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The derivation given here was first published by J. J. Thomson (discoverer of the electron) in 1907. It is derived for the special case where the final velocity of the particle is zero but the Larmor formula is true for any sort of accelerated motion provided that the speed of the particle is always much less than the speed of light.
The energy per unit volume stored in an electric field is
Energy/vol =
Neglecting the radial component of the field:
Energy/vol =
If the direction in which the energy goes is not important, we can average the energy over all directions. Using a mathematical device, introduce a
coordinate system with the origin at the center of the sphere and the x axis along the particle’s original direction of motion. Then for any point (x, y, z) on the spherical shell, cos θ = x/R.
Using angle brackets to denote an average over all points on the shell,
.
Now since the origin is at the center of the sphere, the average value of x^2 is the same as the average value of y^2 or z^2:
But this implies that
since,
and R is constant over the whole shell. Combining equations gives
So the average energy per unit volume stored in the transverse electric field is
To obtain the total energy stored in the
transverse electric field, we must multiply equation by the volume of the spherical shell. The surface area of the shell is 4πR2 and its thickness is ct0, so its volume is the product of these factors. Therefore the total energy is
Total energy in electric field =
The total energy is independent of R; that is, the shell carries away a fixed amount of energy that is not diminished as it expands.
There is also a magnetic field, which carries away an equal amount of energy. Many details about magnetic fields have been omitted. A factor of 2 needs inserting. Thus the total energy carried away by the pulse of radiation is twice that of the previous equation or
Total energy in pulse =
Divide both sides of this equation by t0, the duration of the particle’s acceleration. The left-hand side then becomes the energy radiated by the particle per unit time, or the power given off during the acceleration: Power radiated
An example is the electric field around an oscillating charge. A map of the electric field lines around a positively charged particle oscillating sinusoidally, up and down, between the two gray regions near the center. Points A and B are one wavelength apart. If you follow a straight line out from the charge at the center of the figure, you will find that the field oscillates back and forth in direction. The distance over which the direction of the field repeats is called the wavelength. For instance, points A and B in the figure are exactly one wavelength apart. The time that it takes the pattern to repeat once is called the period of the wave, and is equal to the time that the source charge takes to repeat one cycle of its motion. The period is also equal to the time that the wave takes to travel a distance of one wavelength. Since it moves at the speed of light, we can infer that the wavelength and the period are related by
Interesting. If this is can be finished and cleaned up, it should moved to the article proper. Especially if this is how Larmor did it.
linas14:38, 4 January 2006 (UTC)reply
Epsilon or varepsilon for permittivity of free space
As of Sept 5 2008, this article used \epsilon rather than \varepsilon for the permittivity of vacuum. I've always see \varepsilon used for this constant. Indeed, this is what is used in
http://en.wikipedia.org/wiki/Permittivity
I changed \epsilon to \varepsilon in this article. If this is incorrect, please describe why \epsilon should be used in this equation. —Preceding
unsigned comment added by
DanHickstein (
talk •
contribs)
18:59, 5 September 2008 (UTC)reply
Bremsstrahlung#Dipole approximation discusses the power radiated from an accelerated charge, but the formulas are different from this article. I don't see how to reconcile them. At the very least I expect the two articles to cross-reference each other and explain the different assumptions that lead to the different formulas. It's also possible that one of the two articles is flat-out wrong. Someone knowledgeable can please help?? Thanks in advance!! --
Steve (
talk)
14:19, 12 December 2012 (UTC)reply
The non-relativistic formula in the lede of
Larmor formula is:
In the non-relativistic limit, the second term in parentheses can be neglected, and the first term reduces to a^2/c^2. Set q = e and gamma = 1 and the formulas become identical. Where's the problem?
Art Carlson (
talk)
11:07, 13 December 2012 (UTC)reply
Bremsstrahlung says "The general expression for the total radiated power is
Larmor says
I'll put Larmor in SI units for a better comparison
I'll pull out gamma^2 in Bremsstrahlung's for a better comparison
Let be the angle between velocity and acceleration...
Usually I can stomach arrow notation for vectors, but symbols such as are truly repulsive. If no one objects, I'm going to change the article's notation to use boldface for vectors.
Zueignung (
talk)
23:51, 30 December 2012 (UTC)reply
Derivation 1 typo?
there is an erroneous equation in the derivation 1 section
I believe the correct formulas are either
or
as taken from Jackson 3rd edition pg 665
I'll change it to the latter to flow with the text before the equation
You are correct: the formula equating dP/dΩ to the expression involving (qa/R)2/c3 must be wrong, based on dimensional analysis alone. dP/dΩ has units of erg s−1 = g cm2 s−3, and (qa/R)2/c3 has units of g s−3.
Zueignung (
talk)
17:31, 16 December 2014 (UTC)reply
Original source by Larmor
I added a new source, where the formula is actually mentioned.
I couldn't find it in "On a dynamical theory of the electric and luminiferous medium", so this hint shows a paper from the right year and the right author, but is a bit misleading.
141.53.32.79 (
talk)
12:01, 7 October 2015 (UTC)reply
non-radiation condition
The last sentence in "Atomic Physics" appears to be a serious misunderstanding of the paper of Haus (found in the link to the wikipedia page "Nonradiation condition"), which says that non-accelerating charges do not radiate. — Preceding
unsigned comment added by
47.23.28.187 (
talk)
00:27, 30 May 2018 (UTC)reply
If electron acceleration is the source of antenna radiation, then at higher frequencies antenna efficiency would increase, which is clearly not the case. One can safely assume that the main thesis of this Wikipedia article is wrong.
80.121.121.188 (
talk)
09:08, 11 August 2019 (UTC)wabireply
The meaning of the formula is independent of a specific system of units, so what is the idea behind the parenthesis and how does it not apply just as well when working in the
cgs system? --
Lambiam18:23, 23 January 2023 (UTC)reply
I think this last term is wrong and should be removed, as this µ0, stands for something like 1 over ϵ_0 c^3, which I think is meaningless and certainly not the expression of a dipole moment. This is already obvious by checking the (SI) units.
Snoopy Urania (
talk)
15:02, 2 April 2023 (UTC)reply
I don't understand this. The
vacuum permittivity is related to the
vacuum permeability by This implies that the last two expressions in the chain of equalities are equivalent – if the last one is wrong, so is the one but last. So that one is then wrong too and should also be removed. But it is clearly equivalent to the one before it, so ...
None of this is relevant to the suggested dependence on the system of units. In the literature some texts have a factor and others don't. The formulas on the first line have
dimension the same as
power, Those of the second line have dimension if I'm not mistaken, in any case another dimension than power. --
Lambiam18:59, 2 April 2023 (UTC)reply
ALL of the equations in that section depend on the system of units used.
If all equations depend on the system of units, the sentence about how the power radiated by a single electron is described in either unit system by the same formula is wrong.
Can you explain how the formula for in cgs units has the
dimension of power? What are the dimensions of and ?
In cgs units, a has the dimension cm/sec^2, c has dimension cm/sec. Then, q^2a^2/c^3 has the dimension q^2/(cm-sec).
q^2/cm has the dimension of energy in ergs, So q^2a^2/c^3 has the dimension of power in ergs/sec. q has the dimension of
'statcoulombs', but that doesn't have to be mentioned if q^2/cm is recognized as energy in ergs. — Preceding
unsigned comment added by
Alfa137 (
talk •
contribs)
01:38, 20 April 2023 (UTC)reply