Larmor formula
In electrodynamics, the Larmor formula is used to calculate the total power radiated by a nonrelativistic point charge as it accelerates.
When any charged particle (such as an electron, a proton, or an ion) accelerates, energy is radiated in the form of electromagnetic waves.
For a particle whose velocity is small relative to the speed of light (i.e., nonrelativistic), the total power that the particle radiates (when considered as a point charge) can be calculated by the Larmor formula:
we integrate the Poynting vector over the surface of a sphere of radius R, to get:[3]
We make a Lorentz transformation to the rest frame of the point charge where
We integrate the rest frame Poynting vector over the surface of a sphere of radius R', to get.
The radiated power can be put back in terms of the original acceleration in the moving frame, to give
The variables in this equation are in the original moving frame, but the rate of energy emission on the left hand side of the equation is still given in terms of the rest frame variables.
This result (in two forms) is the same as Liénard's relativistic extension[4] of Larmor's formula, and is given here with all variables at the present time.
Its nonrelativistic reduction reduces to Larmor's original formula.
For high-energies, it appears that the power radiated for acceleration parallel to the velocity is a factor
However, writing the Liénard formula in terms of the velocity gives a misleading implication.
The radiated power is actually a Lorentz scalar, given in covariant form as
To show this, we reduce the four-vector scalar product to vector notation.
With this expression for the scalar product, the manifestly invariant form for the power agrees with the vector form above, demonstrating that the radiated power is a Lorentz scalar The angular distribution of radiated power is given by a general formula, applicable whether or not the particle is relativistic.
In the case of linear motion (velocity parallel to acceleration), this simplifies to[6]
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.
The radiation reaction phenomenon is one of the key problems and consequences of the Larmor formula.
According to classical electrodynamics, a charged particle produces electromagnetic radiation as it accelerates.
The particle loses momentum and energy as a result of the radiation, which is carrying it away from it.
The dynamics of charged particles are significantly impacted by the existence of this force.
In particular, it causes a change in their motion that may be accounted for by the Larmor formula, a factor in the Lorentz-Dirac equation.
According to the Lorentz-Dirac equation, a charged particle's velocity will be influenced by a "self-force" resulting from its own radiation.
Such non-physical behavior as runaway solutions, when the particle's velocity or energy become infinite in a finite amount of time, might result from this self-force.
[7] The invention of quantum physics, notably the Bohr model of the atom, was able to explain this gap between the classical prediction and the actual reality.
The Bohr model proposed that transitions between distinct energy levels, which electrons could only inhabit, might account for the observed spectral lines of atoms.
The Liénard-Wiechert potential is a more comprehensive formula that must be employed for particles travelling at relativistic speeds.
In certain situations, more intricate calculations including numerical techniques or perturbation theory could be necessary to precisely compute the radiation the charged particle emits.
