propagation of electromagnetic waves through a medium or in a vacuum. It is a three-dimensional form of the wave equation. The homogeneous form ...
*
Maxwell's equations in curved spacetime
In physics, Maxwell's equations in curved spacetime govern the dynamics of the electromagnetic field in curved spacetime (where the metric may not be the Minkowski metric) or where one uses an arbitrary (not necessarily Cartesian) coordinate ...
*
Radiation reaction
In physics, radiation is the emission or transmission of energy in the form of waves or particles through space or through a material medium. This includes:
* ''electromagnetic radiation'', such as radio waves, microwaves, infrared, visi ...
*
Wave equation
The (two-way) wave equation is a second-order linear partial differential equation for the description of waves or standing wave fields — as they occur in classical physics — such as mechanical waves (e.g. water waves, sound waves and s ...
*
Wheeler–Feynman absorber theory
The Wheeler–Feynman absorber theory (also called the Wheeler–Feynman time-symmetric theory), named after its originators, the physicists Richard Feynman and John Archibald Wheeler, is an interpretation of electrodynamics derived from the assump ...
Notes
References
* J. Larmor, "On a dynamical theory of the electric and luminiferous medium", ''Philosophical Transactions of the Royal Society'' 190, (1897) pp. 205–300 ''(Third and last in a series of papers with the same name).''
* (Section 14.2ff)
*
*
{{DEFAULTSORT:Larmor Formula
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