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Help Understanding Wakefield When You Integrate Through Path 
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#1
Jun1914, 07:40 PM

P: 28

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: [tex]\frac{dE}{sds}(z) [/tex] Which to my understanding (which I am very certain of) describes the [tex]\frac{dE}{ds}[/tex] (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 gaussianly 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: [tex]\frac{dE}{ds}(z,θ) [/tex] Where this describes the [tex]\frac{dE}{ds}[/tex] (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 [tex]\frac{dE}{ds}[/tex] 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: [tex] Etotal(z,θ) [/tex]. 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 [tex]\frac{dE}{ds}(z,θ) [/tex] wrt to θ from 0 to θ (thet total travesrved angle of teh eletcron beam)? We would have something like [tex]\frac{dE}{ds}(z) [/tex] which is still a function of ds (the path travesersed). 2. We integrate [tex]\frac{dE}{ds}(z,θ) [/tex] wrt to the path ds? We would now have something like [tex]Etotal(z,θ) [/tex] which is still a function of θ. Any help on conceptually understanding this would be greatly appreciated! I have been at it for days! 


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