Help Understanding Wakefield When You Integrate Through Path

[jasonpatel]

Hi All,

I am trying to understand the some of the properties of wakefields, namely the energy change. So, as a preface I am interested in primarily the wakefield due to electron beams as they progress through a curved section (the eletcrons radiate strongly when they are in circular motion). There are two main/simple regimes:

1. When the wakefield is constant and not dependent on how far Δθ the electrons have traversed, we have a wakefield like so:

$$\frac{dE}{sds}(z)$$

Which to my understanding (which I am very certain of) describes the $$\frac{dE}{ds}$$ (the change in energy per distance traveled along its curved trajectory) for a given z (position along the eletcron beam, where zero is defined as the certer of the eletcron beam which we can consider to be gaussian-ly distributed).

2. When the wakefield is not constant and is dependent on how far Δθ the eletcrons have traversed, we have the wakefield like so:

$$\frac{dE}{ds}(z,θ)$$

Where this describes the $$\frac{dE}{ds}$$ (the change in energy per distance traveled along the curved trajectory) for a given z (position along the eletcron beam) and θ (the amount the eletcrons have traversed). Now, the main difference between 1 and 2 is the fact that for two $$\frac{dE}{ds}$$ is changing wrt θ (is some function of theta).

If you were to integrate regime 1, wrt ds from 0 to L (the total path length), then you would get the total energy change through the curved region as a function of z: $$Etotal(z,θ)$$.

Now this is where I start getting confused: If we turn our attention to regime two with θ dependence.

WHat do we have when:

1. We integrate $$\frac{dE}{ds}(z,θ)$$ wrt to θ from 0 to θ (thet total travesrved angle of teh eletcron beam)? We would have something like $$\frac{dE}{ds}(z)$$ which is still a function of ds (the path travesersed).

2. We integrate $$\frac{dE}{ds}(z,θ)$$ wrt to the path ds? We would now have something like $$Etotal(z,θ)$$ which is still a function of θ.

Any help on conceptually understanding this would be greatly appreciated! I have been at it for days!

 

[AndyW]

Hi there,

Thank you for your post and for your interest in wakefields. I can definitely help you understand the properties of wakefields and how they relate to the energy change of electron beams.

Firstly, it is important to understand that wakefields are electric fields that are created by the motion of charged particles, such as electrons. These fields can either accelerate or decelerate particles as they pass through them, resulting in a change in energy.

Now, let's focus on the two regimes that you have described. In regime 1, the wakefield is constant and does not depend on the angle the electrons have traversed. This means that the change in energy per distance traveled, \frac{dE}{ds}, is the same for all angles. This is what the expression \frac{dE}{sds}(z) represents - the change in energy per distance traveled at a specific position along the electron beam.

In regime 2, the wakefield is not constant and is dependent on the angle the electrons have traversed. This means that the change in energy per distance traveled, \frac{dE}{ds}, is now a function of both z (position along the electron beam) and θ (angle traversed). This is what the expression \frac{dE}{ds}(z,θ) represents.

Now, to answer your questions:

1. When we integrate \frac{dE}{ds}(z,θ) with respect to θ from 0 to θ, we are essentially looking at the total change in energy as the electrons traverse a specific angle. This can be thought of as the change in energy per distance traveled, but now it is specific to a certain angle. So, we would have something like \frac{dE}{ds}(z), where z is the position along the electron beam.

2. When we integrate \frac{dE}{ds}(z,θ) with respect to the path ds, we are essentially looking at the total change in energy as the electrons travel a specific distance. This can be thought of as the change in energy per angle traversed, but now it is specific to a certain distance. So, we would have something like Etotal(z,θ), where θ is the angle traversed.

I hope this helps clarify things for you. If you have any more questions, please don't hesitate to ask. Keep up the good work and happy researching