Larmor formula

In electrodynamics, the Larmor formula is used to calculate the total power radiated by a non relativistic 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 charged particle (such as an electron, a proton, or an ion) accelerates, it radiates away energy in the form of electromagnetic waves. For velocities that are small relative to the speed of light, the total power radiated is given by the Larmor formula:
where formula_3 or formula_4 is the proper acceleration, formula_5 is the charge, and formula_6 is the speed of light. A relativistic generalization is given by the Liénard–Wiechert potentials.
In either unit system, the power radiated by a single electron can be expressed in terms of the classical electron radius and electron mass as:
One implication is that an electron orbiting around a nucleus, as in the Bohr model, should lose energy, fall to the nucleus and the atom should collapse. This puzzle was not solved until quantum theory 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)
and
where formula_10 is the charge's velocity divided by formula_11, formula_12 is the charge's acceleration divided by "c", formula_13 is a unit vector in the formula_14 direction, formula_15 is the magnitude of formula_16, formula_17 is the charge's location, and formula_18. The terms on the right are evaluated at the retarded time formula_19.
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 formula_10 while the acceleration field depends on both formula_10 and formula_12 and the angular relationship between the two. Since the velocity field is proportional to formula_23, it falls off very quickly with distance. On the other hand, the acceleration field is proportional to formula_24, which means that it falls much 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 flux density of the radiation field by computing its Poynting vector:
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 formula_26 and simplifying gives
If we let the angle between the acceleration and the observation vector be equal to formula_28, and we introduce the acceleration formula_29, then the power radiated per unit solid angle is
The total power radiated is found by integrating this quantity over all solid angles (that is, over formula_28 and formula_32). This gives
which is the Larmor result for a non-relativistic 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 formula_34
(at distance formula_15
from the charge), always emerging from the future position of the charge,
and there is no tangential component of the electric field formula_36.
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 formula_34 emerges from a new
position. Given the fact that the electric field must be continuous, a
non-zero tangential component of the electric field formula_38 appears,
which decreases like formula_24 (unlike the radial component which
decreases like formula_23).
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 formula_23 cannot radiate, because the Poynting vector
associated with them will behave like formula_42.
The tangential component comes out (SI units):
And to obtain the Larmour formula, one has to integrate over all angles, at
large distance formula_15 from the charge, the
Poynting vector associated with formula_38, which is:
giving (SI units)
This is mathematically equivalent to:
Since formula_49, we recover the result quoted at the top of the article, namely
Relativistic generalization.
Covariant form.
Written in terms of momentum, , the non-relativistic Larmor formula is (in CGS units)
The power can be shown to be Lorentz invariant. Any relativistic generalization of the Larmor formula must therefore relate to some other Lorentz invariant quantity. The quantity formula_52 appearing in the non-relativistic formula suggests that the relativistically correct formula should include the Lorentz scalar found by taking the inner product of the four-acceleration with itself [here 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 formula_54, 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 formula_55 means that when the Lorentz factor formula_56 is very close to one (i.e. formula_57) the radiation emitted by the particle is likely to be negligible. However, as formula_58 the radiation grows like formula_55 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 formula_60, i.e. the factor formula_55 becomes formula_62. 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 formula_64 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 formula_28 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 the Abraham–Lorentz force in the nonrelativistic limit and the Abraham–Lorentz–Dirac force in the relativistic setting.
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 description of atomic physics, 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.