When a beam of charge particles pass through an RF cavity it leave wake behind it and in fundamental mode the induced voltage is known as beam loading voltage . The amplitude of beam loading voltage is depends upon the total charge and charge distribution (form factor) and cavity parameters (2pi*f*R/Q). But there is no dependence on beam energy or beta (v/c).
It looks surprising as different energy particles induce same voltage which means same energy loss.
It is not my main field but it seems to me the voltage due to one electron going by a cavity mostly only depends on its position, not its speed. There may be second order effects from speed due to delayed potential. Work is force times distance so it would seem that to a first order the same amount of energy is transferred independent of the electron speed. However, the time in which it is delivered, and so the power, does depend on speed, so the power goes up with beam energy.
For relativistic beams the electron field is distorted so there may be increase in power with beam energy even though the speed approaches c, unless the distortion reduces the coupling or has no effect on it.
I'm interested to see what corrections there may be to my answer!
Sorry I disagree: There is definitely a strong dependency on the velocity of the bunch; we call this transit time factor and very slow beams leave very little field in the cavity. As for the position dependency : this depends on the mode we are considering. For an accelerating TM 0 type mode its rather small for a deflecting mode TM1 type is quite strong. This concept is used for resonant beam position monitors. For further details you may consult the CERN electronic library (CDS CERN open access to anyone) or also find related papers via JACOW (also open access..) just type JACOW in Google and then search in the conference reports (do not use keyword search)
Dear Fritz, I also think, that the beam induced voltage must have a dependency on velocity of beam. As you mentioned it should be similar to transient time factor. But I did not find literature reporting the same.
My confusion is that If we take the beam induced voltage as Vb= Vb0*T, where Vb0 is the peak voltage induced and T is transient time factor. When this voltage act on trilling bunch then trilling bunch should see as Vb1=Vb*T=Vb0*T^2. In typical TM010 like mode the T~ 0.7, hence the Vb1=0.5*Vb0, which means trilling bunch will see only half of beam induced voltage. This looks odd to me.
Could you please through light on it, or provide some reference?
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