What is the formula relating acceleration of charged particle to wavelength of radiation emitted?

Question

When you accelerate a charged particle, as when electrons slam into the anode in an X-ray tube, or the beam of charged particles is wiggled by a non-uniform magnetic field in a synchrotron, electromagnetic radiation is generated. What's the formula to calculate its wavelength?

Answer 1

Suppose a charged particle moves at a more or less constant velocity v1 followed by a finite time interval t of acceleration resulting in a new velocity v2. The radiation emitted always spans a large range of frequencies which (as a rough approximation) goes from frequencies of about 0 to 1/t, dropping to zero at frequencies much higher than 1/t.

For the acceleration events in your examples: For an electron passing through a sharp local magetic field, t ~ a/v where a is the size of the region with the magnetic field and v the electron velocity. For a collision with a nucleus or other charged particle t ~ b/v where b is the impact parameter (the closest distance between the straight line trajectory of the electron and the particle it's colliding with).

For purposes of order of magnitude calculations, the total energy per frequency interval radiated by a particle with charge e (in gaussian cgs units) is roughly:

d Energy/d omega ~ e^2 * |v2-v1|^2/c^3

for frequencies in the region 0 < omega < 1/t.

There is never one single frequency emitted except when there's a sustained pure sinusoidal varying current as in a radio antenna. Of course atomic and nuclear transitions produce photons with precise frequencies (wavelengths) but those are not really the result of accelerating charges in the usual sense.

Answer 2

Since e=fh, that is, energy is equal to frequency times Plancks constant, and e=V/r^2, that is, energy is equal to the accelerating voltage devided by the distance squared, then fh=V/r^2 or f=v/r^2h. So the frequency of radiation obtained by slamming a particle into an anode in a field V over a distance r is V/r^2h. This is off the top of my head, if my memory serves me.

Answer 3

Stars act like a black physique, which emit radiation at shorter wavelengths with increasing floor temperature. the main significant physique of the megastar acts like a sparkling sturdy merchandise with the aid of fact it truly is made up of dense, intense rigidity gases below it truly is environment. the best stars at the same time with carbon stars, "late sort" pink giants and pink dwarfs are cool adequate for molecules at the same time with molecular carbon, water, and titanium oxide to variety. those stars emit extra often than not infrared radiation and in case you ought to work out interior the infrared area of the spectrum they could outshine all different stars. Yellow and white stars emit extra often than not seen gentle with the aid of fact they are lots warmer. The extra advantageous floor temperature shifts the peak wavelengths to the yellow and green area of the spectrum. the main well liked blue and blue white stars are so warm they emit extra often than not ultraviolet gentle. those stars have floor temperatures of as much as 500,000 levels Celsius and often are lots fainter interior the seen area of the spectrum while in comparison with their brightness in ultraviolet radiation. the main well liked stars are collapsed stars at the same time with white dwarfs and neutron stars that have been the two basically shaped or are sucking in gas from a close-by better half. those gadgets could have floor temperatures of thousands and thousands of levels Celsius and emit maximum of their capacity as lively x-rays. in short, the better the megastar's floor temperature, the shorter the dominant wavelengths of electromagnetic radiation it emits basically as a bite of warm steel glows with a pink colour at 800 levels Celsius yet glows white in colour at a million,six hundred levels Celsius. Doubling the exterior temperature of an merchandise additionally will boost the entire emissions of capacity by making use of a element of sixteen. a megastar with a floor temperature of 30,000 levels will supply off sixteen situations extra capacity than yet another megastar of the comparable actual length yet a floor temperature of 15,000 levels.