Total Power Radiated by Ultra-relativistic Particle

In summary, the task is to evaluate the total power radiated to all angles by an ultra relativistic particle, keeping only the leading power of gamma in the power formula. However, integrating the formula proves to be challenging and it is unclear how to use the given information effectively. One approach could be to rearrange the formula and consider a power expansion, keeping the lowest order terms as gamma gets large. However, careful manipulation of theta is required and the integration process remains unclear.
  • #1
jameson2
53
0

Homework Statement



Given the formula for power radiated into a solid angle, evaluate the total power radiated to all angles by an ultra relativistic particle, keeping the leading power of [tex] \gamma [/tex] only.

Homework Equations


The power formula:
[tex] \frac{dP'}{d\Omega}=\frac{q^2 \alpha^2}{\pi^2 c}\frac{2\gamma^{10}\theta^2}{(\gamma^2 \theta^2 +1)^5} [/tex]

The Attempt at a Solution



Basically, I can't integrate this:

[tex] P'= \int^{2\pi}_0 \int^\pi_0 \frac{q^2 \alpha^2}{\pi^2 c}\frac{2\gamma^{10}\theta^2}{(\gamma^2 \theta^2 +1)^5} sin(\theta)d\theta d\phi [/tex]

I was thinking that since gamma will be large, you can ignore the 1 on the bottom line, but that doesn't get me anywhere. Possibly I'm just missing how I can use the fact that the question says "keeping only the leading power of gamma", but it's not clear to me at all.Thanks for any hints.
 
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  • #2
how about re-arrange as follows
[tex]
Psingle-quote
= \int^{2\pi}_0 \int^\pi_0 \frac{q^2 \alpha^2}{\pi^2 c}\frac{2\gamma^{10}\theta^2}{(\gamma^2 \theta^2 +1)^5} sin(\theta)d\theta d\phi

= \int^{2\pi}_0 \int^\pi_0 \frac{q^2 \alpha^2}{\pi^2 c}\frac{2\theta^2}{( \theta^2 +\frac{1}{\gamma^2})^5} sin(\theta)d\theta d\phi
[/tex]

then consider a power expansion
 
  • #3
So just get
[tex] \theta^{10} + 5\theta^8\frac{1}{\gamma^2} [/tex]
on the bottom line? Is this what it means by only keep the leading power of gamma?

I'm still not sure how to go about integrating this though.
 
  • #4
as gamma, gets large, 1/gamma gets small, so I would try to expand in power series in terms of 1/gamma and only keep the lowest order terms. geometric series may be handy here, though you will need to be careful on how you manipulate theta...
 

1. What is total power radiated by ultra-relativistic particle?

The total power radiated by an ultra-relativistic particle refers to the amount of electromagnetic energy that is emitted by a charged particle as it moves at extremely high speeds, close to or at the speed of light. This emission of energy is known as synchrotron radiation and is an important phenomenon in astrophysics, particle accelerators, and other fields.

2. How is total power radiated calculated?

The total power radiated by an ultra-relativistic particle can be calculated using the Larmor formula, which takes into account the particle's charge, velocity, and acceleration. This formula is given by P = (2/3) * (q^2 * a^2)/c^3, where P is the power, q is the particle's charge, a is its acceleration, and c is the speed of light.

3. What factors affect the total power radiated by an ultra-relativistic particle?

The total power radiated by an ultra-relativistic particle is affected by several factors, including the particle's charge, velocity, and acceleration. Additionally, the strength of the magnetic field the particle is moving through and the angle at which the particle's velocity and the magnetic field are aligned also play a role in determining the amount of power radiated.

4. Why is the total power radiated by ultra-relativistic particles important?

The total power radiated by ultra-relativistic particles plays a crucial role in various fields of science, including astrophysics, particle physics, and plasma physics. It helps us understand the behavior of charged particles moving at high speeds and the effects of strong magnetic fields on these particles. This knowledge is essential for studying and analyzing phenomena such as cosmic rays, black holes, and particle accelerators.

5. Can total power radiated by ultra-relativistic particles be observed?

Yes, total power radiated by ultra-relativistic particles can be observed through the emission of synchrotron radiation. This radiation can be detected using telescopes and other instruments that are sensitive to electromagnetic waves in the radio, infrared, and X-ray regions of the spectrum. By studying this radiation, scientists can gather valuable information about the properties of the particles emitting it.

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