Larmor formula

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A Yagi-Uda antenna. Radio waves can be radiated from an antenna by accelerating electrons in the antenna.

In physics, in the area of 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 accelerating or decelerating, any charged particle (such as an electron) 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:

$P = \frac{e^2 a^2}{6 \pi \epsilon_0 c^3} \mbox{ (SI units)}$

$P = {2 \over 3} \frac{e^2 a^2}{ c^3} \mbox{ (cgs units)}$

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.

1 Derivation
- 1.1 Derivation 1: Fields of a moving charge
  - 1.1.1 Retarded potential solutions
  - 1.1.2 Electric and magnetic fields
- 1.2 Derivation 2: Using Edward M. Purcell approach
2 Issues and implications
3 See also
4 References

[edit] Derivation

[edit] Derivation 1: Fields of a moving charge

M87's Energetic Jet. The glow is caused by synchrotron radiation, high-energy electrons spiraling along magnetic field lines, and was first detected in 1956 by Geoffrey R. Burbidge in M87 confirming a prediction by Hannes Alfvén and Nicolai Herlofson in 1950, and Iosif S. Shklovskii in 1953.

[edit] 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)

$\varphi (\mathbf{r}, t) = \int { { \delta \left ( t' + { { \left | \mathbf{r} - \mathbf{r}' \right | } \over c } - t \right ) } \over { { \left | \mathbf{r} - \mathbf{r}' \right | } } } \rho (\mathbf{r}', t') d^3r' dt'$

and

$\mathbf{A} (\mathbf{r}, t) = \int { { \delta \left ( t' + { { \left | \mathbf{r} - \mathbf{r}' \right | } \over c } - t \right ) } \over { { \left | \mathbf{r} - \mathbf{r}' \right | } } } { \mathbf{J} (\mathbf{r}', t')\over c} d^3r' dt'$

where

${ \delta \left ( t' + { { \left | \mathbf{r} - \mathbf{r}' \right | } \over c } - t \right ) }$

is a Dirac delta function and the current and charge densities are

$\mathbf{J} (\mathbf{r}', t') = e \mathbf{v}_0(t') \delta \left ( \mathbf{r}' - \mathbf{r}_0(t') \right )$

$\rho (\mathbf{r}', t') = e \delta \left ( \mathbf{r}' - \mathbf{r}_0 (t') \right )$

for a particle at $\mathbf{r}_0(t')$ traveling with velocity $\mathbf{v}_0(t')$ .

[edit] Electric and magnetic fields

The scalar and vector potentials are related to the electric and magnetic fields by

$\mathbf{E} = - \nabla \varphi - {1 \over c} {\partial \mathbf{A} \over \partial t}$

$\mathbf{B} = \nabla \times \mathbf{A}$ .

The fields can be written

$\mathbf{E}(\mathbf{r}, t) = e \left [ { { \left ( \mathbf{n} - { \mathbf{v}_0 \over c } \right ) \left ( 1-\beta^2 \right ) } \over { \kappa^3 R^2 } } \right ]_{\mbox{ret} } + {e \over c} \left [ { { \mathbf{n} \times \left ( \mathbf{n} - { \mathbf{v}_0 \over c } \right ) \times { \mathbf{a} \over c } } \over { \kappa^3 R } } \right ]_{\mbox{ret} }$

$\mathbf{B}(\mathbf{r}, t) = \mathbf{n} \times \mathbf{E}(\mathbf{r}, t)$

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 $\mathbf{r} - \mathbf{r}_0$ ,

$\kappa \ \stackrel{\mathrm{def}}{=}\ 1 - \mathbf{n} \cdot { \mathbf{v}_0 \over c }$

$\beta^2 \ \stackrel{\mathrm{def}}{=}\ {v_0^2 \over c^2 }$

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.

[edit] Derivation 2: Using Edward M. Purcell approach

The full derivation can be found in 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 with constant velocity has a radial electric field $E r$ (at distance $R$ from 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 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 $E r$ emerges from a new position. Given the fact that the electric field must be continuous, a non-zero tangential component of the electric field $E t$ appears, which decreases like $1 / R$ (unlike the radial component which decreases like $1 / R 2$ ).

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 $1 / R 2$ cannot radiate, because the Poynting vector associated with them will behave like $1 / R 4$ .

The tangential component comes out (SI units):

$E_t = {{e a \sin(\theta)} \over {4 \pi \epsilon_0 c^2 R}}$ .

And to obtain the Larmour formula, one has to integrate over all angles, at large distance $R$ from the charge, the Poynting vector associated with $E t$ , which is:

$\mathbf{S} = {{E_t^2 \over \eta_0}}\mathbf{\hat{r}} = {{e^2 a^2 \sin^2(\theta)} \over {16 \pi^2 \epsilon_0 c^3 R^2}} \mathbf{\hat{r}}$

giving (SI units)

$P = {{e^2 a^2} \over {6 \pi \epsilon_0 c^3}}$

[edit] Issues and implications

[edit] Energy flux

The energy flux from the electromagnetic wave is given by the Poynting vector (cgs units).

$\mathbf{S} = { c \over {4 \pi } } {\mathbf{E} \times \mathbf{B} }$

Integration of the power over the surface of a sphere centered on the emitting particle yields the Larmor power formula in the nonrelativistic limit.

[edit] 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 limit.

[edit] 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.

[edit] See also

[edit] 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).

Jackson, John D. (1998). Classical Electrodynamics (3rd ed.). Wiley. ISBN 0-471-30932-X.

Misner, Charles; Thorne, Kip S. & Wheeler, John Archibald (1973). Gravitation. San Francisco: W. H. Freeman. ISBN 0-7167-0344-0.

R. P. Feynman, F. B. Moringo, and W. G. Wagner (1995). Feynman Lectures on Gravitation. Addison-Wesley. ISBN 0-201-62734-5.