In the
physics
Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which ...
of
electromagnetism, the Abraham–Lorentz force (also Lorentz–Abraham force) is the
recoil force on an
accelerating charged particle caused by the particle emitting
electromagnetic radiation
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 ...
by self-interaction. It is also called the radiation reaction force, radiation damping force
or the self-force. It is named after the physicists
Max Abraham and
Hendrik Lorentz.
The formula although predating the theory of
special relativity, was initially calculated for non-relativistic velocity approximations was extended to arbitrary velocities by
Max Abraham and was shown to be physically consistent by
George Adolphus Schott. The non-relativistic form is called Lorentz self-force while the relativistic version is called Lorentz-Dirac force or Abraham–Lorentz–Dirac force.
The equations are in the domain of
classical physics, not
quantum physics, and therefore may not be valid at distances of roughly the
Compton wavelength or below.
[Fritz Rohrlich: ''The dynamics of a charged sphere and the electron'', Am. J. Phys. 65 (11) p. 1051 (1997)]
"The dynamics of point charges is an excellent example of the importance of obeying the validity limits of a physical theory. When these limits are exceeded the predictions of the theory may be incorrect or even patently absurd. In the present case, the classical equations of motion have their validity limits where quantum mechanics becomes important: they can no longer be trusted at distances of the order of (or below) the Compton wavelength… Only when all distances involved are in the classical domain is classical dynamics acceptable for electrons." There are, however, two analogs of the formula that are both fully quantum and relativistic: one is called the "Abraham–Lorentz–Dirac–Langevin equation", the other is the self-force on a moving mirror.
The force is proportional to the square of the object's
charge, times the
jerk (rate of change of acceleration) that it is experiencing. The force points in the direction of the jerk. For example, in a
cyclotron, where the jerk points opposite to the velocity, the radiation reaction is directed opposite to the velocity of the particle, providing a braking action. The Abraham–Lorentz force is the source of the
radiation resistance of a radio
antenna
Antenna ( antennas or antennae) may refer to:
Science and engineering
* Antenna (radio), also known as an aerial, a transducer designed to transmit or receive electromagnetic (e.g., TV or radio) waves
* Antennae Galaxies, the name of two collid ...
radiating
radio waves.
There are pathological solutions of the Abraham–Lorentz–Dirac equation in which a particle accelerates ''in advance'' of the application of a force, so-called ''pre-acceleration'' solutions. Since this would represent an effect occurring before its cause (
retrocausality), some theories have speculated that the equation allows signals to travel backward in time, thus challenging the physical principle of
causality. One resolution of this problem was discussed by
Arthur D. Yaghjian[
] and is further discussed by
Fritz Rohrlich
Fritz Rohrlich (May 12, 1921 – November 14, 2018) was an American theoretical physicist and educator who published in the fields of quantum electrodynamics, classical electrodynamics of charged particles, and the philosophy of science.
Life ...
and Rodrigo Medina.
Definition and description
Mathematically, the Lorentz-self force derived for non-relativistic velocity approximation
, is given in
SI units by:
:
or in
Gaussian units by
:
where
is the force,
is the derivative of
acceleration
In mechanics, acceleration is the rate of change of the velocity of an object with respect to time. Accelerations are vector quantities (in that they have magnitude and direction). The orientation of an object's acceleration is given by ...
, or the third derivative of
displacement, also called
jerk, ''μ''
0 is the
magnetic constant, ''ε''
0 is the
electric constant, ''c'' is the
speed of light in
free space, and ''q'' is the
electric charge
Electric charge is the physical property of matter that causes charged matter to experience a force when placed in an electromagnetic field. Electric charge can be ''positive'' or ''negative'' (commonly carried by protons and electrons res ...
of the particle.
Physically, an accelerating charge emits radiation (according to the
Larmor formula
In electrodynamics, the Larmor formula is used to calculate the total power 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.
When any charge ...
), which carries
momentum away from the charge. Since momentum is conserved, the charge is pushed in the direction opposite the direction of the emitted radiation. In fact the formula above for radiation force can be ''derived'' from the Larmor formula, as shown
below
Below may refer to:
*Earth
* Ground (disambiguation)
*Soil
*Floor
* Bottom (disambiguation)
*Less than
*Temperatures below freezing
*Hell or underworld
People with the surname
*Ernst von Below (1863–1955), German World War I general
*Fred Below ...
.
The Abraham-Lorentz force, a generalization of Lorentz self-force for arbitrary velocities is given by:
Where γ is the Lorentz factor associated with v, velocity of particle. The formula is consistent with special relativity and reduces to Lorentz's self-force expression for low velocity limit.
The covariant form of radiation reaction deduced by Dirac for arbitrary shape of elementary charges is found to be: