Talk:Larmor formula

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Just got to get the equations correct. Wont be too long. Please bear with me!--Light current 01:14, 23 October 2005 (UTC) TEMP HOME

[edit] Derivation of formula

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 = 1/2 \epsilon_0\|E|^2 Neglecting the radial component of the field:


Energy/vol = {q^2 a^2 \sin^2 \theta} \over {32 \pi^2 \epsilon_0 c^4 R^2}

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,


\sin^2 \theta = {1 - cos^2 \theta} = 1 - {x^2 \over R^2} \.

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:

x^2 = y^2 = z^2 \

But this implies that

x^2 = \frac{1}{3}(x^2 + y^2 + z^2) = \frac{1}{3} R^2 = \frac{R^2}{3}

since,

x^2 + y^2 + z^2 = R^2 \

and R is constant over the whole shell. Combining equations gives

\sin^2 \theta = {1 - R^2 \over 3R^2} = \frac{2}{3}

So the average energy per unit volume stored in the transverse electric field is

{{q^2 a^2} \over {48 \pi^2 \epsilon_0 c^4 R^2}}

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 = E = \frac{e^2 a^2}{6 \pi \epsilon_0 c^3}

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 = E = \frac{e^2 a^2}{6 \pi \epsilon_0 c^3}

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

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

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. linas 14:38, 4 January 2006 (UTC)

[edit] Equivalence principle

What is your source for this? --EMS | Talk 22:13, 31 July 2006 (UTC)

Added references Complexica 20:29, 8 August 2006 (UTC)
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