- Larmor formula
In
physics , in the area ofelectrodynamics , 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 thewave theory of light .When accelerating or decelerating, any charged particle (such as an
electron ) radiates away energy in the form ofelectromagnetic wave s. For velocities that are small relative to thespeed of light , the total power radiated is given by the Larmor formula::P = frac{e^2 a^2}{6 pi varepsilon_0 c^3} mbox{ (SI units)}
:P = {2 over 3} frac{e^2 a^2}{ c^3} mbox{ (cgs units)}
where a is the acceleration, e is the charge, and c is the speed of light. A relativistic generalization is given by the
Liénard-Wiechert Potentials .Derivation
Derivation 1: Fields of a moving charge
Retarded potential solutions
In the case that there are no boundaries surrounding the sources, the retarded solutions for the scalar and vector potentials (cgs units) of the nonhomogeneous wave equations are (see
Nonhomogeneous electromagnetic wave equation ):varphi (mathbf{r}, t) = int { { delta left ( t' + { { left | mathbf{r} - mathbf{r}' ight | } over c } - t ight ) } over { { left | mathbf{r} - mathbf{r}' ight | } } } ho (mathbf{r}', t') d^3r' dt'
and
:mathbf{A} (mathbf{r}, t) = int { { delta left ( t' + { { left | mathbf{r} - mathbf{r}' ight | } over c } - t ight ) } over { { left | mathbf{r} - mathbf{r}' ight | } } } { mathbf{J} (mathbf{r}', t')over c} d^3r' dt'
where
:delta left ( t' + { { left | mathbf{r} - mathbf{r}' ight | } over c } - t ight ) }
is a
Dirac delta function and the current and charge densities are:mathbf{J} (mathbf{r}', t') = e mathbf{v}_0(t') delta left ( mathbf{r}' - mathbf{r}_0(t') ight )
:ho (mathbf{r}', t') = e delta left ( mathbf{r}' - mathbf{r}_0 (t') ight )
for a particle at mathbf{r}_0(t') traveling with velocity mathbf{v}_0(t') .
Electric and magnetic fields
The scalar and vector potentials are related to the electric and magnetic fields by
:mathbf{E} = - abla varphi - {1 over c} {partial mathbf{A} over partial t}
:mathbf{B} = abla imes mathbf{A} .
The fields can be written:mathbf{E}(mathbf{r}, t) =
e left [ { { left ( mathbf{n} - { mathbf{v}_0 over c } ight ) left ( 1-eta^2 ight ) } over { kappa^3 R^2 } } ight ] _{mbox{ret} }
+ {e over c} left [ { { mathbf{n} imes left ( mathbf{n} - { mathbf{v}_0 over c } ight ) imes { mathbf{a} over c } } over { kappa^3 R } } ight ] _{mbox{ret} }
:mathbf{B}(mathbf{r}, t) = mathbf{n} imes mathbf{E}(mathbf{r}, t)
where
:mathbf{a} is the acceleration,
:mathbf{n} is a unit vector in the mathbf{r} - mathbf{r}_0 direction,
:R is the magnitude of mathbf{r} - mathbf{r}_0 ,
:kappa stackrel{mathrm{def{=} 1 - mathbf{n} cdot { mathbf{v}_0 over c }
:eta^2 stackrel{mathrm{def{=} {v_0^2 over c^2 }
and the terms on the right are evaluated at the retarded time
:t' = t - {R over c} .
The second term, proportional to the acceleration, represents a spherically moving light wave. The first term falls of as the square of the distance and represents a wave that decays with distance.
Derivation 2: Using Edward M. Purcell approach
The full derivation can be found in [http://physics.weber.edu/schroeder/mrr/MRRtalk.html http://physics.weber.edu/schroeder/mrr/MRRtalk.html]
Here is an explanation which can help understanding the above page.
This approach is based on the finite speed of light. A charge moving withconstant velocity has a radial electric field E_r(at distance Rfrom the charge), always emerging from the future position of the charge,and there is no tangential component of the electric field E_t=0).This future position is completely deterministic as long as the velocityis constant. When the velocity of the charge changes, (say it bounces backduring a short time) the future position "jumps", so from this moment andon, the radial electric field E_r emerges from a newposition. Given the fact that the electric field must be continuous, anon-zero tangential component of the electric field E_t appears,which decreases like 1/R (unlike the radial component whichdecreases like 1/R^2).
Hence, at large distances from the charge, the radial component is negligiblerelative to the tangential component, and in addition to that, fields whichbehave like 1/R^2 cannot radiate, because the Poynting vectorassociated with them will behave like 1/R^4.
The tangential component comes out (SI units):
:E_t = e a sin( heta)} over {4 pi varepsilon_0 c^2 R .
And to obtain the Larmour formula, one has to integrate over all angles, atlarge distance R from the charge, the
Poynting vector associated with E_t, which is::mathbf{S} = E_t^2 over eta_0mathbf{hat{r = e^2 a^2 sin^2( heta)} over {16 pi^2 varepsilon_0 c^3 R^2 mathbf{hat{r
giving (SI units)
:P = e^2 a^2} over {6 pi varepsilon_0 c^3
Issues and implications
Energy flux
The energy flux from the electromagnetic wave is given by the
Poynting vector (cgs units).:mathbf{S} = { c over {4 pi } } {mathbf{E} imes mathbf{B} }
Integration of the power over the surface of a sphere centered on the emitting particle yields the Larmor power formula in the nonrelativistic limit.
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 theAbraham-Lorentz-Dirac force in the relativistic limit.Atomic physics
A classical electron 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 .ee also
*
Atomic theory
*Cyclotron radiation
*Electromagnetic wave equation
*Maxwell's equations in curved spacetime
*Radiation reaction
*Wave equation 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)."
*cite book |author=Jackson, John D.|title=Classical Electrodynamics (3rd ed.)|publisher=Wiley|year=1998|id=ISBN 0-471-30932-X
*
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